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B.} \lastname{Pozzetti}]{\firstname{Maria Beatrice} \lastname{Pozzetti}} \address{Università di Bologna\\ Dipartimento di Matematica\\ Piazza di Porta San Donato 5\\ 40126 Bologna (Italy)} \email{beatrice.pozzetti@unibo.it} \author[\initial{A.} \lastname{Sambarino}]{\firstname{Andrés} \lastname{Sambarino}} \address{Sorbonne Université\\ Institut de Mathématiques de Jussieu\\ Paris Rive Gauche (IMJ-PRG)\\ CNRS UMR 7586\\ 4 place Jussieu\\ 75005 Paris (France)} \email{andres.sambarino@imj-prg.fr} \thanks{B.~P. is supported by the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Cluster of Excellence), and acknowledges further support by DFG grant 338644254 (within the framework of SPP2026) and 427903332 (in the Emmy Noether program). A.~S. was partially financed by ANR DynGeo ANR-16-CE40-0025. Part of this work was carried out in Oberwolfach, we thank the institute for its great hospitality. B.~P. additionally thanks Prof. Farkas and Humboldt University for their hospitality, the Familienzimmer des mathematiches Institut made it possible to work on this paper while taking care of a~small child.} \CDRGrant[EXC-2181/1]{390900948} \CDRGrant[DFG]{338644254} \CDRGrant[DFG]{SPP2026} \CDRGrant[DFG]{427903332} \CDRGrant[DynGeo]{ANR-16-CE40-0025} \begin{abstract} We interpret the Hilbert entropy of a convex projective structure on a closed higher-genus surface as the Hausdorff dimension of the non-differentiability points of the limit set in the full flag space $\mathcal{F}(\mathbb{R}^3)$. Generalizations for regularity properties of boundary maps between locally conformal representations are also discussed. An ingredient for the proofs is the concept of \emph{hyperplane conicality} that we introduce for a $\theta$-Anosov representation into a~reductive real-algebraic Lie group $\mathsf{G}$. In contrast with directional conicality, hyperplane-conical points always have full mass for the corresponding Patterson--Sullivan measure. \end{abstract} \begin{altabstract} On interprète l'entropie de Hilbert d'une structure projective convexe sur une surface de genre minoré par 2 comme la dimension de Hausdorff des points de non-differentiabilité de l'ensemble limite dans l'espace des drapeaux complets $\mathcal{F}(\mathbb{R}^3)$. Des généralisations concernant des propriétés de régularité des applications limites au bord à l'infini pour des représentations localement conformes sont également traitées. Un ingrédient de la démonstration de ces résultats est le concept de \emph{point limite conique suivant un hyperplan} que l'on introduit pour une représentation $\theta$-anosovienne à valeurs dans un groupe réductif réel-algébrique $\mathsf{G}$. Contrairement aux points coniques suivant une direction, l'ensemble des points coniques suivant un hyperplan est toujours de masse totale pour la mesure de Patterson--Sullivan correspondante. \end{altabstract} \datereceived{2024-01-26} \daterevised{2025-10-23} \dateaccepted{2025-11-24} \editors{C.~Sorger and V.~Colin} %\begin{DefTralics} %\renewcommand{\t}{\theta} %\newcommand{\sfG}{\sf{G}} %\newcommand{\R}{\mathbb{R}} %\newcommand{\calF}{\mathcal{F}} %\renewcommand{\sf}[1]{{\mathsf{#1}}} %\end{DefTralics} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \setcounter{tocdepth}{1} \setcounter{secnumdepth}{3} \hypersetup{bookmarksdepth = 3} \begin{document} \maketitle \tableofcontents \section{Introduction}\label{intro} Consider a closed connected orientable surface $S$ of genus at least two, and let $\rho:\piS\to\PSL(3,\R)$ be a faithful representation preserving an open convex set $\Omega=\Omega_\rho\subset\P(\R^3)$, properly contained in an affine chart. The group $\rho(\piS)$ is necessarily discrete and acts co-compactly on $\Omega$: one says that $\rho$ \emph{divides} $\Omega$. The geometry of such convex set $\Omega$ is well studied, by Benoist~\cite{convexes1} it is strictly convex with $\class^{1+\nu}$ boundary $\bord\Omega$ (that is not $\class^2$ unless it is an ellipse), and the Hilbert metric of $\Omega$ is Gromov-hyperbolic. The geodesic flow of $\Omega/\rho(\piS)$ is an Anosov flow and its topological entropy, the \emph{Hilbert entropy} $\hJ{\sf H}={(\hJ{\sf H})}_\rho$, satisfies \[ \hJ{\sf H}\leq1, \] an inequality proved by Crampon~\cite{crampon} that is strict if $\Omega$ is not an ellipse. A consequence of Theorem~\ref{tutti} below is a new geometric interpretation of the Hilbert entropy which we now explain. For each $x\in\bord\Omega$ let $\Xi(x)\in\grass_{2}(\R^3)$ be the unique plane whose projectivization is tangent to $\bord\Omega$ at $x$. By~\cite{convexes1}, the image curve $\Xi(\bord\Omega)\subset\grass_2(\R^3)\simeq\P((\R^3)^*)$ is also the boundary of a strictly convex divisible set $\Omega^*$ and is thus again a $\class^{1+\nu}$-circle. The \emph{full-flag-curve} \[ \{(x,\Xi(x)):x\in\bord\Omega\}\subset\calF(\R^3), \] is the graph of a monotone map between $\class^1$ circles and thus is a Lipschitz submanifold that is therefore differentiable almost everywhere. We establish the following: \begin{coro}\label{hilbert} Let $\rho:\piS\to\PSL(3,\R)$ divide a strictly convex set that is not an ellipse. Then, the set of non-differentiability points of the full flag curve has Hausdorff dimension ${(\hJ{\sf H})}_\rho$. \end{coro} Throughout the paper the Hausdorff dimension is computed with respect to a(ny) Riemannian metric on the flag space. When $\Omega$ is an ellipse the result does not apply as the associated curve is differentiable everywhere while $\hJ{\sf H}=1$. A classical result by Choi--Goldman~\cite{choigoldman} states that the space of representations dividing a convex set forms a connected component of the character variety $\fkX(\piS,\PSL(3,\R))$ of homomorphisms up to conjugation. This component is known today as \emph{the Hitchin component} of $\PSL(3,\R)$ and is diffeomorphic to a ball of dimension $-8\chi(S)$. Nie~\cite{Nie0} and Zhang~\cite{ZhangSL3} have found paths $(\rho_t)$ in this Hitchin component such that $({\hJ{\sf H}})_{\rho_t}\to 0$ as $t\to\infty$. Together with Corollary~\ref{hilbert} this suggest that the closer $\Omega$ is to being an ellipse (the \emph{Fuchsian locus}), the less differentiable the flag curve is whilst the furthest away from Fuchsian locus, the more regular the flag curve becomes. The proof of Corollary~\ref{hilbert} is outlined in Section~\ref{corAProof} and serves as a guide path for the strategy on the general case (Theorems~\ref{LCdiff} and~\ref{tutti}). \subsection{Locally conformal representations and concavity properties}\label{2hyper} Let $\K$ be $\R$, $\C$ or the non-commutative field of Hamilton's quaternions $\H$. Denote by \[ \a=\left\{(a_1,\dots,a_d)\in\R^d:\sum_i a_i=0\right\} \] the Cartan subspace of the real-algebraic group $\SL(d,\K)$, by \begin{equation}\label{e.slroot} \slroot_i(a_1,\dots,a_d)=a_i-a_{i+1} \end{equation} the $i^{\, \rm th}$ simple root and by $\a^+\subset \a$ the Weyl chamber whose associated set of simple roots is $\simple=\{\slroot_i:i\in\lb1,d-1\rb\}$. Let $\cartan:\SL(d,\K)\to\a^+$ be the \emph{Cartan projection} with respect to the choice of an inner (or Hermitian) product on $\K^d$. The $e^{\cartan_i(g)}$'s are the \emph{singular values} of the matrix $g$, namely the square roots of the modulus of the eigenvalues of the matrix $gg^*$. We also let $d_\P$ denote the distance on $\P(\K^d)$ induced by the chosen Hermitian product. Let $\G$ be a finitely generated word-hyperbolic group, consider a finite symmetric generating set and let $|\,|$ be the associated word-length. For $k\in\lb1,d-1\rb$, a representation $\rho:\G\to\SL(d,\K)$ is $\{\slroot_k\}$-\emph{Anosov} if there exist positive constants $\mu$ and $c$ such that for all $\g\in\G$ one has \[ \slroot_k\bigl(\cartan(\rho(\g))\bigr)\geq\mu|\g|-c. \] A $\{\slroot_k\}$-{Anosov} representation is also $\{\slroot_{d-k}\}$-Anosov. Under such assumption there exists an equivariant H\"older-continuous map \[ \xi^{k}_\rho:\bord\G\to\grass_k(\K^d), \] called the \emph{limit map} in the Grassmannian $\grass_k(\K^d)$ of $k$-dimensional subspaces of~$\K^d$, which is a homeomorphism onto its image. If $k\leq l\in\lb1,d-1\rb$ and $\rho$ is also\linebreak $\{\slroot_l\}$-Anosov then the limit maps are {compatible}, i.e. $\xi^k_\rho(x)\subset\xi^l_\rho(x)$ $\forall x$, see Section~\ref{Anosov} for references and details. \begin{defi}\label{d.hyp} Fix $p\in\lb2,d-1\rb$. A $\{\slroot_1,\slroot_{d-p}\}$-Anosov representation\linebreak $\rho:\G\to\SL(d,\K)$ is \emph{$(1,1,p)$-hyperconvex} if for every pairwise distinct triple $x,y,\linebreak z\in\bord\G$ one has \begin{equation}\label{hypdefi} \bigl(\xi^{1}_\rho(x)+\xi^{1}_\rho(y)\bigr)\cap\xi^{{d-p}}_\rho(z)=\{0\}. \end{equation} If in addition one has $\cartan_2(\rho(\g))=\cartan_{p}(\rho(\g))$ $\forall\g$, we say that $\rho$ is \emph{locally conformal}. \end{defi} Hyperconvex representations form an open subset of the character variety \[ \fkX\bigl(\G,\SL(d,\K)\bigr)=\hom\bigl(\G,\SL(d,\K)\bigr)/\SL(d,\K) \] and appear naturally. For example, when $\K=\R$, strictly convex divisible sets give rise to $(1,1,d-1)$-hyperconvex representations, while higher rank Teichm\"uller theory provides many examples of $(1,1,2)$-hyperconvex representations of surface groups, see Example~\ref{ex.hyperconvex}. When $p=2$ the second part of the definition is trivially true, so $(1,1,2)$-hypercon\-vex representations over $\K$ \emph{are} locally conformal, when $p>2$ the assumption constrains the Zariski closure of $\rho(\G)$. However, Zariski-dense locally conformal representations exist (and form open sets) for the groups locally isomorphic to $\SL(n,\R)$, $\SL(n,\C)$, $\SL(n,\H)$, $\SU(1,n)$, $\Sp(1,n)$, $\SO(p,q)$, see~\cite[Section~8]{PSW1} %P.-S.-Wienhard for details, and, of course, $\SO(1,n)$ where every convex co-compact representation is locally conformal. A concrete example in $\SU(1,n)$ consists on considering a convex co-compact group in $\H^n_\C$ whose limit set intersects the projectivization of any complex line in at most 2~points. These subgroups are locally conformal (\cite[Proposition~8.3]{PSW1}) and their limit set (though fractal) is tangent to the contact distribution of $\bord\H^n_\C$. Consider also $\bar{\K}\in\{\R,\C,\H\}$ and positive integers $d$ and $\ovd$. Throughout the paper we mainly deal with a pair of locally conformal representations \[ \rho:\G\to\SL(d,\K)\quad\text{and}\quad\ovrho:\G\to\SL(\ovd,\bar{\K}), \] with equivariant maps $\xi=\xi^1_\rho$ and $\ovxi=\xi^1_{\ovrho}$, and we study regularity properties of the equivariant H\"older-continuous homeomorphism \[ \Xi=\ovxi\circ\xi^{-1}:\xi(\bord\G)\to\ov{\xi}(\bord\G). \] To avoid confusion we denote the simple roots of $\SL(\ovd,\bar{\K})$ by $\{\ov{\slroot}_i:i\in\lb1,\ovd-1\rb\}$, and to simplify notation we identify $\g$ with $\rho(\g)$ and we let $\ov\g=\ovrho(\g)$. We consider also the graph of $\Xi$, or equivalently the \emph{graph map}, \[ \varXi:\bigl(\xi,\ov{\xi}\bigr):\bord\G\to\P(\K^d)\times\P(\bar{\K}{}^{\ovd}). \] \pagebreak{} \begin{defi} Fix $\hol\in(0,1]$. We will say that $\Xi$ is $\hol$\emph{-concave} at $x\in\bord\G$, or that $x$ is a $\hol$\emph{-concavity point for} $\Xi$, if there exists a sequence $(y_k)$ converging to $x$ as $k\to\infty$ such that the incremental quotient \begin{equation}\label{incri} \frac{d_\P\bigl(\ovxi(x),\ovxi(y_k)\bigr)}{d_\P\bigl(\xi(x),\xi(y_k)\bigr)^\hol} \end{equation} is bounded away from $\{0,\infty\}$. The set of $\hol$-concavity points is denoted by $\Ext_{\rho,\ovrho}^\hol$. \end{defi} Observe that $\Xi$ can be $\hol$-concave at $x$ for several $\hol$'s and that it is a $1$-concave point if one has $y_k\to x$ such that $d(\xi(x),\xi(y_k))$ and $d_\P(\ovxi(x),\ovxi(y_k))$ are comparable. \begin{figure}[h] \centering \begin{tikzpicture} \begin{scope}[scale=1.5] \begin{scope}[rotate=-90] \draw[name path=Q] (0,0) parabola bend (0,0) (-1.5,1.5); \fill[color=gray!20] (0,0) parabola bend (0,0) (-1.5,1.5)--(-1,1.5)--cycle; \fill[color=white] (0,0) parabola bend (0,0) (-1,1.5)--cycle; \draw (0,0) parabola bend (0,0) (-1,1.5); \draw[name path=Q] (0,0) parabola bend (0,0) (-1.5,1.5); \node at (-1.5,1.5) [above right] {$Cs^\hol$}; \node at (-1,1.5) [below right] {$cs^\hol$}; \end{scope} \draw[gray!60] (0.8,0)--(0.8,0.8)--(0,0.8); \draw[gray!60] (0.4,0)--(0.4,0.67)--(0,0.67); \draw[gray!60] (0.1,0)--(0.1,0.3)--(0,0.3); \draw[gray!60] (0.05,0)--(0.05,0.215)--(0,0.215); \draw (0,0)--(2,0); \node[below] at (2,0) {$d_\P(\xi(x),\xi(z))$}; \draw(0,0)--(0,2); \node[left] at (0,2) {$d_\P(\ovxi(x),\ovxi(z))$}; \draw [fill] (0.8,0) circle [radius = 0.02]; \draw [fill] (0.8,0.8) circle [radius = 0.02]; \draw [fill] (0,0.80) circle [radius = 0.02]; \draw [fill] (0.4,0) circle [radius = 0.02]; \draw [fill] (0.4,0.67) circle [radius = 0.02]; \draw [fill] (0,0.67) circle [radius = 0.02]; \draw [fill] (0.1,0.3) circle [radius = 0.017]; \draw [fill] (0.1,0) circle [radius = 0.017]; \draw [fill] (0,0.3) circle [radius = 0.017]; \draw [fill] (0.05,0) circle [radius = 0.008]; \draw [fill] (0.05,0.215) circle [radius = 0.008]; \draw [fill] (0,0.215) circle [radius = 0.008]; \begin{scope}[rotate=-90] \draw (0,0) parabola bend (0,0) (-1.5,1.5); \draw (0,0) parabola bend (0,0) (-1,1.5); \end{scope} \end{scope} \end{tikzpicture} \caption{A $\hol$-concave point $x$. The marked points on the axis' represent $d_\P(\xi(x),\xi(y_k))$ and $d_\P(\ovxi(x),\ovxi(y_k))$ respectively.} \end{figure} In what follows we will compute the Hausdorff dimension of $\varXi(\Ext_{\rho,\ovrho}^\hol)$ with respect to the product metric on $\P(\K^d)\times\P(\bar{\K}{}^{\ovd})$ for $\hol$ lying on an interval that we now define. The \emph{dynamical intersection} between $\rho$ and $\ovrho$ with respect to $\ovsloot_1$ and $\slroot_1$ is defined by \[ \II_{\ovsloot_1}(\slroot_1)=\lim_{t\to\infty}\frac1{\#\sf R_t(\ovsloot_1)}\sum_{\g\in \sf R_t(\ovsloot_1)}\frac{\slroot_1(\lambda(\g))}{\ovsloot_1(\lambda(\ov\g))}, \] where $\sfR_t(\ovsloot_1)=\{[\g]\in[\G]:\ovsloot_1(\lambda(\ov\g))\leq t\}$ and $\lambda:\SL(d,\K)\to\a^+$ is the Jordan projection. This concept (from Bridgeman et al.~\cite{pressure}, %--Canary--Labourie--Sambarino Burger~\cite{burger}, Knieper~\cite{kni95}, among others) generalizes Bonahon's intersection number between two elements in Teichm\"uller space. Let us say that $\rho$ and $\ovrho$ are \emph{gap-isospectral} if for all $\g\in\G$ one has \[ \slroot_1\bigl(\lambda(\g)\bigr)=\ovsloot_1\bigl(\lambda(\ov\g)\bigr). \] Corollary~\ref{corraiz} (a consequence of~\cite{pressure} together with Proposition~\ref{nonA}) implies that if $\rho$ and $\ovrho$ are not gap-isospectral, then $\II_{\slroot_1}(\ovsloot_1)>(\II_{\ovsloot_1}(\slroot_1))^{-1}$. We will study $\hol$-concavity for any $\hol\in(0,1]$ with \[ \II_{\slroot_1}(\ovsloot_1)>\hol>\bigl(\II_{\ovsloot_1}(\slroot_1)\bigr)^{-1}. \] Finally, consider the critical exponents \begin{alignat*}{2}\hC{\slroot_1} & =\lim_{t\to\infty}\frac1t\log\#\bigl\{\g\in\G:\slroot_1(\cartan(\g))\leq t\bigr\},\\ \hC{\infty,\hol} & =\lim_{t\to\infty}\frac1t\log\#\bigl\{\g\in\G:\max\bigl\{\hol\slroot_1(\cartan(\g)),\ov{\slroot}_1(\cartan(\ov\g))\bigr\}\leq t\bigr\}. \end{alignat*} \begin{theo}[Theorem~\ref{tutti.LCdiff}]\label{LCdiff} Let $\{\K,\bar{\K}\}\subset\{\R,\C\}$ and let $\rho:\G\to\SL(d,\K)$ and $\ovrho:\G\to\SL(\ovd,\bar{\K})$ be locally conformal, $\R$-irreducible and not gap-isospectral. Then for any $\hol\in(0,1]$ with $\II_{\slroot_1}(\ovsloot_1)>\hol>(\II_{\ovsloot_1}(\slroot_1))^{-1}$, one has \begin{align*} \hol\hC{\infty,\hol}\leq\Hff\left(\varXi\left({\Ext}_{\rho,\ovrho}^\hol\right)\right) &\leq\min\left\{\hC{\infty,\hol},\hol\hC{\infty,\hol}+1-\hol \right\}\\ &<\min\left\{\hC{\ovsloot_1},\hC{\slroot_1}/\hol\right\}\\ &\leq\Hff(\varXi(\bord\G))\\ &=\max\left\{\hC{\slroot_1},\hC{\ovsloot_1}\right\}. \end{align*} If $\K=\H$ (resp. $\bar{\K}=\H$) we further assume that the Zariski closure if $\rho$ (resp. $\ovrho$) does not have compact factors, then the same conclusion holds. \end{theo} The proof of the above Theorem is completed in Section~\ref{ProofA}. For representations in $\PSL(2,\C)$ we can furthermore give a geometric interpretation of the 1-weakly-bi-H\"older points, see Section~\ref{klein}. \subsection{Surface-group representations} Observe that the first line of inequalities in Theorem~\ref{LCdiff} becomes an equality when $\hol=1$. We pursue now this situation while further restricting the source and ambient groups. Let then $\K=\R$ and assume $\bord\G$ is homeomorphic to a circle. Real representations of $\G$ that are $(1,1,2)$-hyperconvex are necessarily locally conformal and form the prototype example of Anosov representations with $\class^1$ limit sets: indeed we have the following result from Pozzetti et al.~\cite{PSW1} and Zhang--Zimmer~\cite{ZZ}. \begin{theo}\label{t.C1} Assume $\bord\G$ is homeomorphic to a circle and let $\rho:\G\to\PGL(d,\R)$ be $\{\slroot_1\}$-Anosov. \begin{itemize} \item \cite{PSW1,ZZ}: If $\rho$ is $(1,1,2)$-hyperconvex, then $\xi^1(\bord\G)\subset\P(\R^{d})$ is\linebreak a~$\class^1$-submanifold tangent at $\xi^1(x)$ to $\xi^2(x)$. \item {\cite{ZZ}}: If $\rho$ is irreducible and $\xi(\bord\G)$ is a $\class^{1+\alpha}$ circle then $\rho$ is $(1,1,2)$-hyperconvex. \end{itemize} \end{theo} The graph map $\varXi=(\xi,\ov{\xi}):\bord\G\to\P(\R^{d})\times\P(\R^{\ovd})$ has image contained in the $\class^{1+\nu}$ torus $\xi(\bord\G)\times\ov{\xi}(\bord\G)$ and $\varXi(\bord\G)$ is the graph of $\Xi$, a H\"older-continuous homeomorphism between $\class^{1+\nu}$-circles. By monotonicity of $\Xi$, $\varXi(\bord\G)$ is a Lipschitz curve and is thus differentiable almost everywhere. We let \[ \Ndiff_{\rho,\ovrho}\subset\varXi(\bord\G) \] be the subset of points where the curve $\varXi(\bord\G)$ is not differentiable. The combination of Lemma~\ref{generalcase} and Corollary~\ref{nondiffbconicalR} establishes that in the current situation (with mild additional assumptions) \[ \varXi\bigl(\Ext^1_{\rho,\ovrho}\bigr)=\Ndiff_{\rho,\ovrho}, \] whence with Theorem~\ref{LCdiff} one obtains the following: \begin{theo}\label{tutti} Assume $\bord\G$ is homeomorphic to a circle and let $\rho:\G\to\SL(d,\R)$ and $\ovrho:\G\to\SL(\ovd,\R)$ be $(1,1,2)$-hyperconvex and not gap-isospectral. Then, \[ \Hff\big(\Ndiff_{\rho,\ovrho}\big)=\hC{\infty,1}<1. \] \end{theo} We emphasize that no irreducibility assumption is made on the representations $\rho$ and $\ovrho$. On the other hand, if the representations are irreducible and gap-isospectral, we show that there exists an isomorphism between the Zariski closures of $\rho(\G)$ and of~$\ovrho(\G)$ intertwining the two representations. It follows then that $\varXi(\bord\G)$ is the diagonal of the $\class^{1+\nu}$ torus, and thus differentiable everywhere. To prove this we give the following preliminary classification of Zariski-closures, established in Section~\ref{s.Zd}. Recall that if $\sfG$ is a semi-simple real-algebraic group of non-compact type, then irreducible proximal representations $\Fund:\sfG\to\PGL(V)$ are determined by their highest restricted weight $\chi_\Fund$. A special subset of dominant weights are the so-called \emph{fundamental weights} $\{\peso_\sroot:\sroot\in\simple\}$, and are indexed by the set of simple roots $\simple$ of~$\sfG$ (see Section~\ref{representaciones} for definitions and details). \begin{theo}\label{t.Zcl} Assume $\bord\G$ is homeomorphic to a circle and let $\rho:\G\to\PGL(d,\R)$ be irreducible and $(1,1,2)$-hyperconvex. Then the Zariski closure $\sfG$ of $\rho(\G)$ is simple and the highest weight of the induced representation $\Fund:\sfG\to\PGL(d,\R)$ is a multiple of a fundamental weight associated to a root whose root-space is one-dimensional. \end{theo} In light of the following examples it is unclear if further restrictions can occur. \begin{exam}\label{ex.hyperconvex} Any pair of representations $\rho:\pi_1S\to\sfG$ and $\Fund:\sfG\to\PGL(V)$ in each of the following classes (and small deformations), gives rise to a $(1,1,2)$-hyperconvex representation via post-composition $\Fund\circ\rho$. In particular the limit set of $\rho$ in the specified flag manifold of $\sfG$ is a $\class^{1+\nu}$ curve: \begin{itemize} \item $\sfG$ is split, $\rho:\piS\to\sfG$ is Hitchin, and $\Fund$ satisfies $\chi_\Fund=n\peso_\sroot$ for any $\sroot\in\simple$ and $n\in\N_{>0}$. This is non-trivial and requires results from Fock--Goncharov~\cite{FG} and Labourie~\cite{labourie} together with Lusztig's canonical basis~\cite[Proposition~3.2]{Lusztig-TP} (see Sambarino~\cite[Section~5.8]{clausurasPos} for details). As a result the limit set of $\rho$ in any maximal flag manifold $\calF_{\{\sroot\}}$ of $\sfG$ is a $\class^{1+\nu}$ curve. \item $\rho:\piS\to\PO(p,q)$ is $\Theta$-positive and $\Fund$ has highest weight $\peso_\sroot$ for any root $\sroot$ in the interior\footnote{I.e. $\sroot$ is only connected to roots in $\Theta$ in the Dynkin diagram of $\simple$.} of $\Theta$ (\cite[Theorem~10.3]{PSW2}, see also Beyrer--Pozzetti~\cite[Remark~4.6]{BP}). In particular the limit set in any flag manifold of the form ${\sf{Is}}_k(\R^{p,q})$ for $k\leq p-2$ is a $\class^{1+\nu}$-curve. When $\rho$ is moreover Zariski-dense, we can consider any $\Fund$ with $\chi_\Fund^+=n\peso_\sroot$ for any $\sroot \in\inte\Theta$ and $n\in\N_{>0}$. \item For all $k\geq 1$, $k$-positive representations $\rho:\piS\to\PSL(d,\R)$ introduced in~\cite{BeyP2} are $(1,1,2)$-hyperconvex. \end{itemize} \end{exam} For these examples also the following applies: \begin{coro}\label{cor.B} Assume $\bord\G$ is homeomorphic to a circle, let $\sfG$ be a simple Lie group and let $\rho:\G\to\sfG$ have Zariski-dense image. Assume there exist $\{\sroot,\bb\}\subset\simple$ distinct such that both $\Fund_\sroot\circ\rho$ and $\Fund_\bb\circ\rho$ are $(1,1,2)$-hyperconvex. Then: \begin{enumerate} \item\label{coro1.9.1} The image of $\xi^{\{\sroot,\bb\}}:\bord\G\to\calF_{\{\sroot,\bb\}}$ is Lipschitz and the Hausdorff dimension of the points where it is non-differentiable is $\hC{\max\{\sroot,\bb\}}$. \item\label{coro1.9.2} If the opposition involution $\ii$ on $\ge$ is non-trivial and $\bb=\ii\sroot$ then \[ \hC{\max\{\sroot,\bb\}}=\hC{(\sroot+\bb)/2}. \] \end{enumerate} \end{coro} \begin{rema} A different approach to Theorem~\ref{tutti}, relying on Theorems~\ref{t.Zcl} and~\ref{t.C1}, would be to code the action of $\piS$ on $\bord\piS$ via Bowen-series and apply Jordan et al.~\cite[Theorem~1.1]{JKPS}. This method, followed by Pollicott--Sharp~\cite{PS} for two representations in the Teichm\"uller space of $S$, is not applicable for groups other than $\piS$, in particular this approach cannot be used in the generality of Theorem~\ref{LCdiff}. \end{rema} \subsection{Hyperplane vs directional conicality} To prove Theorems~\ref{LCdiff} and~\ref{tutti} we introduce the concept of \emph{hyperplane conicality}, a~generalization of directional conicality from Burger et al.~\cite{BLLO}. Let $\sfG$ be a real-algebraic semi-simple Lie group of non-compact type, $\a\subset\ge$ a~Cartan subspace, $\roots\subset\a^*$ the associated root system and $\simple\subset\roots$ a choice of simple roots with associated Weyl chamber $\a^+$. Consider a non-empty $\t\subset\simple$ and let $\a_\t$ be the associated Levi space. Fix a $\t$-Anosov representation $\rho:\G\to\sfG$ and denote by $\cal L_{\t,\rho}\subset\a_\t$ its $\t$-limit cone. We will recall in Section~\ref{rfr} that, when $\rho(\G)$ is Zariski-dense, there are natural bijections \begin{align*} \inte\P(\cal L_{\t,\rho})& \leftrightarrow\calQ_{\t,\rho}= \left\{\varphi \in(\a_\t)^*:\hJ\varphi=1\right\} \\ &\leftrightarrow \left\{\text{Patterson--Sullivan measures supported on $\xi^\t(\bord\G)$}\right\}. \end{align*} For $\varphi\in\calQ_{\t,\rho}$ we let $\sf u_\varphi\in \inte\P(\cal L_{\t,\rho})$ be the associated direction and $\mu^\varphi$ the associated Patterson--Sullivan measure. Consider now a hyperplane $\sf W\subset\a_\t$ and assume, for the notion to be interesting, that $\sf W$ intersects the relative interior of $\calL_{\t,\rho}$. Then $x\in\bord\G$ is \emph{$\sf W$-conical} if there exists a conical sequence $(\g_n)_0^\infty\subset\G$ converging to $x$, a constant $K$ and a sequence $(w_n)_0^\infty\in\sf W$ such that for all $n$ one has \[ \bigl\|\cartan_\t(\rho(\g_n))- w_n\bigr\|\leq K, \] where $\cartan_\t:\sfG\to\a^+_\t$ is the $\t$-Cartan projection. The set of $W$-conical points will be denoted by $\bord_{\sf W,\rho}\G=\bord_{\sf W}\G$. Inspired by~\cite{BLLO}, in Theorem~\ref{ps-sandwich} we show the following. \begin{theo}\label{Ahyper} Let $\rho:\G\to\sfG$ be a Zariski-dense $\t$-Anosov representation and $\sf W$ be a hyperplane of $\a_\t$ intersecting non-trivially the interior of $\calL_{\t,\rho}$. Then for every $\varphi\in\calQ_{\t,\rho}$ with $\sf u_\varphi\in\P(\sf W)$ one has $\mu^\varphi(\bord_\sf W\G)=1$. \end{theo} \subsection{Strategy of the proof of Corollary~\ref{hilbert}}\label{specialcases} \label{corAProof} Corollary~\ref{hilbert} is a consequence of Theorem~\ref{tutti} where $\ovrho$ is the dual representation of~$\rho$. We sketch a direct proof of Corollary~\ref{hilbert} serving as a guide-path for the general result. Let $\rho:\piS\to\SL(3,\R)$ be the holonomy of a strictly convex projective structure dividing the convex set $\Omega$. We consider the $L^\infty$ distance on the product $(\P(\R^3), d_\P)\times\linebreak(\P((\R^3)^*),d_{\P})$, which is equivalent to the Riemannian distance, and thus induces the same Hausdorff dimension. As a replacement of Sullivan's shadows we use \emph{coarse cone type at infinity}, inspired by Cannon's work on \emph{cone types}~\cite{CannonCones} (see also Section~\ref{cont}). Fix a finite symmetric generating set on $\piS$ and let $|\,|$ be the associated word length. For $\g\in\piS$ and $c>0$, the \emph{coarse cone type at infinity} $\cone^{c}_\infty(\g)$ of $\gamma$ is the set of endpoints at infinity of $(c,c)$-quasi geodesic rays based at $\g^{-1}$ passing through the identity. See Figure~\ref{figure:conetype}. \begin{figure}[hh] \centering \begin{tikzpicture} \begin{scope}[scale=0.4, rotate=75] \draw circle [radius=.5]; \draw [thick] circle [radius = 3]; \draw [fill] circle [radius = 0.03]; \draw[thick] (-0.1,-0.3) -- (0.5,0.1); \node [below left] at (0,0) {$\g$}; \node [below left] at (0,1.2) {$\g\eta$}; \node [above] at (1.2,3.2) {$\g\cone^{c}_{\infty}(\g)$}; \draw [thick] (0.004,-1.3) -- (0.1,-0.7) -- (-0.1,-0.3) -- (0,0); \draw [thick] (0,0) -- (-0.3,1) -- (0.2,2) -- (0,3); \draw [thick] (0.2,2) -- (0.4, 2.7) -- (0.7,2.91); \draw [thick] (0.2,2) -- (0.4, 2.7); \draw [thick] (1,0.2) -- (0.89,1.37); \draw [thick] (1.8,0.8) -- (2.7,1.3); \draw [thick] (0,0) -- (1,0.2) -- (1.8,0.8) -- (2.846,0.948); \node [below] at (0.004,-1.3) {$e$}; \draw [fill] (0.7,2.91) circle [radius = 0.03]; \draw [fill] (2.7,1.3) circle [radius = 0.03]; \draw [fill] (0.004,-1.3) circle [radius = 0.03]; \draw [fill] (2.12,2.12) circle [radius = 0.03]; \draw [fill] (0,3) circle [radius = 0.03]; \draw [fill] (2.846,0.948) circle [radius = 0.03]; \draw [thick] (0,0) -- (0.89,1.37) -- (1.2,2) -- (2.12,2.12); \draw [thick] (1.2,2) -- (1.4,2.65); \draw [fill] (1.4,2.65) circle [radius = 0.03]; \node [right] at (3,-2) {$\piS$}; \filldraw [red](0,3) circle [radius=2pt]; \filldraw [red](0.7,2.91) circle [radius=2pt]; \filldraw [red](2.846,0.948) circle [radius=2pt]; \filldraw [red](2.7,1.3) circle [radius=2pt]; \filldraw [red](2.12,2.12) circle [radius=2pt]; \filldraw [red] (1.4,2.65) circle [radius=2pt]; \end{scope} \begin{scope}[scale=0.4,shift = {(-12,0)}] \draw [thick] circle [radius = 3]; \draw circle [radius=.75]; \draw [fill] circle [radius = 0.03]; \node [left] at (0,0) {$e$}; \draw[thick] (-0.1,-0.3) -- (0.5,0.1); \draw [thick] (0.004,-1.3) -- (0.1,-0.7) -- (-0.1,-0.3) -- (0,0); \draw [thick] (0,0) -- (-0.3,1) -- (0.2,2) -- (0,3); \draw [thick] (0.2,2) -- (0.4, 2.7) -- (0.7,2.91); \draw [thick] (1,0.2) -- (0.89,1.37); \draw [thick] (0.2,2) -- (0.4, 2.7); \draw [thick] (1.8,0.8) -- (2.7,1.3); \draw [thick] (0,0) -- (1,0.2) -- (1.8,0.8) -- (2.846,0.948); \node [below] at (0.004,-1.3) {$\g^{-1}$}; \draw [fill] (0.7,2.91) circle [radius = 0.03]; \draw [fill] (2.7,1.3) circle [radius = 0.03]; \draw [fill] (0.004,-1.3) circle [radius = 0.03]; \draw [fill] (2.12,2.12) circle [radius = 0.03]; \draw [fill] (0,3) circle [radius = 0.03]; \draw [fill] (2.846,0.948) circle [radius = 0.03]; \draw [thick] (0,0) -- (0.89,1.37) -- (1.2,2) -- (2.12,2.12); \draw [thick] (1.2,2) -- (1.4,2.65); \draw [fill] (1.4,2.65) circle [radius = 0.03]; \node [right] at (3,-2) {$\piS$}; \draw [->] (3.5,3) to [out = 180,in = 90] (1.5,1.2); \node [right] at (3.5,3) {$\cone^{c}(\g)$}; \end{scope} \end{tikzpicture} \caption{The coarse cone type of $\g\in\G$ (left). The set $\g\cdot\cone^{c}_\infty(\g)$ (right). Pictures from~\cite{PSW1}. %Pozzetti--Sambarino--Wienhard }\label{figure:conetype} \end{figure} We let $\xi:\bord\piS\to\bord\Omega$ be the natural identification via the action of $\rho(\piS)$ on~$\Omega$, and analogously $\ovxi:\piS\to\bord\Omega^*$. We denote by $\varXi:=(\xi,\ovxi): \piS\to\bord\Omega\times\bord\Omega^*$ the flag curve. Consider $x\in\bord\piS$ and let $\alpha_i\to x$ be a geodesic ray on $\piS$. The following fact is a consequence of Proposition~\ref{conetypesBalls}. \begin{enonce} {Fact} For big enough $i$, the subset $\xi(\alpha_i\cone^{c}_\infty(\alpha_i))\subset\bord\Omega$ is coarsely the intersection of a ball of radius $e^{-\slroot_1(\alpha_i)}$ about $\xi(x)$ with $\bord\Omega$. By duality, one has $\ovxi(\alpha_i\cone^{c}_\infty(\alpha_i))\subset\bord\Omega^*$ is coarsely the intersection of a ball of radius $e^{-\slroot_2(\alpha_i)}$ about $\ovxi(x)$ with $\bord\Omega^*$. \end{enonce} The coarse constants and the minimal length $i$ required in the above statement depend only on the representation and not on the point $x$. For any point $x\in\bord\piS$ we distinguish two complementary situations that don't depend on the choice of the geodesic ray $(\alpha_i)_{i\in\N}$ converging to $x$: \begin{enumerate}\romanenumi \item For all $R>0$ there exists $N\in\N$ with $|\slroot_1(\cartan(\alpha_i))-\slroot_2(\cartan(\alpha_i))|\geq R$ for all $i\geq N;$ \item There exists $R>0$ and an infinite set of indices $\I\subset\N$ such that for all $k\in\I$ one has $|\slroot_1(\cartan(\alpha_k))-\slroot_2(\cartan(\alpha_k))|\leq R$. We say in this case that $x$ is $\flat$\emph{-conical} ($\flat$ stands for 'barycenter of the chamber'). \end{enumerate} \begin{figure}[!h] \centering \begin{tikzpicture} \begin{scope}[rotate=0,scale=0.7,shift={(-3,-2)}] \draw (0,0) rectangle (6,4); \fill (3,0) circle (2pt); \fill (0,2) circle (2pt); \fill (3,2) circle (2pt); \draw[name path=c1,-] (0,0).. controls (1,2) and (2,1.7).. (3,2); \draw[name path=c2,-] (3,2).. controls (4.5,2.5) and (4.5,3).. (6,4); \draw[name path=l1,white] (2,0.2)--(2,3.8); \draw[name path=l2,white] (4,0.2)--(4,3.8); \draw[name intersections={of=c1 and l1, by={a}}, name intersections={of=c2 and l2, by={b}}] (a) rectangle (b); \node [above left] at (b) {\footnotesize{$e^{-\slroot_1(\rho\alpha_i)}$}}; \node [below right] at (b) {\footnotesize{$e^{-\slroot_2(\rho\alpha_i)}$}}; \begin{scope}[shift={(3,2)}] \end{scope} \node [above right] at (3,0) {$\footnotesize{\xi(x)}$}; \node [above right] at (0,2) {$\footnotesize{\ovxi(x)}$}; \end{scope} \end{tikzpicture} \caption{The image of the cone type $\alpha_i\cone^{c}_\infty(\alpha_i)$ by the graph curve $\varXi$ in the $\class^{1+\nu}$-torus $\bord\Omega\times\bord\Omega^*$.}\label{figg} \end{figure} In the first case one is easily convinced by looking at Figure~\ref{figg} that the rectangle becomes flatter along one of its sides (see Section~\ref{s.Hffproof} for details in the general case). Furthermore, since $\slroot_1(\cartan(\alpha_i))-\slroot_1(\cartan(\alpha_{i+1}))$ is uniformly bounded, its sign is eventually constant, and thus the longer side only depends on the point. As a result $x$ is necessarily a differentiability point of the graph curve $\varXi$, with either horizontal or vertical derivative. We are thus bound to understand the set of $\flat$-conical points. We show (see Corollary~\ref{nondiffbconicalR}): \begin{enonce}{Fact} The non-differentiability points of the curve $\varXi(\bord\piS)$ and the $\flat$-conical points coincide. \end{enonce} The main idea for this is to show that if a $\flat$-conical point $x$ were a differentiability point, then the derivative could not be horizontal nor vertical, and thus (by Proposition~\ref{c.indepPer}) $\Xi$ would be bi-Lipschitz. In turn, this would force the periods of the two roots to agree, which in turn would imply that the representation is Fuchsian, contradicting the assumption that $\Omega$ is not an ellipse. It remains to understand the Hausdorff dimension of the set of $\flat$-conical points. The upper bound (Proposition~\ref{upper}) \begin{equation}\label{u} \Hff(\{\flat-\text{conical}\})\leq\hC{\max\{\slroot_1,\slroot_2\}} \end{equation} follows readily: since for a $\flat$-conical point the lengths $e^{-\slroot_1(\alpha_k)}$ and $e^{-\slroot_2(\alpha_k)}$ are comparable independently on $k\in\I$, one can replace the rectangle in Figure~\ref{figg} by the (smaller) square of length \[ e^{-\max\{\slroot_1(\cartan(\alpha_k)),\slroot_2(\cartan(\alpha_k))\}} \] and still get a covering\footnote{Choosing the longer side $e^{-\min\{\slroot_1(\cartan(\alpha_k)),\slroot_2(\cartan(\alpha_k))\}}$ gives the bound {$\Hff\varXi(\bord\piS)\leq 1$.}} (this time by balls on the $L^\infty$ metric) of the set $\{\flat-\text{conical}\}$. Standard arguments on Hausdorff dimension give Eq.~\eqref{u}. Finding a lower bound for the Hausdorff dimension is more subtle; we use here an appropriate Patterson--Sullivan measure to study how the mass of a ball of radius $r$ scales with $r$. Since $\varXi(\bord\piS)$ is a subset the full flag space $\calF(\R^3)$ and \[ \|v\|_\infty:=\max\{|\slroot_1(v)|,|\slroot_2(v)|\} \] is a norm on $\a_{\PSL(3,\R)}$, we can apply results by Quint~\cite{Quint-Div} to determine a linear form $\varphi^\infty_\flat\in\a^*$ whose associated growth direction is the barycenter $\scr{b}=\ker(\slroot_1-\slroot_2)$. By~\cite[Proposition~3.3.3]{Quint-Div} \[ \hC{\max\{\slroot_1,\slroot_2\}}=\|\varphi^\infty_\flat\|^1, \] where $\|\,\|^1$ is the operator norm on $\a^*$ defined by $\|\,\|_\infty$, which turns out to be the $L^1$ norm $\|a\slroot_1+b\slroot_2\|^1=|a|+|b|$. The form $\varphi^\infty_\flat$ additionally admits an associated \emph{Patterson--Sullivan} probability measure, namely a measure $\ps^\infty$ such that for all $\g\in\piS$ one has (see Corollary~\ref{existe}) \begin{equation}\label{radi} \ps^\infty\bigl(\varXi(\g\cone^{c}_\infty(\g))\bigr)\leq Ce^{-\varphi^\infty_\flat(\cartan(\g))}. \end{equation} A key extra information available in the case of $\PSL(3,\R)$ is that the form $\varphi^\infty_\flat$ is explicit and, up to scaling, doesn't depend on $\rho$. For this we need a small parenthesis on the \emph{critical hypersurface} $\calQ_\rho$ of $\rho$, depicted in Figure~\ref{calQ}, and characterized by \[ \calQ_\rho=\bigl\{\varphi\in\a^*: \hC\varphi =1\bigr\}, \] where the \emph{critical exponent} of a functional $\varphi\in\a^*$ is \[ \hC\varphi:=\lim_{t\to\infty}\frac1t\log\#\bigl\{\g\in\piS:\varphi(\cartan(\g))\leq t\bigr\}\in(0,\infty]. \] The critical hypersurface $\calQ_\rho\subset\a^*$ is a closed analytic curve that bounds a strictly convex set (\cite{exponentecritico,exponential}), %(Sambarino~\cite{exponential} and Potrie--Sambarino~\cite{exponentecritico}), and thus by~\cite{Quint-Div}, the linear form $\varphi^\infty_\flat$ is uniquely determined by \begin{equation}\label{vi=norm} \|\varphi^\infty_\flat\|^1=\inf\big\{\|\varphi\|^1:\varphi\in\calQ_\rho\big\}. \end{equation} \begin{figure}[h]\centering \begin{tikzpicture} \begin{scope}[scale=0.7] \begin{scope}[rotate=-30,scale=0.5] \begin{scope}[shift={(0,3)},scale=0.85] \draw[name path global=Q, thick] (-13,3).. controls (0,-2.5).. (13,3); \node at (13,3) [right] {$\calQ_\rho$}; \end{scope} \draw[name path global=ii, dashed] (0,0) -- (0,6); \fill[name intersections={of=Q and ii, by={J}}] (J) circle (1pt); \begin{scope}[shift={(J)}] \draw [blue] (-6,0)--(6,0); \end{scope} \draw[<->] (-1,6) to [bend left] (1,6); \node at (0,7) {$\ii$}; \end{scope} \fill (J) circle (2pt); \draw [name path=slroot2] (0,0) -- (120:5cm); \draw (0,0) -- (180:2cm); \draw (0,0) -- (300:2cm); \draw [name path=slroot1] (0,0) -- (360:5cm); \node at (5,.5) [above] {$\a^*$}; \fill [name intersections={of=Q and slroot2, by={a}}] (a) circle (2pt); \node [above right] at (a) {$\slroot_2$}; \fill [name intersections={of=Q and slroot1, by={b}}] (b) circle (2pt); \node [below] at (b) {$\slroot_1$}; \draw [name path=borde, blue] (a) -- (b); \fill [name intersections={of=borde and ii, by={c}}] (c) circle (2pt); \node [right] at (c) {\footnotesize{$\sf H=\frac{\slroot_1+\slroot_2}2$}}; \node [left] (h') at (J) {}; \node [above left] (def) at (-3.5,0.5) {$\varphi^\infty_\flat=\hJ{\sf H}\sf H$}; \draw[->, orange] (def).. controls (2,2.5) and (-3.5,-.5).. (h'); \end{scope} \end{tikzpicture} \caption{The critical hypersurface of a strictly convex projective structure on $S$. Since $\sf H$ is a convex combination of $\{\slroot_1,\slroot_2\}$ one has $\|\sf H\|^1=1$ and thus $\|\varphi^\infty_\flat\|^1=\hC{\sf H}$.}\label{calQ} \end{figure} \pagebreak{} Again by~\cite{exponentecritico} one has $\{\slroot_1,\slroot_2\}\subset\calQ_\rho$. Since both $\calQ_\rho$ and the norm $\|\,\|^1$ are invariant by the opposition involution $\ii$ (see again Figure~\ref{calQ}) we deduce that, if we let $\sf H=(\slroot_1+\slroot_2)/2$, then \begin{equation}\label{>min} \varphi^\infty_\flat=\hC{\sf H}\cdot\sf H\geq \hC{\sf H}\min\{\slroot_1,\slroot_2\}. \end{equation} In particular, using Eq.~\eqref{vi=norm}, we obtain that $\hC{\max\{\slroot_1,\slroot_2\}}=\hC{\sf H}$. Moreover, since the geodesic flow is Anosov (by Benoist~\cite{convexes1}) we can apply Bowen's characterization of entropy~\cite{axiomA} (and Remark~\ref{r.entropy=crit}) to obtain that the Hilbert entropy $\hJ{\sf H}=\hC{\sf H}$. After this small parenthesis on the critical hypersurface, we come back to the lower bound on the Hausdorff dimension. Since $\varXi$ is a graph, $\varXi(\bord\piS)$ has the same intersection with the rectangle in Figure~\ref{figg} than with the larger square of size \[ e^{-\min\{\slroot_1(\cartan(\alpha_i)),\slroot_2(\cartan(\alpha_i))\}}; \] this square is now a ball (for the $L^\infty$ metric) of radius $e^{-\min\{\slroot_1(\cartan(\alpha_i)),\slroot_2(\cartan(\alpha_i))\}}$. Thus for all $i$, $\varXi(\alpha_i\cone^{c}_\infty(\alpha_i))$ is coarsely a ball of the latter radius and one has \begin{align*} \ps^\infty\bigl(B(\varXi(x),e^{-\min\{\slroot_1(\cartan(\alpha_i)),\slroot_2(\cartan(\alpha_i))\}}\bigr) & \leq \ps^\infty\bigl(\varXi(\alpha_i\cone^{c}_\infty(\alpha_i))\bigr)\leq Ce^{-\varphi^\infty_\flat(\cartan(\alpha_i))}\\ & \leq C\bigl(e^{-\min\{\slroot_1(\cartan(\alpha_i)),\slroot_2(\cartan(\alpha_i))\}}\bigr)^{\hC{\sf H}}, \end{align*} \noindent where the last inequalities follow from Eqs.~\eqref{radi} and~\eqref{>min}. This gives a possibly bigger constant $C'$ such that, for all $r$, \[ \ps^\infty\big(B(\varXi(x),r)\big)\leq C'r^{\hC{\sf H}}. \] Again, classical Hausdorff dimension arguments (c.f. Corollary~\ref{edgar} below) give that, for any measurable subset $E\subset\varXi(\bord\piS)$ with full $\ps^\infty$ mass, one has $\Hff(E)\geq \hC{\sf H}$. Since $\PSL(3,\R)$ has rank smaller than 3 and $\rho$ is $\simple$-Anosov we can apply~\cite[Theorem~1.6]{BLLO} %Burger--Landesberg--Lee--Oh to obtain that $\ps^\infty(\{\flat\text{-conical}\})=1$ and thus we have the desired lower bound \[ \Hff\bigl(\{\flat\text{-conical}\}\bigr)\geq \hC{\sf H}, \] which combined with the upper bound~\eqref{u} and the equality $\hC{\max\{\slroot_1,\slroot_2\}}=\hC{\sf H}$, gives the proof of Corollary~\ref{hilbert}. In the general case~\cite[Theorem~1.6]{BLLO} is not applicable and we replace it with Theorem~\ref{Ahyper}. \qed \medskip \subsection*{Structure of the paper} The preliminaries of the paper are standard facts about linear algebraic groups, recalled in Section~\ref{Liegroups}, the work of Sambarino in~\cite{dichotomy} about linear cocycles over the boundary of a hyperbolic group (in Section~\ref{cocycles}), as well as basic facts about Anosov representations and their Patterson--Sullivan theory recalled from~\cite{BPS, GW, PSW1, dichotomy} in the first part of Section~\ref{Anosov}. In the rest of Section~\ref{Anosov} we prove Theorem~\ref{hyper} a more precise statement than Theorem~\ref{Ahyper}, discussing the Patterson--Sullivan measure of $(\sf W,\varphi)$-conical points. The heart of the proof is to construct and study a rank 2 flow whose recurrence set is related to $(\sf W,\varphi)$-conical points. In Section~\ref{s.Hffbcon} we consider two locally conformal representations. We prove Theorem~\ref{Hffconical}, stating that for such a pair the Hausdorff dimension of the set of $\hol$-conical points belongs to \[ \left[\hol\hC{\infty,\hol},\min\left\{\hC{\infty,\hol},\hol\hC{\infty,\hol}+1-\hol\right\}\right]. \] The lower bound is obtained by analyzing properties of the linear form $\vi$ whose associated growth direction is $(\hol,1)$; its Patterson--Sullivan measure $\mu^{\vi}$ gives full mass to the set of $\hol$-conical points thanks to Theorem~\ref{hyper}. Using cone-types we can show that for a fine set of balls $\mu^{\vi}(B(x,r))\leq Cr^{-\hol\hC{\infty,\hol}}$. The upper bound uses results of~\cite{PSW1} to construct a fine covering of the set of $\hol$-conical points with balls of radius $e^{-\max\{\hol\slroot,\ovsloot\}}$. In Section~\ref{teoLCdiff} we prove Theorem~\ref{LCdiff}. In Section~\ref{s.6} we prove that if the graph map between $\R$-hyperconvex representations has an oblique derivative, then the map is bi-Lipschitz (Proposition~\ref{c.indepPer}). This only relies on basic properties of hyperconvex representations, and is crucial for the proof of Theorem~\ref{tutti}, achieved in Section~\ref{s.Hffproof}, as it allows the identification of $\flat$-conical points and points of non-differentiability. \subsection*{Acknowledgments} We thank Katie~Mann, Gabriele~Viaggi, Anna~Wienhard and Maxime~Wolff for insightful conversations and Andrés~Navas for pointing us to useful literature. \section{Linear algebraic groups}\label{Liegroups} Throughout the text $\sfG$ will denote a real-algebraic semi-simple Lie group of non-compact type and $\ge$ its Lie algebra. \subsection{Linear algebraic groups} Fix a Cartan involution $o:\ge\to\ge$ with associated Cartan decomposition $\ge=\k\oplus\p$. Let $\a\subset\p$ be a maximal abelian subspace and let $\roots\subset\a^*$ be the set of restricted roots of $\a$ in $\ge$. For $\sroot\in\roots$, we denote by \[ \ge_\sroot=\{u\in\ge:[a,u]=\sroot(a)u\;\forall a\in\a\} \] its associated root space. The (restricted) root space decomposition is $\ge=\ge_0\oplus\bigoplus_{\sroot\in\roots}\ge_\sroot$, where $\ge_0$ is the centralizer of $\a$. Fix a Weyl chamber $\a^+$ of $\a$ and let $\roots^+$ and $\simple$ be, respectively, the associated sets of positive and simple roots. Let $\Weyl$ be the Weyl group of $\roots$ and $\ii:\a\to\a$ be the opposition involution: if $u:\a\to\a$ is the unique element in $\Weyl$ with $u(\a^+)=-\a^+$ then $\ii=-u$. We denote by $(\cdot,\cdot)$ both the Killing form of $\ge$, its restriction to $\a$, and its associated dual form on $\a^*$, the dual of $\a$. For $\chi,\psi\in\a^*$ let \[ \<\chi,\psi\>=2\frac{(\chi,\psi)}{(\psi,\psi)}. \] The \emph{restricted weight lattice} is defined by \[ \poids=\left\{\varphi\in\a^*:\<\varphi,\sroot\>\in\Z\;\forall\sroot\in\roots\right\}. \] It is spanned by the \emph{fundamental weights} $\{\peso_\sroot:\sroot\in\simple\}$, defined by \begin{equation}\label{pesoFund} \<\peso_\sroot,\bb\>=d_\sroot\delta_{\sroot\bb} \end{equation} for every $\sroot,\bb\in\simple$, where $d_\sroot=1$ if $2\sroot\notin\roots^+$ and $d_\sroot=2$ otherwise. A subset $\t\subset\simple$ determines a pair of opposite parabolic subgroups $\sf P_\t$ and $\check{\sf P}_\t$ whose Lie algebras are \begin{align*} \p_\t &=\bigoplus_{\sroot\in\roots^+\cup\{0\}}\ge_{\sroot} \oplus\bigoplus_{\sroot\in\<\simple-\t\>}\ge_{-\sroot},\\ \check{\p_\t} &=\bigoplus_{\sroot\in\roots^+\cup\{0\}}\ge_{-\sroot} \oplus\bigoplus_{\sroot\in\<\simple-\t\>}\ge_{\sroot}. \end{align*} The group $\check{\sf P}_\t$ is conjugated to the parabolic group $\sf P_{\ii\t}$. We denote the \emph{flag space} associated to $\t$ by $\calF_\t=\sfG/\sf P_\t$. The $\sfG$ orbit of the pair $([\sf P_{\t}],[\check{\sf P}_{\t}])$ is the unique open orbit for the action of $\sfG$ in the product $\calF_\t\times\calF_{\ii\t}$ and is denoted by $\posgen_\t$. \subsection{Cartan and Jordan projection} Denote by $\sf K=\exp\k$ and $\sf A=\exp\a$. The \emph{Cartan decomposition} asserts the existence of a continuous map $\cartan:\sfG \to \a^+$, called the \emph{Cartan projection}, such that every $g\in\sfG$ can be written as $g=ke^{a(g)}l$ for some $k,l\in\sf K$. We will need the following uniform continuity of the Cartan projection: \begin{prop}[{\cite[Proposition~5.1]{Benoist-HomRed}}]\label{p.Ben} For any compact $ L\subset \sfG$ there exists a compact set $H\subset \a$ such that, for every $g\in \sfG$, one has \[ \cartan(LgL)\subset \cartan(g)+H. \] \end{prop} By the Jordan's decomposition, every element $g\in\sfG$ can be uniquely written as a commuting product $g=g_eg_{ss}g_u$ where $g_e$ is conjugate to an element in $\sf K$, $g_{ss}$ is conjugate to an element in $\exp(\a^+)$ and $g_u$ is unipotent. The \emph{Jordan projection} $\lambda=\lambda_{\sfG}:\sfG\to\a^+$ is the unique map such that $g_{ss}$ is conjugated to $\exp\big(\lambda(g)\big)$. \begin{defi} Let $\Gamma\subset\sfG$ be a discrete subgroup, then its \emph{limit cone} $\cal L_\Gamma$ is the smallest closed cone of the closed Weyl chamber $\a^+$ that contains $\{\lambda(g):g\in\Gamma\}$. \end{defi} We will need the following results by Benoist. \begin{theo}[{Benoist~\cite{limite,benoist2}}]\label{densidad} Let $\Gamma\subset\sfG$ be a Zariski-dense sub-semigroup, then its limit cone $\cal L_\Gamma$ has non-empty interior. Moreover, the group generated by the Jordan projections $\{\lambda(g):g\in\Gamma\}$ is dense in $\a$. \end{theo} \subsection{Representations of \texorpdfstring{$\sfG$}{sf G}}\label{representaciones} The standard references for the following are Fulton--Harris~\cite{FultonHarris}, Humphreys \cite{james} and Tits~\cite{Tits}. Let $\Fund:\sfG\to\PGL(V)$ be a finite dimensional rational\footnote{Namely a rational map between algebraic varieties.} irreducible representation and denote by $\phi_\Fund:\ge\to\fksl(V)$ the Lie algebra homomorphism associated to $\Fund$. The \emph{weight space} associated to $\chi\in\a^*$ is the vector space \[ V_\chi=\{v\in V:\phi_\Fund(a) v=\chi(a) v\ \forall a\in\a \}. \] \noindent We say that $\chi\in\a^*$ is a \emph{restricted weight} of $\Fund$ if $V_\chi\neq0$. Theorem~7.2 from~\cite{Tits} states that the set of weights has a unique maximal element with respect to the partial order $\chi\succ\psi$ if $\chi-\psi$ is a $\N$-linear combination of positive roots. This is called \emph{the highest weight} of $\Fund$ and denoted by $\chi_\Fund$. By definition, for every $g\in\sfG$ one has \begin{equation}\label{eq:spectralrep} \lambda_1(\Fund(g))=\chi_\Fund(\lambda(g)), \end{equation} where $\lambda_1$ is the logarithm of the spectral radius of $\Fund(g)$. We denote by $\poids(\phi)$ the set of restricted weights of the representation $\phi_\Fund$ \[ \poids(\phi)=\bigl\{\chi\in\a^*:V_\chi\neq\{0\}\bigr\}, \] these are all bounded above by $\chi_\Fund$ (see for example~\cite[Section~13.4, Lemma~B]{james}), namely every weight $\chi\in\poids(\phi)$ has the form \[ \chi_\Fund-\sum_{\sroot\in\simple} n_\sroot\sroot\quad \text{for}\quad n_\sroot\in\N. \] The \emph{level} of a weight $\chi$ is the integer $\sum_\sroot n_\sroot$, the highest weight is thus the only weight of level zero. Additionally, if $\chi\in\poids(\phi_\Fund)$ and $\sroot\in\roots^+$ then the elements of $\poids(\phi_\Fund)$ of the form $\chi+j\sroot,\, j\in\Z$ form an unbroken string \[ \chi+j\sroot,\: j\in\lb-r,q\rb \] and $r-q=\<\chi,\sroot\>$. One can then recover algorithmically the set $\poids(\phi_\Fund)$ level by level starting from $\chi_\Fund$, as follows: \begin{itemize} \item Assume the set of weights of level at most $ k$ is known and consider a weight $\chi$ of level $k$. \item For each $\sroot\in\simple$ compute $\<\chi,\sroot\>$, this gives the length $r-q$ of the $\sroot$-string through $\chi$. The weights of the form $\chi +j\sroot$, for positive $j$, have level smaller than $k$ and are thus known, thus we can decide whether $\chi-\sroot$ is a weight or not, determining the set of weights of level $k+1$. \end{itemize} The following lemma follows at once from the algorithmic description above. Let $\ge=\bigoplus_i\ge_i$ be the decomposition in simple factors of a semi-simple real Lie algebra of non-compact type. Recall that if $\a_i \subset\ge_i$ is a Cartan subspace, then $\a=\bigoplus_i\a_i$ is a Cartan subspace of $\ge$. Any $\varphi\in(\a_i)^*$ extends to a functional on $\a$, still denoted~$\varphi$, by vanishing on the remaining factors. The restricted root system of $\ge$ is then $\simple_\ge=\bigcup \simple_{\ge_i}$. The \emph{associated simple factor} to $\sroot\in\simple_\ge$ is $\ge_i$ such that $\sroot\in\simple_i$. \begin{lemm}\label{simple} Let $\ge$ be a semi-simple real Lie algebra of non-compact type and $\phi$ be an irreducible representation of $\ge$ whose highest restricted weight is a multiple of a fundamental weight, $\chi_\phi=k\peso_\sroot$ for some $\sroot\in\simple$. Then $\phi$ factors as a representation of the simple factor associated to $\sroot$. \end{lemm} \begin{proof} Proceeding by induction on the levels of $\phi$, one readily sees that for every $\slroot\in\simple_j$ for $j\neq i$ and all $\chi\in\poids(\phi)$ one has $\<\chi,\slroot\>=0$. Thus the associated root space $(\ge_j)_{-\slroot}$ acts trivially on every weight space of $\phi$ and so the whole factor $\ge_j$ acts trivially. \end{proof} The following set of simple roots plays a special role in representation theory. \begin{defi}\label{trep} Let $\Fund:\sfG\to\PGL(V)$ be a representation. We denote by $\t_\Fund$ the set of simple roots $\sroot\in\simple$ such that $\chi_\Fund-\sroot$ is still a weight of $\Fund$. Equivalently \begin{equation}\label{peso<} \t_\Fund=\bigl\{\sroot\in\simple:\<\chi_\Fund,\sroot\>\neq0 \bigr\}. \end{equation} \end{defi} The following lemma will be needed in the proof of Theorem~\ref{t.Zcl}. \begin{lemm}\label{not} Let $\ge$ be semi-simple of non-compact type and $\phi:\ge\to\fkgl(V)$ an irreducible representation. For $\sroot\in\t_\phi$ and $v\in V_{\chi_\phi}-\{0\}$, the map $n\mapsto \phi(n)v$ is injective when defined on $\ge_{-\sroot}$. \end{lemm} \begin{proof} By definition of $\chi_\phi$ every $n\in\ge_{\sroot}$ acts trivially on $V_{\chi_\phi}$. For $y\in\ge_{-\sroot}-\{0\}$, there exists $x\in\ge_\sroot$ such that $\{x,y,h_\sroot\}$ spans a Lie algebra isomorphic to $\sl_2(\R)$, where $h_\sroot$ is defined by $\varphi(h_\sroot)=\<\varphi,\sroot\>$ for all $\varphi\in\a^*$. If $\phi(y)v=0$ then, since $\phi(x)V_{\chi_\phi}=0$ one concludes $\phi(h_\sroot)v=0$ and since $V_{\chi_\phi}$ is a weight-space one has $\phi(h_\sroot)V_{\chi_\phi}=0$. This in turn implies that \[ \<\chi_\phi,\sroot\>=\chi_\phi(h_\sroot)=0, \] contradicting that $\sroot\in\t_\phi$. \end{proof} We denote by $\|\,\|_\Fund$ an Euclidean norm on $V$ invariant under $\Fund \sf K$ and such that $\Fund\sf A $ is self-adjoint, see for example Benoist--Quint's book~\cite[Lemma~6.33]{BQLibro}. By definition of $\chi_\Fund$ and $\|\,\|_\Fund$, and Eq.~\eqref{eq:spectralrep} one has, for every $g\in \sfG, $ that \begin{equation}\label{eq:normayrep} \log\|\Fund g\|_\Fund=\chi_\Fund(\cartan(g)). \end{equation} Here, with a slight abuse of notation, we denote by $\|\,\|_\Fund$ also the induced operator norm, which doesn't depend on the scale of $\|\,\|_\Fund$. Denote by $W_{\chi_\Fund}$ the $\Fund\sf A $-invariant complement of $V_{\chi_\Fund}$. The stabilizer in $\sfG $ of $W_{\chi_\Fund}$ is $\wk{\sf P}_{\t_\Fund}$, and thus one has a map of flag spaces \begin{equation}\label{maps} (\zeta_\Fund,\zeta^*_\Fund):\calF_{\t_\Fund}^{(2)}(\sfG)\to \grass_{\dim V_{\chi_\Fund}}^{(2)}(V). \end{equation} This is a proper embedding which is an homeomorphism onto its image. Here, as above, $\grass_{\dim V_{\chi_\Fund}}^{(2)}(V)$ is the open $\PGL (V)$-orbit in the product of the Grassmannian of $(\dim V_{\chi_\Fund})$-dimensional subspaces and the Grassmannian of $(\dim V-\dim V_{\chi_\Fund})$-dimensional subspaces. One has the following proposition (see also~\cite[Chapter~XI]{LAG}). \begin{prop}[{\cite{Tits}}]\label{prop:titss} For each $\sroot\in\simple$ there exists a finite dimensional rational irreducible representation $\Fund_\sroot:\sfG\to\PSL(V_\sroot)$, such that $\chi_{\Fund_\sroot}$ is an integer multiple $l_\sroot\peso_\sroot$ of the fundamental weight and $\dim V_{\chi_{\Fund_\sroot}}=1$. \end{prop} We will fix from now on such a set of representations and call them, for each $\sroot\in\simple$, the \emph{Tits representation associated to $\sroot$}. \subsection{The center of the Levi group \texorpdfstring{$\sf P_{\t}\cap\check{\sf P}_{\t}$}{sf P\_ t cap check sf P\_ t}}\label{s.Levi} We now consider the vector subspace \[ \a_\t=\bigcap_{\sroot\in\simple-\t}\ker\sroot. \] Denoting by $\Weyl_\t=\{w\in \Weyl:w(v)=v\quad \forall v\in\a_\t\}$ the subgroup of the Weyl group generated by reflections associated to roots in $\simple-\t$, there is a unique projection $\pi_\t:\a\to\a_\t$ invariant under $\Weyl_\t$. The dual $(\a_\t)^*$ is canonically identified with the subspace of $\a^*$ of $\pi_\t$-invariant linear forms. Such space is spanned by the fundamental weights of roots in $\t$, \[ (\a_\t)^*=\left\{\varphi\in\a^*:\varphi\circ\pi_\t=\varphi \right\}=\left\<\peso_\sroot|\a_\t:\sroot\in\t\right\>. \] We will denote, respectively, by \begin{alignat*}{2}\cartan_\t& =\pi_\t\circ \cartan:\sfG\to \a_\t\ \\ \lambda_\t & =\pi_\t\circ\lambda:\sfG\to \a_\t, \end{alignat*} the compositions of the Cartan and Jordan projections with $\pi_\t$. \subsection{The Busemann--Iwasawa cocycle} The \emph{Iwasawa decomposition} of $\sfG$ states that every $g\in\sfG$ can be written uniquely as a product $lzu$ with $l\in\sf K$, $z\in\sf A$ and $u\in\sf U_\simple$, where $\sf U_\simple$ is the unipotent radical of $\sf P_{\simple}$. The \emph{Busemann--Iwasawa cocycle} of $\sfG$ is the map $\bus:\sfG\times\calF\to\a$ such that, for all $g\in\sfG$ and $k[\sf P_\simple]\in\calF$, \[ \bus(g,k[\sf P_\simple])=\log(z) \] where $\log:\sf A\to \a$ denotes the inverse of the exponential map, and $gk=lzu$ is the Iwasawa decomposition of $gk$. Lemmes~6.1 and 6.2 from~\cite{Quint-PS} proved that the function $\bus_\t=\pi_\t\circ\bus$ factors as a cocycle $\bus_\t:\sfG\times \calF_\t\to\a_\t$. The Busemann--Iwasawa cocycle can also be read from the representations of $\sfG$. Indeed, \cite[Lemme~6.4]{Quint-PS} shows that for every $g\in\sfG$ and $x\in\calF_\t$ one has \begin{equation}\label{busnorma} l_\sroot\peso_\sroot(\bus(g,x))=\log\frac{\|\Fund_\sroot(g)v\|_\Fund}{\|v\|_\Fund}, \end{equation} where $v\in\zeta_{\Fund_\sroot}(x)\in\P(\sf V_\sroot)$ is non-zero, and $l_\sroot$ is as in Proposition~\ref{prop:titss}. \subsection{Gromov product and Cartan attractors}\label{GryBus} Let $\K$ be either $\C$ or $\R$. For a decomposition $\K^d=\ell\oplus V$ into a line $\ell$ and a~hyperplane $V$ together with an inner (Hermitian) product $o$ on $\K^d$, one defines the \emph{Gromov product} by \[ \Gr(\ell,V)=\Gr^o(\ell,V):=\log\frac{|\varphi(v)|}{\|\varphi\|\|v\|}=\log\sin\angle_o(\ell,V), \] for any non-vanishing $v\in\ell$ and $\varphi\in (\K^d)^*$ with $\ker\varphi=V$. This induces, for any semisimple Lie group $\sfG$ and subset $\t<\Delta$, a \emph{Gromov product} $\Gr_\t:\posgen_\t\to\a_\t$ defined, for every $(x,y)\in\posgen_\t$ and $\sroot\in\t$, by \[ l_\sroot\peso_\sroot(\Gr_\t(x,y))= \Gr^{\Fund_\sroot}(\zeta_{\Fund_\sroot} x,\zeta_{\Fund_\sroot}^*y)=\log\sin\angle_o\left(\zeta_{\Fund_\sroot} y,\zeta_{\Fund_\sroot}^*x\right), \] where $\zeta^*_{\Fund_\sroot}$ and $\zeta_{\Fund_\sroot}$ are the equivariant maps from Eq.~\eqref{maps}, and the Hermitian product~$o$ is induced by an Euclidean norm $\|\,\|_{\Fund_\sroot}$ invariant under $\Fund_\sroot\sf K$. From~\cite[Lemma~4.12]{orbitalcounting} one has, for all $g\in\sfG$ and $(x,y)\in\posgen_\t$, \begin{equation}\label{forGr} \Gr_\t(gx,gy)-\Gr_\t(x,y)=-\big(\bus_\t(g,x)+\ii\bus_{\ii\t}(g,y)\big). \end{equation} If $g=k\exp(\cartan(g))l$ is a Cartan decomposition of $g\in\sfG$ we define its $\t$-\emph{Cartan attractor} (resp. \emph{repeller}) by \[ U_\t(g)=k[\sf P_\t]\in\calF_\t\quad\text{and}\quad U_{\ii\t}(g^{-1})=l^{-1}[\check{\sf P}_\t]\in\calF_{\ii\t}. \] The \emph{Cartan basin of $g$} is defined, for $\alpha>0$, by \begin{equation}\label{e.CartBasin} B_{\t,\alpha}(g)= \left\{x\in\calF_\t:\peso_\sroot \Gr_\t\left(x,U_{\ii\t}(g^{-1})\right)>-\alpha,\; \forall \sroot\in\theta\right\}. \end{equation} \begin{rema}\label{fty} Observe that a statement of the form $\peso_\sroot\Gr_\t(x,y) \geq-\kappa$ for all $\sroot\in\t$ is a quantitative version (depending on the choice of $\sf K$) of the transversality between $x$ and $y;$ in particular it implies that $x$ and $y$ are transverse. \end{rema} Neither the Cartan attractor nor its basin are uniquely defined unless for all $\sroot\in\t$ one has $\sroot\big(\cartan(g)\big)>0$, regardless one has the following: \begin{rema} Given $\alpha>0$ there exists a constant $K_\alpha$ such that if $y\in\calF_{\t}$ belongs to $B_{\t,\alpha}(g)$ then one has \begin{equation}\label{comparision-shadow} \big\|\cartan_\t(g)-\bus_\t(g,y)\big\|\leq K_\alpha. \end{equation} Indeed, using Tits's representations of $\sfG$ and Eqs.~\eqref{eq:normayrep} and~\eqref{busnorma} this boils down to the elementary fact that if $A\in\GL_d(\R)$ verifies\footnote{Recall from Eq.~\eqref{e.slroot} that we denote by $\slroot_i$ the simple roots of $\GL_d(\RR)$.} $\slroot_1(\cartan(A))>0$ then for every $v\in\R^d$ one has \[ \log\frac{\|Av\|}{\|v\|}\geq \log\|A\|+\log\sin\angle\big(\R\cdot v,U_{d-1}(A^{-1})\big) \] (see for example~\cite[Lemma~A.3]{BPS}). \end{rema} \section{H\"older cocycles on \texorpdfstring{$\bord\G$}{bord G}}\label{cocycles} Let $\G$ be a finitely generated group, and fix a finite generating set $S$. A group $\G$ is \emph{Gromov hyperbolic} if its Cayley graph $\sf{Cay}(\G, S)$ is a Gromov hyperbolic geodesic metric space. In this case we denote by $\bord\G$ its Gromov boundary, namely the equivalence classes of (quasi)-geodesic rays. It is well known that, up to H\"older homeomorphism, $\bord\G$ doesn't depend on the choice of the generating set $S$. We will denote by $\bord^2\G$ the set of distinct pairs in $\bord\G$: \[ \bord^2\G:=\{(x,y)\in\bord\G\times\bord\G|\, x\neq y\}. \] For a finitely generated, non-elementary, word-hyperbolic group $\G$ we denote by $\sf g=(\sf g_t:\sf U\G\to\sf U\G)_{t\in\R}$ the \emph{Gromov--Mineyev geodesic} flow of $\G$ (see~\cite{gromov, mineyev}). %Gromov~\cite{gromov} and Mineyev~\cite{mineyev} Throughout this section we will have the same assumptions as in~\cite[Section~3]{dichotomy}, namely that $\sf g$ is metric-Anosov and that the lamination induced on the quotient by $\widetilde{\cal W}^{cu}=\{(x,\cdot,\cdot)\in\widetilde{\sf U\G}\}$ is the central-unstable lamination of $\sf g$. Since we will mostly recall needed results % S. from~\cite[Section~3]{dichotomy} we do not overcharge the paper with the definitions of metric-Anosov and central-unstable lamination: by~\cite{pressure}, %Bridgeman--Canary--Labourie--Sambarino word-hyperbolic groups admitting an Anosov representation verify the required assumptions. \begin{defi} Let $V$ be a finite dimensional real vector space. A \emph{H\"older cocycle} is a function $c:\G\times\bord\G\to V$ such that: \begin{itemize} \item for all $\g,h\in \G$ one has $c(\g h,x)=c(h,x)+c(\g,h(x))$, \item there exists $\alpha\in(0,1]$ such that for every $\g\in\G$ the map $c(\g,\cdot)$ is $\alpha$-H\"older continuous. \end{itemize} \end{defi} Recall that every \emph{hyperbolic element}\footnote{Namely an infinite order element.} $\g\in\G$ has two fixed points on $\bord\G$, the attracting $\g_+$ and the repelling $\g_-$. If $x\in\bord\G-\{\g_-\}$ then $\g^nx\to\g_+$ as $n\to\infty$. The \emph{period} of a H\"older cocycle for a hyperbolic $\g\in\G$ is $\ell_c(\g):=c(\g,\g^+)$. A cocycle $c^*:\G\times\bord\G\to\R$ is \emph{dual to $c$} if for every hyperbolic $\g\in\G$ one has \[ \ell_{c^*}(\g)=\ell_c(\g^{-1}). \] \subsection{Real-valued cocycles}\label{Sreal} Assume now $V=\R$ and consider a cocycle $\kappa$ with non-negative (and not all vanishing) periods. We denote by $[\G]$ the set of conjugacy classes in $\G$ and, for an element $\g\in\G$ we denote by $[\g]$ its conjugacy class. For $t>0$ we let \[ \sf R_t(\kappa)=\bigl\{[\g]\in[\G]\text{ hyperbolic}:\ell_\kappa(\g)\leq t\bigr\} \] and define the \emph{entropy} of $\kappa$ by \[ \hJ\kappa=\limsup_{t\to\infty}\frac1t\log\#\sf R_t(\kappa)\in(0,\infty]. \] For such a cocycle consider the action of $\G$ on $\bord^2\G\times\R$ via $\kappa$: \begin{equation}\label{gaction} \g\cdot(x,y,t)=\left(\g x,\g y,t-\kappa\left(\g,y\right)\right). \end{equation} The following is a straightforward consequence of~\cite[Theorem~3.2.2]{dichotomy}. \begin{prop}\label{-infty} Let $\kappa$ be a H\"older cocycle with non-negative periods and finite entropy. Then, the above action of $\G$ on $\bord^2\G\times\R$ is properly-discontinuous and co-compact. If moreover $c$ is another H\"older cocycle with non-negative periods and finite entropy then there exists a $\G$-equivariant bi-H\"older-continuous homeomorphism $E:\bord^2\G\times\R\to\bord^2\G\times\R$ which is an orbit equivalence between the $\R$-translation actions. \end{prop} We recall the notion of \emph{dynamical intersection}, a~concept from~\cite{pressure} %Bridgeman--Canary--Labourie--Sambarino for H\"older functions over a metric-Anosov flow, that can be pulled back to this setting via the existence of the \emph{Ledrappier potential} of $\kappa$ from~\cite[Section~3.1]{dichotomy}. The \emph{dynamical intersection} of two real valued cocycles $\kappa, c$ is \begin{equation}\label{defiII} \II(\kappa,c)=\lim_{t\to\infty}\frac1{\sf R_t(\kappa)}\sum_{\g\in\sf R_t(\kappa)}\frac{\ell_c(\g)}{\ell_\kappa(\g)}. \end{equation} We record in the following proposition various needed facts about $\II$: \begin{prop}[{\cite[Section~3]{pressure}}]\label{ineq} The dynamical intersection defined above is well defined, linear in the second variable and for all positive $s$ satisfies $\II(s\kappa,c)=\II(\kappa,c)/s$. If also $c$ has non-negative periods and finite entropy then $\II(\kappa,c)\geq \hJ\kappa/\hJ c$. Moreover, if $\II(\kappa,c)= \hJ\kappa/\hJ c$ then for every $\g\in\G$ one has $\hJ\kappa\ell_\kappa(\g)=\hJ c\ell_c(\g)$. \end{prop} We will also need the following definitions. \begin{defi}\leavevmode \begin{itemize} \item A \emph{Patterson--Sullivan measure for $\kappa$ of exponent} $\delta\in\R_+$ is a probability measure $\ps$ on $\bord\G$ such that for every $\g\in\G$ one has \begin{equation}\label{defP} \frac{d \g_*\ps}{d\ps}(\cdot)=e^{-\delta\cdot \kappa\left(\g^{-1},\,\cdot\,\right)}. \end{equation} \item Let $\kappa^*$ be a cocycle dual to $\kappa$, then a \emph{Gromov product} for the ordered pair $(\kappa^*, \kappa)$ is a function $[\cdot,\cdot]:\bord^2\G\to\R$ such that for all $\g\in\G$ and $(x,y)\in\bord^2\G$ one has \[ [\g x,\g y]-[x,y]=-\bigl(\kappa^*(\g,x)+\kappa(\g,y)\bigr). \] \end{itemize} \end{defi} \subsection{The critical hypersurface and intersection}\label{QQ} Let now $c:\G\times\bord\G\to V$ be a H\"older cocycle. Its \emph{limit cone} is denoted by \[ \calL_c=\overline{\bigcup_{\g\in\G}\R_+\cdot\ell_c(\g)} \] and its \emph{dual cone} by $(\calL_c)^*=\{\psi\in V^*:\psi|_{\calL_c}\geq0\}$. Observe that for every $\varphi\in\inte(\calL_c)^*$, $\varphi\circ c$ is a real-valued cocycle, so the concepts from Section~\ref{Sreal} apply. We denote by \begin{alignat}{2}\label{qh1} \calQ_c & =\left\{\varphi\in\inte\cdc{c}:\hJ{\varphi\circ c}=1\right\},\\ \cal D_c & = \left\{\varphi\in\inte\cdc{c}:\hJ{\varphi\circ c}\in(0,1)\right\}, \end{alignat} respectively the \emph{critical hypersurface} and the \emph{convergence domain} of $c$. For $\varphi\in\inte(\calL_c)^*$ we consider the linear map $\II_{\varphi}=\II_{\varphi}^c:V^*\to\R$ defined by \[ \II_{\varphi}^c(\psi):= \II(\varphi\circ c,\psi\circ c), \] as in Eq.~\eqref{defiII}. The natural identification between the set of hyperplanes in $V^*$ and $\P(V)$ is used in the next proposition. \begin{coro}[{\cite[Corollary~3.4.3]{dichotomy}}] \label{strictly} Assume $\calL_c$ has non-empty interior and that there exists $\psi\in(\calL_c)^*$ such that $\hJ{\psi}<\infty$. Then $\cal D_c$ is a strictly convex set with boundary $\calQ_c$. The latter is an analytic co-dimension one sub-manifold of~$V^*$. The map $\sf u^c:\calQ_c\to\P(V)$ defined by \[ \varphi\mapsto\sf u^c_\varphi:=\sf T_\varphi\calQ_c=\ker\II_\varphi \] is an analytic diffeomorphism between $\calQ_c$ and $\inte(\P(\calL_c))$. \end{coro} \subsection{Ergodicity of directional flows}\label{directional} It follows from Proposition~\ref{-infty} that if there exists $\psi\in(\calL_c)^*$ with $\hJ\psi<\infty$ then the $\G$-action $\bord^2\G\times V$ \[ \g(x,y,v)=\big(\g x,\g y,v-c(\g,y)\big) \] is properly discontinuous. \begin{defi} A H\"older cocycle $c$ is \emph{non-arithmetic} if the periods of $c$ generate a dense subgroup in $V$. \end{defi} We fix $\varphi\in\calQ_c$ and denote by $u_\varphi\in\sf u_\varphi$ the unique vector in $\cal L_c\cap\sf u_\varphi$ with $\varphi(u_\varphi)=1$. We define then the \emph{directional flow} \begin{align*} \df^\varphi &=\left(\df^\varphi_t:\G\/\left(\bord^2\G\times V\right)\to\G\/\left(\bord^2\G\times V\right)\right)_{t\in\R}\\ \intertext{by} t\cdot(x,y,v) &=(x,y,v-tu_\varphi). \end{align*} \begin{enonce}{Assumption}\label{Pat-SulEx} We assume there exists: \begin{itemize} \item a dual cocycle $(\varphi\circ c)^*$, \item a Gromov product $[\,,\,]_\varphi$ for such a pair, \item Patterson--Sullivan measures, $\ps^\varphi$ and ${\ov\ps}^\varphi$, respectively for each of the cocycles; (the exponent of both measures is then necessarily $\hJ\varphi=1$~\cite[Proposition~3.3.2]{dichotomy}). \end{itemize} \end{enonce} Consider then the $\varphi$-\emph{Bowen--Margulis} measure $\BM^\varphi$ on $\G\/(\bord^2\G\times V)$ defined as the measure induced on the quotient by the measure \begin{equation}\label{forBM} e^{-[\cdot,\cdot]_\varphi}{\ov\ps}^\varphi\otimes\ps^\varphi\otimes \Fundeb_{V}, \end{equation} for a $V$-invariant Lebesgue measure on $V$. We denote by $\cal K(\df^\varphi)$ the \emph{recurrence set} of the directional flow $\df^\varphi$: \[ \cal K(\df^\varphi):=\left\{p\in\G\/\left(\bord^2\G\times V\right)\,\middle|\, \exists B \text{ open bounded},\; t_n\to \infty \;\text{ with }\; \df^\varphi_{t_n}(p)\in B \right\}. \] \begin{coro}[{\cite[Corollary~3.6.1]{dichotomy}}] \label{d=2} Assume that $c$ is non-arithmetic, and that there exists $\varphi\in\calQ_c$ satisfying Assumptions~\ref{Pat-SulEx}. If $\dim V\leq2$ then the directional flow $\df^\varphi$ is $\BM^\varphi$-ergodic, and $\cal K(\df^\varphi)$ has total mass. If $\dim V\geq4$ then $\cal K(\df^\varphi)$ has measure zero. \end{coro} \section{Subspace conicality for Anosov representations: Theorem~\ref{Ahyper}}\label{Anosov} \subsection{Gromov hyperbolic groups and cone types}\label{cont} Let $\G=\langle S\rangle$ be a finitely generated non-elementary Gromov hyperbolic group, and recall from Section~\ref{cocycles} that we denote by $\bord^2\G$ the set of distinct pairs in its Gromov boundary~$\bord\G$. \begin{defi}\label{conicalsequence} A divergent sequence $\{\g_n\}_{n\in\N}\subset\G$ converges to a point $x\in\bord\G$ \emph{conically} if for every $y\in\bord\G-\{x\}$ the sequence $(\g_n^{-1}y,\g_n^{-1}x)$ remains on a~compact set of $\bord^2\G$. \end{defi} \begin{rema}\label{r.conicalgeodesic} It is easy to verify that a sequence $\{\g_n\}_{n\in\N}$ converges conically to $x\in\bord\G$ if and only if it lies in an uniform neighborhood of any geodesic ray $(\alpha_n)_0^\infty$ converging to $x$, namely there exists $K>0$ and a subsequence $\{\alpha_{n_k}\}$ such that for all $k$ one has $d_\G(\alpha_{n_k},\g_{k})0$ is given, then there exist $\ov L\in\N$ and $\delta_{\rho,c}>0$, depending only $c$ and the domination constants of $\rho$, such that for every $(c,c)$-quasi-geodesic segment through the identity $\{\alpha_i\}_{-m}^k$ with $k,m\geq L$ one has, for all $\sroot\in\t, $ that \[ \peso_\sroot\Gr_\t\bigl(U_\t(\rho\alpha_k), \;U_{\ii\t}(\alpha_{-m})\bigr)\geq\log\delta_{\rho,c}. \] \end{prop} Combining Proposition~\ref{conical} and Proposition~\ref{fundamentalConstant} we obtain: \begin{coro}\label{c.fundamentalConstant} Up to decreasing $\delta_{\rho,c}$, for every $\g\in\G$ and every $x\in\cone_\infty^c(\g)$ one has \[ \peso_\sroot\Gr_\t\bigl(\xi^\t_\rho(x),\;U_{\ii\t}(\g^{-1})\bigr)\geq\log\delta_{\rho,c}. \] In particular, if we let $\alpha=-\log\delta_{\rho,c}$ then (recall Eq.~\eqref{e.CartBasin}) \begin{equation}\label{e.conesinCartan} \xi^\theta_\rho(\cone_{\infty}^{c}(\g))\subset B_{\theta,\alpha}(\g). \end{equation} \end{coro} \begin{defi}\label{FConstant} Let $\rho:\G\to\sfG$ be $\t$-Anosov and $c>0$, then the constant $\delta_{\rho,c}$ verifying both Proposition~\ref{fundamentalConstant} and Corollary~\ref{c.fundamentalConstant} will be called \emph{the fundamental constant} of $\rho$ and $c$. If we consider geodesics instead of quasi-geodesics (i.e. $(c,C)=(1,0)$) we let $\delta_\rho$ be the \emph{fundamental constant} associated to $\rho$. \end{defi} The following two results will be needed in Section~\ref{s.6.1}. \begin{prop}[{cfr.~\cite[Section~5.1]{PSW1}}]%P.-S.-Wienhard~ \label{p.coneinBall} \label{Lipschitz-compatibleA} Let $\rho:\G\to\SL(d,\K)$ be projective Anosov and consider $c>0$. Then there exists a constant $K$, depending on $c$ and on $\rho$ such that for every large enough $\g\in\G$ one has \[ \xi^1_\rho\bigl(\g \cone_\infty^{c}(\g)\bigr)\subset B\left(U_1(\g),\; Ke^{-\slroot_1(\cartan(\g))}\right). \] \end{prop} \begin{proof} Using Corollary~\ref{c.fundamentalConstant} for $\theta=\{\slroot_1\}$, the result follows as in~ \cite[Section~5.1]{PSW1}.% P.--S.--Wienhard \end{proof} \begin{prop}\label{Lipschitz-compatibleB} Let $\rho:\G\to\SL(d,\K)$ be projective Anosov. For every $\alpha>0$ there exist $C$ and $\mu>0$ such that for every $\ell_1,\ell_2\in\P(\K^d)$ with \[ \Gr\left(\ell_i,U_{d-1}(\g^{-1})\right) \geq-\alpha, \quad i=1,2 \] it holds $d_\P(\rho(\g)\ell_1,\rho(\g)\ell_2)\leq Ce^{-\mu|\g|}d(\ell_1,\ell_2)$. \end{prop} \begin{proof} For an Hermitian product on $\C^d$, and every $\alpha>0$ there exists $C>0$ such that if $h\in\GL(d,\C)$ is such that $\slroot_1(\cartan(h))>0$, then for all $\ell_1,\ell_2\in\P(\C^d)$ with {$\angle(\ell_i,U_{d-1}(h^{-1}))>\alpha$} one has \[ d_\P(h\ell_1,h\ell_2)\leq Ce^{-\slroot_1(\cartan(h))}d_\P(\ell_1,\ell_2), \] (a proof follows, for instance, by applying~\cite[Lemma~2.8]{PSW1} to $g=h^{-1}$, $P=U_1(h)$ and $Q=hU_{d-1}(h)$). The result then follows by applying Definition~\ref{AnosovDefi}. \end{proof} The following technical result will be useful in the proof of Proposition~\ref{conK}. Given an Anosov representation, we can use the Gromov product to determine the endpoint of a conical sequence (recall Definition~\ref{conicalsequence}): \begin{lemm}\label{UdU1} Let $\rho:\G\to\sfG$ be $\t$-Anosov. If $\{\g_n\}\subset\G$ is a conical sequence, $x\in\bord\G$, and there exists $\sroot\in\t$ such that $\peso_\sroot\Gr_\t(\xi^\t(x),U_{\ii\t}(\g_n))\to-\infty$, then $\g_n\to x$. \end{lemm} \begin{proof} We denote by $y$ the endpoint of the conical sequence $\g_n$. Proposition~\ref{conical} and Remark~\ref{r.conicalgeodesic} imply that $U_{\ii\t}(\g_n)\to\xi^{\ii\t}(y)$. Since, however, $\peso_\sroot\Gr_\t(\xi^\t(x),U_{\ii\t}(\g_n))\to-\infty$, we deduce that $\xi^{\ii\t}(y)$ is not transverse to $\xi^\t(x)$ (recall Remark~\ref{fty}). Since $\xi^\t$ is transverse, we deduce that $x=y$. \end{proof} It will be useful in the proof of Proposition~\ref{conK} to know that the endpoints of conical sequences belong to pushed Cartan basins: \begin{lemm}\label{l.conicbasin} Let $\rho:\G\to G$ be $\theta$-Anosov, $x\in\bord\G$. If $\g_n\to x$ conically, then there exists $\alpha$ only depending on the sequence and the representation $\rho$ such that for every $n$, $\xi^\theta(x)\in \g_nB_{\t,\alpha}(\g_n)$. \end{lemm} \begin{proof} We know from Remark~\ref{r.conicalgeodesic} that $\g_n$ is contained in a neighbourhood of a geodesic ray to $x$, or equivalently there exist a constant $c$ such that $\gamma^{-1}x\in\cone^{c}_\infty(\gamma_n)$. The result is then a consequence of Eq.~\eqref{e.conesinCartan}. \end{proof} \subsection{Patterson--Sullivan theory of Anosov representations}\label{rfr} If $\rho$ is a $\t$-Anosov representation, then we can pullback the Busemann--Iwasawa cocycle of $\sfG$ using the equivariant maps: the \emph{refraction cocycle} associated to a $\t$-Anosov representation $\rho:\G\to\sfG$ is $\rfr:\G\times\bord\G\to\a_\t$ given by \[ \rfr(\g,x)=\rfr_{\t,\rho}(\g,x)=\bus_\t\bigl(\rho(\g),\;\xi^\t_\rho(x)\bigr). \] %Bridgeman--Canary--Labourie-S. Theorem~1.10 in~\cite{pressure} show that the Mineyev geodesic flow of a group $\G$ admitting an Anosov representations is metric-Anosov, and thus Section~\ref{cocycles} applies to~$\rfr$. Moreover, the following fact places $\rfr$ in the assumptions required in Sections~\ref{Sreal} and~\ref{QQ}, see~\cite{dichotomy} for details. \begin{enonce}{Fact} The periods of the refraction cocycle equal the $\theta$-Jordan projection: $\rfr(\g,\g^+)=\lambda_\t(\g)$. For any $\sroot\in\t$ the real valued cocycle $\peso_\sroot\rfr$ has finite entropy. \end{enonce} We import the following concepts of cocycles to the setting of Anosov representations: \begin{itemize} \item The limit cone of $\rfr$ will be denoted by $\calL_{\t,\rho}$ and referred to as \emph{the $\t$-limit cone of $\rho$}; it is the smallest closed cone that contains the projected Jordan projections $\{\lambda_\t(\g):\g\in\G\}$. \item The \emph{interior of the dual cone} $\inte(\calL_{\t,\rho})^*\subset\a_\t^*$ consists of linear forms whose \emph{entropy} \[ \hJ\varphi= \lim_{t\to\infty}\frac1t\log\#\bigl\{[\g]\in[\G]:\varphi(\lambda_\t(\g))\leq t\bigr\} \] is finite. \item The $\t$-\emph{critical hypersurface}, resp. $\t$-\emph{convergence domain}, of $\rfr$ will be denoted by \begin{align*}\label{qh2} \calQ_{\t,\rho} & =\left\{\varphi\in\inte\cdc{\t,\rho}:\hJ\varphi=1\right\},\\ \cal D_{\t,\rho} & = \left\{\varphi\in\inte\cdc{\t,\rho}:\hJ\varphi\in(0,1)\right\}. \end{align*} \item If $\calL_{\t,\rho}$ has non-empty interior, then we have a \emph{duality} diffeomorphism between $\calQ_{\t,\rho}$ and $\inte\P(\calL_{\t,\rho})$ given by \[ \varphi\mapsto\sf u_\varphi=\sf T_\varphi\calQ_\rho. \] \end{itemize} Observe that $\sf u_\varphi\in\inte\P(\calL_{\t,\rho})$ and $\varphi|\calL_{\t,\rho}-\{0\}>0$ so $\varphi(\sf u_\varphi)\neq0$. More information on these objets can be found on~\cite[Section~5.9]{dichotomy}. \begin{rema}\label{r.entropy=crit} It it proven in~\cite[Theorem~2.31$\MK$(2)]{GMT} %Glorieux--Monclair--Tholozan (see also~\cite[Corollary~5.5.3]{dichotomy}) that if $\rho$ is $\t$-Anosov then for every $\varphi\in\inte(\calL_{\t,\rho})^*$ the entropy $\hJ\varphi$ equals the critical exponent \[ \hC\varphi:= \lim_{t\to\infty}\frac1t\log\#\left\{\g\in\G:\varphi(\cartan(\g))\leq t\right\}. \] In particular the $\t$-convergence domain is also given by \[ \cal D_{\t,\rho} =\left\{\varphi\in (\a_\t)^*:\sum_{\g\in\G}e^{-\varphi(\cartan(\g))}<\infty\right\}, \] see~\cite[Section~5.7.2]{dichotomy}. \end{rema} The following is essentially a consequence of~\cite[Proposition~3.3.3]{Quint-Div}, %Quint's, we outline a~proof. \begin{coro}\label{quintgrowththeta} Consider a norm $N$ on $\a_\t$, then the critical exponent $h^N_\rho$ of the series $s\mapsto\sum_{\g\in\G}e^{-sN(\cartan_\t(\g))}$ is given by \[ h^N_\rho=\inf\left\{N^*(\varphi):\varphi\in\calQ_{\t,\rho}\right\}, \] where $N^*$ is the associated dual norm on $\a_\t^*$. \end{coro} \begin{proof} We consider the counting measure $\nu=\sum_{\g\in\G}\delta_{\cartan_\t(\g)}$ on $\a_\t$. We then have, in the notation of~\cite[Section~3]{Quint-Div}, that $\tau^N_\nu=h^N_\rho$ and, by Remark~\ref{r.entropy=crit}, $\sigma^N_\nu=\inf_{\varphi\in\calQ_{\t,\rho}}N^*(\varphi)$. Thus, in order to deduce the corollary from~\cite[Proposition~3.3.3]{Quint-Div} it is enough to verify that the counting measure $\nu$ is of concave growth as in~\cite[Section~3.2]{Quint-Div}. In turn this is a consequence of Lemma~\ref{l.Quint} below, an adaptation of~\cite[Proposition~2.3.1]{Quint-Div} (see also~\cite[Lemma~3.8]{KimOhWang} %Kim--Oh--Wang where similar arguments are explained for the $\a_\t$ counting measure). \end{proof} \begin{lemm}\label{l.Quint} Let $\|\,\|$ be a norm on $\a_\t$. Let $\grupo<\sfG$ be Zariski-dense and $\t$-Anosov. Then there exists a product map $m:\grupo\times\grupo\to\grupo$ with the following properties: \begin{enumerate} \item\label{lemm4.16.1} there exists a real number $\kappa\geq 0$ such that, for all $\g_1,\g_2\in\grupo$, \[ \bigl\|\cartan_\t(m(\g_1,\g_2))-\cartan_\t(\g_1)-\cartan_\t(\g_2)\bigr\|\leq\kappa; \] \item\label{lemm4.16.2} for every real $R\geq 0$ there exists a finite subset $H$ of $\grupo$ such that, for $\g_1,\g_2\g_1',\g_2'$ in $\grupo$ with $\|\cartan_\t(\g_i)-\cartan_\t(\g_i')\|0$, the $r$-\emph{tube about $\ell$} by \[ \T_r(\ell):=\left\{v\in V\,\middle|\, \exists w\in\ell, \|v-w\|0$ and a~conical sequence $\{\g_n\}_0^\infty\subset\G$ with $\g_n\to y$ such that for all $n$ one has $\Pi(\cartan_\theta(\rho(\g_n)))\in\T_r(\sf u_\varphi^\cc)$. \end{defi} The next statement follows from the definitions. \begin{lemm} A point $y\in\bord\G$ is $\sf W$-conical if and only if it is $\sf u^\cc_\varphi$-conical. \end{lemm} If we are allowed to worsen the constants, we can replace, in Definition~\ref{defCon}, the conical sequence $(\g_n)$ with an infinite subset of a geodesic ray: \begin{lemm}\label{conGeo} A point $y\in\bord\G$ is $\sf u_\varphi^\cc$-conical if and only if there exists $r>0$, a~geodesic ray $(\alpha_i)_0^\infty$ converging to $y$ and an infinite set of indices $\I\subset\N$ such that for all $k\in\I$ one has \[ \Pi(\cartan_\t(\alpha_k))\in\T_r\left(\sf u_\varphi^\cc\right). \] \end{lemm} \begin{proof} Assume $y$ is $\sf u_\varphi^\cc$-conical, then since $\{\g_n\}_0^\infty$ is conical, for any geodesic ray $(\alpha_n)_0^\infty$ converging to $y$ there exists $K>0$ and a subsequence $\{\alpha_{n_k}\}$ such that for all $k$ one has $d_\G(\alpha_{n_k},\g_{k})0$ small enough so that the quotient projection $\sf p:\bord^2\G\times V\to\G\backslash\bord^2\G\times V$ is injective on $\tilde{\rmB}=A^-\times A^+\times B(0,T)$. We can thus use Eq.~\eqref{forBM} to compute the measure of $\rmB=\sf p(\tilde{\rmB})$. For $\tilde{\cal K}({\df}^\varphi)=\sf p^{-1}(\cal K({\df}^\varphi))$, Proposition~\ref{conK} asserts \[ A^-\times\left(A^+\cap\bord_{\sf W,\rho}\G\right)\times B(0,T)=\tilde{\cal K}\left({\df}^\varphi\right)\cap\tilde{\rmB}. \] If $\codim \sf W=1$ by Corollary~\ref{dicoCC} $\BM^\varphi(\tilde{\rmB})=\BM^\varphi(\tilde{\cal K}({\df}^\varphi)\cap\tilde{\rmB})$, which implies that $\mu^\varphi(A^+\setminus\bord_{\sf W,\rho}\G)=0$ and thus $\mu^\varphi(\bord_{\sf W,\rho}\G)=1$. On the other hand, if $\codim\sf W\geq3$, then we have $\BM^\varphi(\tilde{\cal K}({\df}^\varphi))=0$ so $\mu^\varphi(A^+\cap\bord_{\sf W,\rho}\G)=0$ and the theorem is proved. \section{Locally conformal representations: Hausdorff dimension of \texorpdfstring{$\hol$}{hol}-conical points}\label{s.Hffbcon} In this section we let $\K=\R, \C$ or $\H$, the non-commutative field of Hamilton's quaternions. A Cartan subspace $\a$ of $\SL(d,\K)$ is the subspace of $\R^d$ consisting of vectors whose coordinates sum 0. For $g\in\SL(d,\K)$ we denote by \[ \cartan(g)=(\cartan_1(g),\dots,\cartan_d(g))\in\a^+ \] the coordinates of the Cartan projection. We recall Definition~\ref{d.hyp}. \begin{defi}\label{locConf} Let $p\in\lb2,d-1\rb$. A $\{\slroot_1,\slroot_{d-p}\}$-Anosov representation $\rho:\G\to\SL(d,\K)$ is \emph{$(1,1,p)$-hyperconvex} if, for every pairwise distinct triple $(x,y,z)\in\bord\G^{(3)}$, one has \[ \left(\xi^1(x)+\xi^1(y)\right)\cap\xi^{d-p}(z)=\{0\}. \] If in addition one has $\cartan_2(\rho(\g))=\cartan_p(\rho(\g))$ $\forall\g$, we say that $\rho$ is \emph{locally conformal}. As before, we identify from now on $\g$ and $\rho(\g)$. \end{defi} The terminology is justified by Proposition~\ref{conetypesBalls} below stating that for such representations pushed coarse cone types are coarsely balls, a small refinement of an analogous result from~\cite{PSW1}. %P.-S.-Wienhard In this section we will study conicality from Section~\ref{sandwich} on a specific situation that we now explain. Later, in Section~\ref{teoLCdiff}, we will relate this section to the notion of $\hol$-concavity and in Section~\ref{s.Hffproof} to differentiability properties of the map $\ovxi\circ\xi^{-1}$. Consider $\bar{\K}\in\{\R,\C,\H\}$ and two locally conformal representations $\rho:\G\to\SL(d,\K)$ and $\ovrho:\G\to\SL(\ovd,\bar{\K})$, with projective equivariant maps \begin{align*} \xi:\bord\G &\to\P\left(\K^d\right)\\ \ovxi:\bord\G &\to\P\left(\bar{\K}{}^{\ovd}\right). \end{align*} The product representation $(\rho,\ovrho):\G\to\SL(d,\K)\times\SL(\ovd,\bar{\K})$ is $\t$-Anosov for $\t=\{\slroot_1,\slroot_p,\ov{\slroot}_1,\ov{\slroot}_p\}$ with $\{\slroot_1,\ov{\slroot_1}\}$-limit map the ``graph map'' \[ \varXi=\left(\xi,\ov{\xi}\right):\bord\G\to\P\left(\K^{d}\right)\times\P\left(\bar{\K}^{\ovd}\right). \] We consider a Cartan subspace of the product group $\SL(d,\K)\times\SL(\ovd,\bar{\K})$ and let $\a_\t$ be the associated Levi space. Its dual $(\a_\t)^*$ is spanned by the fundamental weights of roots in $\t$. We let \begin{align*} \slroot &:=\frac{ p\peso_{\slroot_1}-\peso_{\slroot_p}}{p-1},\\ \ovsloot &:=\frac {p\peso_{\ovsloot_1}-\peso_{\ovsloot_p}}{p-1}. \end{align*} Both $\slroot,\ovsloot\in(\a_\t)^*$ and under the assumption $\cartan_2(\g)=\cartan_p(\g)$ for all $\g$ of Definition~\ref{locConf}, it holds on $\calL_{\rho} $ that $\slroot_1=\slroot$ and $\ovsloot=\ov{\slroot}_1$ (if $p=2$ the equality holds on $\a$). \begin{defi}\label{flat} Fix $\hol\in(0,1]$. A point $x\in\bord\G$ is \emph{$\hol$-conical} if it is conical as in Definition~\ref{W-conical} for the product representation $(\rho,\ovrho)$ with respect to the hyperplane \[ \{v\in\a_\t:\hol\slroot(v)=\ovsloot(v)\}=\ker(\hol\slroot-\ovsloot). \] Equivalently, there exist $R$, a geodesic ray $(\alpha_j)_0^\infty\subset\G$ with $\alpha_j\to x$, and a subsequence $\{j_k\}$ such that for all $k$ one has \[ \bigl|\hol\slroot\left(\cartan(\alpha_{j_k})\right)-\ov{\slroot}\left(\cartan(\ov\alpha_{j_k})\right)\bigr|\leq R. \] \end{defi} Consider also the critical exponent \[ \hC{\infty,\hol}=\lim_{t\to\infty}\frac 1t\log\#\bigl\{\g\in\G:\max\left\{\hol\slroot(\cartan(\g)),\ovsloot(\cartan(\ov\g))\right\}\leq t\bigr\}, \] and recall from Eq.~\eqref{defiII} the dynamical intersection defined by \begin{equation}\label{IIreps} \II_{\slroot}(\ovsloot)=\lim_{t\to\infty}\frac1{\#\sf R_t(\slroot)}\sum_{\g\in \sf R_t(\slroot)}\frac{\ovsloot(\lambda(\ov\g))}{\slroot(\lambda(\g))}, \end{equation} where $\sfR_t(\slroot)=\{[\g]\in[\G]:\slroot(\lambda(\g))\leq t\}$. In this section we compute the Hausdorff dimension of the image under the graph map $\varXi$ of the set of $\hol$-conical points with respect to a Riemannian metric: \begin{theo}\label{Hffconical} Let $\rho,\ovrho$ be locally conformal representations over $\K$ and $\bar{\K}$ respectively. Assume the group generated by $\{(\slroot(\lambda(\g)),\ovsloot(\lambda(\ov\g))):\g\in\G\}$ is dense in~$\R^2$. Then, for every $\hol\in(0,1]$ with \[ \II_{\slroot}(\ovsloot)>\hol>1/\II_{\ovsloot}(\slroot), \] one has \begin{align*} \hol\hC{\infty,\hol}\leq\Hff\varXi(\{\hol\text{-conical\ points}\}) &\leq\min\left\{\hC{\infty,\hol},\hol\hC{\infty,\hol}+(1-\hol) \right\}\\ &<\min\left\{\hJ{\ovsloot},\hJ{\slroot}/\hol\right\}\\ &\leq\Hff(\varXi(\bord\G))\\ & =\max\left\{\hJ{\slroot},\hJ{\ovsloot}\right\}. \end{align*} \end{theo} The proof of the above result is completed in Section~\ref{proofHffconical}. Recall that if $\hC{\slroot_1}=\hC{\ovsloot_1}$ and the representations are not gap-isospectral, then Proposition~\ref{ineq} gives $\II_{\ovsloot_1}(\slroot_1)>1$. Theorem~\ref{Hffconical} studies then $\hol$-conical points for any $\hol$ with $\II_{\ovsloot_1}(\slroot_1)>1/\hol\geq1$. As the following result shows, the equality between entropies is rather natural for $\K=\R$. \begin{theo}[{\cite{PSW1}}]\label{hyperh=1} Let $\rho:\G\to\SL(d,\K)$ be locally conformal, then \[ \hJ{\slroot}=\Hff(\xi(\bord\G)). \] Moreover, when $\K=\R$ and $\bord\G$ is homeomorphic to a $p-1$-dimensional sphere, $\scr h_{\slroot}=p-1$. \end{theo} When $\G$ is a surface group we can also weaken the assumption on the density of periods: \begin{coro}\label{flatsurface} Assume $\bord\G$ is homeomorphic to a circle and let $\rho$ and $\ovrho$ be non-gap-isospectral real $(1,1,2)$-hyperconvex representations of $\G$. Then \[ \Hff\varXi\left(\{1\text{-conical\ points}\}\right)=\hC{\infty,1}<1. \] \end{coro} \begin{proof} Proposition~\ref{nonA} below states that under our assumptions the group generated by $\{(\slroot(\lambda(\g)),\ovsloot(\lambda(\ov\g))):\g\in\G\}$ is dense in $\R^2$. Theorem~\ref{hyperh=1} guarantees that $\hC{\slroot}=\hC{\ovsloot}$ and Proposition~\ref{ineq} then gives $\II_{\ovsloot}(\slroot)>1$. The equality thus follows from Theorem~\ref{Hffconical}. \end{proof} Kim--Minsky--Oh~\cite{KMO-HffDir} have established related Hausdorff dimension computations when $\rho$ and $\ovrho$ are convex co-compact representations in $\SO(n,1)$ without any assumption on $\II$. \subsection{Cone types are coarsely balls} In~\cite{PSW1} Pozzetti, Sambarino and Wienhard gave a concrete description of the images under the boundary map of the cone types at infinity. We discuss here a slight extension of that result adapted to our needs. We denote by $d_\P$ the distance on $\P(\K^d)$ induced by the choice of an inner (Hermitian) product on $\K^d$ and by $B(\ell,r)\subset\P(\K^d)$ the associated ball of radius $r$ about $\ell$. \begin{prop}\label{conetypesBalls} Let $\rho:\G\to\SL(d,\K)$ be locally conformal. Then there exist positive constants $ c, k_1,k_2$ and $L\in\N$ such that for every $x\in\bord\G$, every geodesic ray $(\alpha_j)_0^\infty$ with endpoint $x$ and every $j>L$ one has \[ B\left(\xi(x),k_1e^{-\slroot_1(\cartan(\alpha_j))}\right)\cap\xi(\bord\G)\subset\xi\left(\alpha_j\cone_\infty^{c}(\alpha_j)\right) \subset B\left(\xi(x),k_2e^{-\slroot_1(\cartan(\alpha_j))}\right). \] \end{prop} \begin{proof} The desired inclusions are proven in~\cite{PSW1} for thickened cone types at infinity. We briefly explain here how to deduce from it the result we need. Following~\cite{PSW1} we denote by $X_\infty(\g)$, for $\g\in\G$, the \emph{thickened cone type at infinity}, namely the tubular neighborhood in $\P(\K^d)$ of $\xi(\cone_\infty(\g))$ of radius $\delta_\rho/2$, where $\delta_\rho$ is the fundamental constant from Definition~\ref{FConstant}. In~\cite[Corollary~5.10]{PSW1} it is established that there exist $c_1>0$ and $L_0>0$ only depending on the domination constants of $\rho$ such that for all $j\geq L_0$ one has \[ B\left(\xi(x),c_1e^{-\slroot_1(\cartan(\alpha_j))}\right)\cap\xi(\bord\G)\subset \alpha_iX_\infty(\alpha_i). \] By definition the thickened cone type $X_\infty(\g)$ is contained in the Cartan basin $B_{\{\slroot_1\},\alpha}(\g)$ for $\alpha=-2\log\delta_\rho$. So~\cite[Proposition~3.3]{PSW2} provides the existence of $c$ and $L_0$ such that for $\g\in\G$ with $|\g|>L_0$, one has \[ X_\infty(\g)\cap\xi(\bord\G)\subset\xi\left(\cone_\infty^{c}(\g)\right). \] Combining both equations one has, for all $j\geq L_0$ that \begin{equation}\label{bola} B\left(\xi(x),c_1e^{-\slroot_1(\cartan(\alpha_j))}\right)\cap\xi(\bord\G)\subset\xi\left(\alpha_j\cone_\infty^{c}(\alpha_j)\right) \subset B\left(\xi(x), Ke^{-\slroot_1(\cartan(\alpha_j))}\right), \end{equation} the second inclusion following from Proposition~\ref{p.coneinBall}. This concludes the proof. \end{proof} \subsection{Hausdorff dimension and related concepts} Recall that, given a metric space $(X,d)$ and a real number $s>0$, the \emph{$s$-capacity} of $X$ is \[ \cal H^s(X,d)=\lim_{\epsilon\to 0}\inf\left\{\sum_{U\in\cal U} \diam U^s\,\middle|\, \cal U \text{ open covering of $X$, } \sup_{U\in\cal U}\diam U<\epsilon\right\} \] and that \begin{equation}\label{HffDef} \Hff(X)=\inf \left\{s\,\middle|\, \cal H^s(X)=0\right\}=\sup \left\{s\,\middle|\,\cal H^s(X)=\infty\right\}. \end{equation} The following can be verified directly from the definition: \begin{lemm}\label{l.Hffunion} If $X=\bigcup_{n\in\N} X_n$ then \[ \Hff(X)=\sup \Hff(X_n). \] \end{lemm} We will use the following consequence of~\cite[Theorem~1.5.14]{Edgar} from Edgar's book: \begin{coro}\label{edgar} Let $E\subset\R^d$ be a measurable subset equipped with a probability measure $\nu$. If the upper density \[ \ov{D}^\alpha(x)=\limsup_{r\to0}\frac{\nu\left(B(x,r)\cap E\right)}{r^\alpha} \] is $\nu$-essentially bounded above, then $\Hff(E)\geq\alpha$. \end{coro} \subsection{The lower bound \texorpdfstring{$\Hff(\varXi\{\hol\text{-conical\ points}\})\geq\scr{b}\hC{\infty,\hol}$}{Hff(varXi{ hol-conical points})>= scrb hC infty, hol}}\label{s.lower} We import some tools from the proof of Theorem~\ref{hyper}. Consider the vector space \[ V^*:=\spa\{\slroot,\ov{\slroot}\} \] together with its radical $\ann(V^*)=\ker\slroot\cap\ker\ovsloot$ and the quotient vector space $V=\a_\t/\ann(V^*)$. Any element of $V^*$ vanishes on $\ann(V^*)$ and thus $V^*$ is naturally identified with the dual space of $V$. Using the preferred basis $\{\slroot,\ovsloot\}$ of $V^*$ we identify $V$ and $\R^2$ via the isomorphism $v\mapsto (\slroot(v),\ovsloot(v))$ and we let \[ \Pi:\a_\t\to\R^2 \] be the quotient projection (composed with the above isomorphism). The image of the hyperplane $\ker\hol\slroot-\ovsloot$ under the composition of $\Pi$ and the identification of $V$ with $\R^2$ is the line passing through $(1,\hol)$, \[ \Pi\left(\ker(\hol\slroot-\ovsloot)\right)=\left\{v\in V:\hol\slroot(v)=\ovsloot(v)\right\}. \] We consider the quadrant \[ V^+=\{\slroot\geq0\}\cap\{\ov{\slroot}\geq0\}. \] Let $\cc=\cc_{(\rho,\ovrho)}:\G\times\bord\G\to V$ be the composition of the refraction cocycle $\rfr_{(\rho,\ovrho)}$ of the pair with $\Pi$. Its periods are \[ \cc(\g,\g_+)=\left(\slroot(\lambda(\g)),\ovsloot(\lambda(\ov\g))\right), \] so by assumption $\cc$ is non-arithmetic. As in Section~\ref{sandwich} one has $\calQ_\cc=V^*\cap\calQ_{\t,(\rho,\ovrho)}$; by non-arithmeticity, the cone $\calL_\cc$ has non-empty interior and thus Corollary~\ref{strictly} gives that $\calQ_\cc$ is a strictly convex curve. We consider the max norm $\|v\|_{\infty,\hol}=\max\{\hol|\slroot(v)|,|\ov \slroot(v)|\}$ on $V$, and its dual (operator) norm on $V^*$ denoted by $\|\,\|^{1,\hol}$. Let $\vi\in\calQ_\cc$ be the unique form such that \[ \left\|\vi\right\|^{1,\hol}=\inf\left\{\|\varphi\|^{1,\hol}\,:\,\varphi\in\calQ_\cc\right\}. \] In the following lemma the role of the assumptions on dynamical intersection in Theorem~\ref{Hffconical} becomes clear: \begin{lemm}\label{paraphi} The functional $\vi/\|\vi\|^{1,\hol}$ is a convex combination $s\hol\slroot+(1-s)\ov\tau$ with $s\in(0,1)$ if and only if \begin{equation}\label{e.Iassumption} \II_\slroot(\ovsloot)> \hol>1/\II_{\ovsloot}(\slroot). \end{equation} In this case one has $\sf T_{\vi}\calQ_\cc=\spa\{ \hol\slroot-\ov{\slroot}\}$. \end{lemm} \begin{proof} Recall from Corollary~\ref{strictly} that $\sf T_{\hJ\slroot\slroot}\calQ_\cc=\ker\II_{\hJ\slroot\slroot}$ and $\calQ_\cc$ is strictly convex. Furthermore, by definition the functional $\vi$ is the point of $\calQ_\cc$, that minimizes the norm $\|\; \|^{1,\hol}$. The level set $\{\|\varphi \|^{1,\hol}=1\}$ is a rhombus with vertices $(\hol\slroot,\ovsloot)$ (in blue in Figure~\ref{phinfty}), the tangent to $\calQ_\cc$ at $\hJ\slroot\slroot$, in red in Figure~\ref{phinfty}, is the level set $\II_{\hJ\slroot\slroot}(\cdot)=1$, whence its intersection with the $\ovsloot$-axis is $\ovsloot/\II_{\hJ\slroot\slroot}(\ovsloot)$, and the tangent to $\calQ_\cc$ at $\hJ{\ovsloot}\ovsloot$ is the level set $\II_{\hJ{\ovsloot}\ovsloot}(\cdot)=1$, and it intersects the $\slroot$-axis is $\slroot/\II_{\hJ{\ovsloot}\ovsloot}(\slroot)$. Eq.~\eqref{e.Iassumption} is thus equivalent to the fact that the slope of the side of the rhombus, equal to $-1/\hol$, is between the slope of the tangent at $\hJ\slroot\slroot$, which is equal to $-\hJ\slroot/\II_{\hJ\slroot\slroot}(\ovsloot)=-1/\II_{\slroot}(\ovsloot)$, and the slope of the tangent at $\hJ{\ovsloot}\ovsloot$, which is equal to $-\II_{\hJ{\ovsloot}\ovsloot}(\slroot)/\hJ{\ovsloot}=-\II_{\ovsloot}(\slroot)$. Strict convexity of $\calQ_\cc$ ensures that this is equivalent to having a unique point in $\calQ_\cc\cap\{t\ovsloot:t>0\}\times\{s\slroot:s>0\}$ tangent to the side of the rhombus, which is the desired functional $\vi$. \end{proof} \begin{figure}[!h] \centering \begin{tikzpicture} \begin{scope}[scale=0.9, rotate=5] \begin{scope}[rotate=-30,scale=0.5] \begin{scope}[shift={(0,3.5)},scale=0.85] \draw[name path=Q, thick] (-13,3).. controls (0,-2.5).. (13,3); \node at (13,3) [right] {$\calQ_\cc$}; \end{scope} \end{scope} \draw [name path=slroot2] (0,0) -- (120:6cm); \draw (0,0) -- (180:2cm); \draw (0,0) -- (300:2cm); \draw [name path=slroot1] (0,0) -- (360:6cm); \node at (5,.5) [above] {$V^*$}; \fill [name intersections={of=Q and slroot2, by={a}}] (a) circle (2pt); \node [above right] at (a) {$\hJ{\ov\tau}\ov\tau$}; \fill [name intersections={of=Q and slroot1, by={b}}] (b) circle (2pt); \node [below] at (b) {$\hJ\slroot\slroot$}; \coordinate (s) at (0.5,0); \node [below] at (s) {$ \hol\slroot$}; \coordinate (tg) at (-1.2,0.9); \draw [name path=tangente, red] (b) -- (tg); \fill [name intersections={of=tangente and slroot2, by={I}}] (I) circle (2pt); \coordinate (hpsi) at (-1.6,2.5); \draw [name path=ig, white] (b) -- (hpsi); \fill [name intersections={of=ig and slroot2, by={hs}}] (hs) circle (2pt); \node [left] at (hs) {$\frac{\hJ\slroot}{\hol}\ov\tau$}; \coordinate (c) at ($(hs)-(b)$); \draw [name path=bar, white] (s)--($(s)+0.5*(c)$); \fill [name intersections={of=bar and slroot2, by={bs}}] (bs) circle (2pt); \node [left] at (bs) {$\ov\tau$}; \draw [blue] (s) --(bs)--($(0,0)-(s)$)--($(0,0)-(bs)$)--(s); \draw [name path=normh, blue] (b) --(hs); \fill (s) circle (2pt); \fill [name intersections={of=normh and Q, by={barpsi}}] (barpsi) circle (2pt); \node [above right] at (barpsi) {$\ov{\psi}$}; \node [above left] (def) at (-3.5,0.5) {$\frac{\hJ\slroot}{\II_\slroot(\ov\tau)}\ov\tau$}; \node [left] (I') at (I) {}; \draw[->, orange] (def).. controls (0,1) and (-3.5,-.5).. (I'); \coordinate (phi) at (3.95,-0.2); \draw [name path=phinfty, blue] (phi)--($(phi)+0.85*(c)$); \fill [name intersections={of=phinfty and Q, by={varphi}}] (varphi) circle (2pt); \node [above] at (varphi) {$\vi$}; \draw [gray] (0,0) -- (varphi); \fill (varphi) circle (2pt); \node [above] (tt) at (tg) {}; \node [above] at (-4,3.4) {$\hJ\slroot\slroot+\sf T_{\hJ\slroot\slroot}\calQ_\cc$}; \draw[->, red] (-4,3.4).. controls (-2,2) and (-3.5,0).. (tt); \fill (hs) circle (2pt); \fill (b) circle (2pt); \fill (bs) circle (2pt); \end{scope} \end{tikzpicture} \caption{The situation of Lemma~\ref{paraphi}.} \label{phinfty} \end{figure} We thus obtain the following key properties of $\vi$: \begin{lemm}\label{phi>min} Under the assumptions of Theorem~\ref{Hffconical} one has \begin{enumerate}\romanenumi \item\label{lemm5.10.1} $\sf u_\vi^\cc=\Pi(\ker(\hol\slroot-\ovsloot))$; \item\label{lemm5.10.2} for any $v\in V^+$ one has \[ \vi(v)\geq \hC{\infty,\hol}\hol\min\{\slroot(v),\ov{\slroot}(v)\}. \] \end{enumerate} Moreover one has $\hC{\infty,\hol}<\min\{\hJ{\ovsloot},\hJ{\slroot}/\hol\}$. \end{lemm} \begin{proof} Lemma~\ref{paraphi} implies that \begin{itemize} \item[\eqref{lemm5.10.1}] $\sf T_{\vi}\calQ_\cc=\spa\{\hol\slroot-\ov{\slroot}\}$ and thus \[ \sf u_\vi^\cc=\Ann\bigl(\R\cdot(\hol\slroot-\ov{\slroot})\bigr)=\Pi(\ker(\hol\slroot-\ovsloot)). \] \item[\eqref{lemm5.10.2}] $\vi/\|\vi\|^{1,\hol}=s\hol\slroot+(1-s)\ov{\slroot}$ for some $s\in(0,1)$ and hence\footnote{Indeed, if $x,y\geq0$, $s\in(0,1)$ and $\hol\in(0,1]$ one has: $s\hol x+(1-s)y\geq\hol\min(x,y)$. Assume for example that $y\geq x$ (the other case follows similarly), then \[ s\hol x+(1-s)y-\hol x\geq (1-s)(1-\hol)x\geq0. \] }, since $\hol\in(0,1]$, \[ \vi(v)\geq\|\vi\|^{1,\hol}\hol\min\{\slroot(v),\ov{\slroot}(v)\} \] for all $v\in V^+$. \end{itemize} In order to prove item~\eqref{lemm5.10.2}, we need to show that $ \hC{\infty,\hol}\leq\|\vi\|^{1,\hol}$. Since \[ \vi\bigl(\cartan_\t((\rho,\ovrho)\g)\bigr)\leq \bigl\|\Pi(\cartan_\t((\rho,\ovrho)\g))\bigr\|_{\infty,\hol}\left\|\vi\right\|^{1,\hol}, \] we deduce, for all $s>\|\vi\|^{1,\hol}$, \[ \sum_{\g\in\G}e^{-s\left\|\Pi(\cartan_\t((\rho,\ovrho)\g))\right\|_{\infty,\hol}}\leq \sum_{\g\in\G}e^{-\left(s/\left\|\vi\right\|^{1,\hol}\right)\vi\left(\cartan_\t((\rho,\ovrho)\g)\right)}<\infty \] where last inequality holds as $\hC\vi=1$ (by Eq.~\eqref{qh1} and Remark~\ref{r.entropy=crit}). The last assertion follows directly from the definitions: \begin{align*} \hC{\infty,\hol} &=\lim_{t\to\infty}\frac 1t\log\#\left\{\g\in\G:\max\bigl\{\hol\slroot(\cartan(\g)),\ovsloot(\cartan(\ov\g))\bigr\}\leq t\right\}\\ &<\lim_{t\to\infty}\frac 1t\log\#\left\{\g\in\G:\bigl\{\hol\slroot(\cartan(\g))\big\}\leq t\right\}\\ &=\hC{\slroot}/\hol=\hJ{\slroot}/\hol, \end{align*} where the strict inequality comes from Corollary~\ref{quintgrowththeta} together with strict convexity of $\calQ_{\t,(\rho,\ovrho)}$ and the last equality follows from Remark~\ref{r.entropy=crit}. The inequality $\hC{\infty,\hol}\leq \hJ{\ovsloot}$ is analogous. \end{proof} Let $\ps^\vi$ be the Patterson--Sullivan measure associated to $\vi$ by Corollary~\ref{existe}. Combining Eqs.~\eqref{e.CartBasin}, \eqref{e.conesinCartan} and Corollary~\ref{existe} we deduce that, for every $\g\in\G$, \begin{equation}\label{shadow} \ps^\vi\left(\g\cone_\infty^{c}(\g)\right)\leq Ce^{-\vi\left(\cartan_\t((\rho,\ovrho)\g)\right)}\leq Ce^{-\hC{\infty,\hol}\hol\min\left\{\slroot(\cartan(\g)),\ov{\slroot}(\cartan(\ov\g))\right\}}, \end{equation} where the last inequality comes from Lemma~\ref{phi>min}. By Proposition~\ref{conetypesBalls} there exist constants $ c,k_1$ and $\ov k_1$ such that if $(\alpha_i)_0^\infty$ is a geodesic ray from $\id$ to $x$ then for all $i$ the subsets \[ \xi\left(\alpha_i\cone_\infty^{c}(\alpha_i)\right)\quad\text{and}\quad\ov{\xi}\left(\alpha_i\cone_\infty^{c}(\alpha_i) \right) \] contain balls on the corresponding projective spaces of radii \[ k_1e^{-\slroot(\cartan(\alpha_i))} \quad\text{and}\quad \ov k_1e^{-\ov{\slroot}(\cartan(\ov\alpha_i))} \] respectively where $k_1, \ov k_1$ depend on the representations but not on $i$. Since $\varXi(\bord\G)$ is a graph, the preceding radius computation implies that the image of the cone type $\varXi(\alpha_i\cone_\infty^{c}(\alpha_i))$ contains the intersection of $\xi(\bord\G)\times\ov{\xi}(\bord\G)$ with a ball, for the product metric on $\P(\K^d)\times\P(\K^{\ovd})$, of radius \begin{equation}\label{radiusgraph} ke^{-\min\left\{\slroot(\cartan(\alpha_i)),\ov{\slroot}(\cartan(\ov\alpha_i))\right\}}, \end{equation} for some uniform constant $k$. This set of balls forms a fine set of neighbourhoods around any point $x\in\partial\G$. Combining this with Eq.~\eqref{shadow} and the fact that $\mu^\vi$ is supported on $\bord\G$, one has, possibly enlarging the constant $C$, that for all $r$ the measure of the ball of radius $r$ about $\varXi(x)$ is \[ \ps^\vi\left(B(x,r)\right)\leq Cr^{-\hC{\infty,\hol}\hol}. \] Since $\dim V^*=2$ and $\cc_{(\rho,\ovrho)}$ is assumed non-arithmetic, Theorem~\ref{ps-sandwich} states that the subset of $\hol$-conical points has full $\ps^\vi$ measure. Applying Corollary~\ref{edgar} one concludes that \[ \Hff\left(\varXi\{\hol-\text{conical points}\}\right)\geq\hol\hC{\infty,\hol}. \] \subsection{The upper bound} We now prove the second inequality. \begin{prop}\label{upper} Let $\rho,\ovrho$ be locally conformal representations over $\K$ and~$\bar{\K}$. For every $\hol\leq 1$, \[ \Hff\bigl(\varXi\{\hol-\text{conical\ points}\}\bigr)\leq\min\left\{\hC{\infty,\hol},\hol\hC{\infty,\hol}+(1-\hol) \right\}. \] \end{prop} \begin{proof} We say that a point $x$ is \emph{$(R,\hol)$-conical} if there exists a geodesic ray $(\alpha_i)_{i\in\N}$ converging to $x$ such that for an infinite subset $\I\subset \N$ of indices and for every $k\in \I$ \begin{equation}\label{e.5.11} \bigl|\hol\slroot\left(\cartan(\alpha_k)\right)-\ov{\slroot}\left(\cartan(\ov\alpha_k)\right)\bigr|\leq R. \end{equation} We denote by $\sf C_\hol^R$ the set of $(R,\hol)$-conical points. By Definition~\ref{flat} one has \[ {\displaystyle\bigcup_{R>0}\sf C_\hol^R=\{x\in\bord\G: x\text{ is }\hol-\text{conical}\}}, \] and thus by Lemma~\ref{l.Hffunion} it suffices to show that for every $R$ one has \[ \Hff\left(\sf C_\hol^{R}\right)\leq \hC{\infty,\hol}. \] For any constant $K>0$ and any $\g\in\G$ we denote by $B_{\g}^{\max, K}$ the open ball of $\P(\K^{d})\times \P(\K^{\ovd})$ given by: \[ B_{\g}^{\max, K}:= B\left(\left(U_1(\g),U_1(\ov\g)\right), K e^{-\max{\{\hol\slroot(\cartan(\g)), \ov{\slroot}(\cartan(\ov\g))\}}}\right), \] and denote by \[ \cal U_T^K:=\big\{B_{\g}^{\max, K}|\, |\g|\geq T\big\}. \] Let $K$, resp. $\ov K$, be the constants given by Proposition~\ref{p.coneinBall} for the representation $\rho$ (resp. $\ovrho$). We first observe that for $C=2e^R\max\{K,\ov K\}$ and every $T>0$, the set $\cal U_T^C$ covers $\varXi(\sf C_\hol ^R)$. Indeed, if $x\in\sf C_\hol^R$ consider the geodesic ray $(\alpha_i)_{i\in\N}$ converging to $x$, and the set $\I$ of indices for which Eq.~\eqref{e.5.11} holds. Then for every $k\in\I$ one has, since $\hol\leq1$, that \begin{align} \slroot\left(\cartan(\rho\alpha_k)\right) &\geq\hol\slroot\left(\cartan(\alpha_k)\right)>\max\left\{\hol\slroot\left(\cartan(\alpha_k)\right), \ovsloot\left(\cartan(\ov\alpha_k)\right)\right\}-R, \label{e.5.12.a}\\ \ovsloot\left(\cartan(\ov\alpha_k)\right) &>\max\left\{\hol\slroot\left(\cartan(\alpha_k)\right), \ovsloot\left(\cartan(\ov\alpha_k)\right)\right\}-R. \label{e.5.12} \end{align} Let now $T$ be fixed and choose $k\in \I$, $k>T$. Since $x\in\alpha_k\cone_\infty^{c}(\alpha_k)$, Proposition~\ref{p.coneinBall} together with Eq.~\eqref{e.5.12} give \begin{align*} d\left(\xi(x),U_1(\alpha_k)\right) &\leq Ce^{-\max\left\{\hol\slroot\left(\cartan(\alpha_k)\right),\, \ovsloot\left(\cartan(\ov\alpha_k)\right)\right\}}\\ d\left(\ovxi(x),U_1(\ov\alpha_k)\right) &\leq Ce^{-\max\left\{\hol\slroot\left(\cartan(\alpha_k)\right),\, \ovsloot\left(\cartan(\ov\alpha_k)\right)\right\}}, \end{align*} as desired. Furthermore, by definition of $\hC{\infty,\hol}$, for every $s>\hC{\infty,\hol}$, \[ \sum_{U\in \cal U_T^C}\diam U^s\leq2^sC^s\sum_{|\g|\geq T}e^{-s\max\left \{\hol\slroot(\cartan(\g)),\,\ovsloot(\cartan(\ov\g))\right\}}<+\infty, \] whence, Eq.~\eqref{HffDef} yields $\Hff(\sf C_\hol^R)\leq\hC{\infty,\hol}$. In order to obtain the second upper bound we observe that, if $\alpha\in\G$ satisfies Eq.~\eqref{e.5.11}, the set $\varXi(\alpha\cone_\infty^{c}(\alpha))$ can be covered with $e^{(1-\hol)\slroot(\cartan(\alpha))}$ balls of radius $2Ce^{-\slroot(\cartan(\alpha))}$. We denote by $\cal U_T$ the collection of open balls, that only take into account elements $\alpha\in\G$ with $|\alpha|>T$ that verify~\eqref{e.5.11}, which in particular covers the set $\sf C_\hol^R$. Using Eq.~\eqref{e.5.12.a} we obtain \begin{align*} \sum_{U\in \cal U_T}\diam U^s &\leq 2^sC^s\sum_{|\g|\geq T}e^{(1-\hol)\slroot(\cartan(\g))}e^{-s\slroot(\cartan(\g))}\\ &\leq 2^sC^s\sum_{|\g|\geq T}e^{-(s-(1-\hol))\slroot(\cartan(\g))}\\ &\leq 2^sC^se^{\frac{R(s-1+\hol)}\hol}\sum_{|\g|\geq T}e^{-\frac{(s-(1-\hol))}\hol\max\left\{\hol\slroot(\cartan(\g)),\,\ovsloot(\cartan(\ov\g))\right\}}. \end{align*} Since the latter quantity is finite whenever $\frac{(s-(1-\hol))}\hol>\hC{\infty,\hol}$, we deduce \[ \Hff\left(\sf C_\hol^R\right)<\hol\hC{\infty,\hol}+(1-\hol). \] \end{proof} We conclude this subsection computing the Hausdorff dimension of the image of the whole boundary through the graph map. See~\cite{HffGraph} for examples of homeomorphisms between Cantor sets for which the Hausdorff dimension of the graph exceeds the maximal Hausdorff dimension of the factors. \begin{prop}\label{p.uppergeneral} Let $\rho:\G\to\SL(d,\K)$, $\ovrho:\G\to\SL(\ovd,\bar{\K})$ be locally conformal. Then \[ \Hff(\varXi(\partial\G))=\max\left\{\hC{\tau},\hC{\ov \tau}\right\} \] \end{prop} \begin{proof} This follows as in the proof of Proposition~\ref{upper} considering the covers of $\varXi(\partial\Gamma)$ given by $\cal U_T^C:=\{B_{\g}^{\min, C}\mid |\g|\geq T\}$ with \[ B_{\g}^{\min, K}:= B\left(\left(U_1(\g),U_1(\ov\g)\right),\; K e^{-\min{\left\{\slroot(\cartan(\g)),\, \ov{\slroot}(\cartan(\ov\g))\right\}}}\right), \] and $C=2\max\{K, \ov K\}$ where $K$ (resp. $\ov K$) is the constant given by Proposition~\ref{p.coneinBall} for the representations $\rho$ (resp. $\ov \rho$). To conclude it is enough to observe that \[ \hC{\min \{\slroot, \ovsloot\}}=\max\left\{\hC{\slroot}, \hC{\ov \slroot}\right\}, \] a fact proven for example in~\cite[Lemma~5.1]{PSW2}. \end{proof} It is easy to generalize Proposition~\ref{p.uppergeneral} to an arbitrary number of factors. as an application we get. \begin{coro} Let $\rho:\G\to\SL(d,\K)$ and $\theta\subset\Delta$ be such that for all $\slroot_i\in\theta$, $\Phi_{\slroot_i}\circ\rho$ is $(1,1,2)$-hyperconvex. Then \[ \Hff\left(\xi^\theta_\rho(\partial\G)\right)=\max_{\slroot_i\in\theta}\hC{\slroot_i}. \] \end{coro} \subsection{Proof of Theorem~\ref{Hffconical}}\label{proofHffconical} The first inequality is established in Section~\ref{s.lower}, the second inequality is proven in Proposition~\ref{upper}, the third inequality follows from Lemma~\ref{phi>min} and the fourth from Theorem~\ref{hyperh=1}.The last equality was established in Proposition~\ref{p.uppergeneral}.\qed \section{\texorpdfstring{$\hol$}{hol}-concavity and \texorpdfstring{$\hol$}{hol}-conicality: Final steps for the proof of Theorem~\ref{LCdiff}}\label{teoLCdiff} The goal of this section is to prove the following more general version of Theorem~\ref{LCdiff}. As before, fix $\{\K,\bar{\K}\}\subset\{\R,\C,\H\}$ together with locally conformal representations $\rho:\G\to\SL(d,\K)$ and $\ovrho:\G\to\SL(\ovd,\bar{\K})$ of an arbitrary word-hyperbolic group~$\G$. For $\hol\in(0,1]$ recall that $\Xi:\xi(\bord\G)\to\ovxi(\bord\G)$ is \emph{$\hol$-concave} at $x\in\bord\G$ if there exists $y_k\to x$ such that the incremental quotients \begin{equation}\label{incrlimit} \frac{d_{\P}\left(\ovxi(x),\ovxi(y_k)\right)}{d_{\P}\left(\xi(x),\xi(y_k)\right)^\hol} \end{equation} are bounded away from $0$ and $\infty$ (independently of $k$). We also let $\Ext_{\rho,\ovrho}^\hol$ be the set of $x\in\bord\G$ that are $\hol$-concavity points of $\Xi$. Finally, recall that $\rho$ and $\ovrho$ are \emph{not gap-isospectral} if there exists $\g\in\G$ such that $\slroot(\cartan(\g))\neq\ovsloot(\cartan(\ov\g))$. \begin{theo}\label{tutti.LCdiff} Let $\rho, \ovrho$ be locally conformal representations acting irreducibly on $\K^d$ and ${\bar{\K}}{}^{\ovd}$ respectively as real vector spaces, that are not gap-isospectral. Consider any $\hol\in(0,1]$ with $\II_{\slroot}(\ovsloot)>\hol>(\II_{\ovsloot}(\slroot))^{-1}$, then: \begin{itemize} \item if $\{\K,\bar{\K}\}\subset\{\R,\C\}$ one has \begin{equation}\label{holdineq} \begin{aligned}[t] \hol\hC{\infty,\hol}\leq\Hff\left(\varXi({\Ext}_{\rho,\ovrho}^\hol)\right) &\leq\min\left\{\hC{\infty,\hol},\hol\hC{\infty,\hol}+(1-\hol) \right\}\\ &<\min\left\{\hJ{\ovsloot},\hJ{\slroot}/\hol\right\}\\ &\leq\Hff(\varXi(\bord\G))\\ &=\max\{\hJ{\slroot},\hJ{\ovsloot}\}; \end{aligned} \end{equation} \item if $\K=\H$ (resp. $\bar{\K}=\H$), Eq.~\eqref{holdineq} holds if we further assume that the real Zariski closure of $\rho(\G)$ (resp. of $\ovrho(\G)$) does not have compact factors. \end{itemize} \end{theo} \subsection{Hyperplane conicality and the concavity condition} We commence with a lemma relating $\hol$-conicality to the desired concavity properties of the equivariant map $\Xi:\xi(\bord\G)\to\ovxi(\bord\G)$. \begin{lemm}\label{generalcase} Let $\rho$ and $\ovrho$ be locally conformal representations over $\K$ and $\bar{\K}$ respectively, and $\hol\in(0,1]$. Then one has $\{\hol\text{-conical points of } (\rho,\ovrho)\}=\Ext_{\rho,\ovrho}^\hol$. \end{lemm} \begin{proof} Let $(\alpha_i)_{i\in\N}$ denote a geodesic ray converging to $x$. Proposition~\ref{conetypesBalls} gives constants $C_1,C_2, \bar{C}_1, \bar{C}_2$ and $L\in\N$ such that, for every $n\in\N$ and every $y_n\in \alpha_n\cone_{\infty}^{c}(\alpha_n)\setminus\alpha_{n+L}\cone_\infty^{c}(\alpha_{n+L})$, it holds \begin{equation}\label{e.nestcone} \begin{aligned} C_1e^{-\slroot(\cartan(\alpha_n))} & n_R$ one has \[ -\hol\slroot(\cartan(\alpha_{n}))+\ov{\slroot}(\cartan(\ov\alpha_{n}))>R. \] In turn this implies, thanks to Eq.~\eqref{e.nestcone}, that for every $y\in \alpha_{n_R}\cone_\infty^{c}(\alpha_{n_R})$, \[ \frac{d_\P\left(\ovxi(y),\ovxi(x)\right)}{d_\P\left(\xi(y),\xi(x)\right)^\hol}\leq e^{-R}\frac {\bar{C}_2}{ {C_1}^\hol}. \] Since $R$ is arbitrary, and the sets $\alpha_{n_R}\cone_\infty^{c}(\alpha_{n_R})$ form a system of neighborhoods of the point $x$, we deduce that the limit in Eq.~\eqref{incrlimit} exists and equals $0$. This concludes the proof of Lemma~\ref{generalcase}. \end{proof} \subsection{Non-arithmeticity of periods} In this section we establish a non-arithme\-ti\-ci\-ty condition, necessary to apply later Theorem~\ref{Hffconical}. This is established in a rather general setting. Recall that a subgroup $\grupo<\SL(d,\K)$ is \emph{$\K$-proximal} if it contains a $\K$-proximal element, i.e. there exists $g\in\grupo$ such that $\slroot_1(\lambda(g))>0$. \begin{prop}\label{nonA} Let $\grupo$ be a finitely generated group. Let $\rho:\grupo\to\SL(d,\K)$ and $\ovrho:\grupo\to\SL(\ovd,\bar{\K})$ be two $\K$-proximal representations that act irreducibly on $\K^d$ and $\bar{\K}{}^{\ovd}$ respectively, as real vector spaces. Assume there exists $\g\in\grupo$ proximal for $\rho$ and $\ovrho$ such that $\slroot_1(\lambda(\rho \g))\neq\ovsloot_1(\lambda(\ov \rho \g))$. If $\{\K,\bar{\K}\}\subset\{\R,\C\}$, then the group generated by the pairs \[ \Bigl\{\bigl(\slroot_1(\lambda(\rho \g)),\ovsloot_1(\lambda(\ov \rho \g))\bigr)\,:\,\g\in\grupo\Bigr\} \] is dense in $\R^2$. If $\K=\H$ we further assume that the Zariski closure over $\R$ of $\rho(\grupo)$ has no compact factors, and the same for $\ovrho(\grupo)$ if moreover $\bar{\K}=\H$, then the same conclusion holds. \end{prop} To prove the proposition we need Lemmas~\ref{semisimple} and~\ref{semi} below. \begin{lemm}\label{semisimple} Let $\K$ be either $\R$ or $\C$. Let $\grupo<\SL(d,\K)$ be a subgroup acting irreducibly on $\K^d$ as a real vector space and assume $\grupo$ contains a $\K$-proximal element. Then the real Zariski closure of $\grupo$ is semi-simple, has finite center and without compact factors. \end{lemm} \begin{proof} If $\K=\R$ Lemma~\ref{semisimple} is the content of~\cite[Lemma~8.6]{entropia} and the proof over $\C$ is a slight modification of the latter. Indeed, let $\sfG$ be the Zariski closure of $\rho(\grupo)$ over the reals, by the irreducibility assumption it is a reductive (real-algebraic) group. By Schur's lemma the elements commuting with $\grupo$ consist only on homotheties, but since we're in special linear group one has that the center of $\sfG$ is finite. The group $\sfG$ is then semi-simple and we let $K$ be the identity component of the product of all the compact simple factors of $\sfG$. We also let $H$ be the identity component of the product of all the non-compact simple factors of $\sfG$. The groups $H$ and $K$ commute and one has $HK$ has finite index in $\sfG$. Consider a proximal $g\in\sfG$, up to a fixed power we may write $g=kh$ with $k\in K$ and $h\in H$. Since $K$ is compact, its eigenvalues have modulus one so we conclude that $h$ is proximal and that $g_+=h_+$. The attracting line of $h$ is thus invariant under~$K$. Since $K$ is connected, an element of $ K$ acts on $h_+$ as multiplication by some element of $\circle$. By irreducibility we may find a basis of $\C^d$ consisting on fixed attracting lines of proximal elements of $H$. This basis simultaneously diagonalizes $K$, so we get an injective map from $K$ to a compact group isomorphic to a $d$-dimensional torus. Consequently $K$ is abelian, and since it commutes with $H$ we conclude that $K$ is contained in the identity component of the center of $\sfG$, which we proved earlier to be trivial. \end{proof} \begin{lemm}\label{semi} Let $G$ be a semi-simple real-algebraic Lie group with finite center and no compact factors. Fix $\vartheta,\ov\vartheta\subset\simple_G$ two non-empty subsets with $\vartheta\cap\ov\vartheta=\emptyset$. Let $\grupo$ be a group and $\scr r:\grupo\to G$ a representation with Zariski-dense image. Then, for every closed cone with non-empty interior $\scr C\subset\inte\calL_{\scr r(\grupo)}$, the group spanned by the pairs \[ \left\{\left(\min_{\sigma\in\vartheta}\sigma(\lambda(\scr r g)),\min_{\ov\sigma\in\ov\vartheta}\ov\sigma(\lambda(\scr r g))\right)\,:\,g\in\grupo\quad\text{and}\quad\lambda(\scr rg)\in\scr C\right\} \] is dense in $\R^2$. \end{lemm} \begin{proof} Define the piecewise linear maps $\slroot,\ovsloot:\a^+\to\R$ by: \begin{align*} \slroot(v) & =\min\bigl\{\sigma(v):\sigma\in\vartheta\bigr\}\\ \ovsloot(v) & =\min\bigl\{\ov\sigma(v):\ov\sigma\in\ov\vartheta\bigr\}. \end{align*} The vanishing set of the difference $\slroot-\ovsloot$ is contained the union of $\ker(\sroot-\bb)$ for arbitrary $\sroot\in\vartheta$ and $\bb\in\ov\vartheta$. Since $\vartheta$ and $\ov\vartheta$ are disjoint, this is a union of hyperplanes of $\a$, from which we deduce that the set of zeroes of $\slroot-\ovsloot$ has empty interior. Since $\scr C\subset \inte\calL_{\scr r(\grupo)}$ has non-empty interior, the difference $\slroot-\ovsloot$ does not identically vanish on $\scr C$. Since $\slroot$ and $\ovsloot$ are piecewise linear, we can choose a possibly smaller closed cone with non-empty interior \[ \scr C'\subset\scr C, \] and $\sroot\in\vartheta$, $\bar{\bb}\in\ov\vartheta$ such that for all $v\in\scr C'$ one has \[ \slroot\times\ovsloot(v):=\left(\slroot(v),\ovsloot(v)\right)=\left(\sroot(v),\bar{\bb}(v)\right). \] Since $\sroot$ and $\bb$ are distinct simple roots the map $ (\sroot,\bb):\a\to\R^2$ is surjective. By~\cite[Proposition~5.1]{limite} there exists a sub-semigroup $\grupo'<\grupo$ such that $\scr r(\grupo')$ is a Zariski-dense Schottky semi-group with $\calL_{\scr r(\grupo')}=\scr C'$. In particular, for all $\g\in\grupo'$ one has \[ \slroot\times\ovsloot(\lambda(\scr r\g))=\left(\sroot(\lambda(\scr r\g)),\bar{\bb}(\lambda(\scr r\g))\right). \] By Benoist's Theorem~\ref{densidad}, stating that the group generated by the Jordan projections $\lambda(\scr r\g)$, for $\g\in\grupo'$, is dense in $\a$, we conclude that the group spanned by \[ \left\{\left(\left(\sroot(\lambda(\scr r\g)),\;\bar{\bb}(\lambda(\scr r\g))\right)\right):\g\in\grupo'\right\} \] is dense in $\R^2$, giving in turn the desired conclusion. \end{proof} \begin{proof}[Proof of Proposition~\ref{nonA}] Denote by $\sfG$ and $\bar{\sfG}$ the Zariski closures of $\rho(\grupo)$ and $\ovrho(\grupo)$ respectively. Both $\sfG$ and $\bar{\sfG}$ are semi-simple, have finite center, and don't have compact factors: if $\{\K,\bar{\K}\}\subset\{\R,\C\}$ then this is the content of Lemma~\ref{semisimple}, if either $\K$ and/or $\bar{\K}$ equals $\H$ then this is an assumption. We let $\iota:\grupo\to\sfG$ and $\ov\iota:\grupo\to\bar{\sfG}$ be the respective inclusions. If we let $\phi:\sfG\to\SL(d,\K)$ and $\ov\phi:\bar{\sfG}\to\SL(\ovd,\bar{\K})$ be the associated real representations, so that $\rho=\phi\circ\iota$ and $\ovrho=\ov\phi\circ\ov\iota$, we have from Section~\ref{representaciones} two subsets of simple roots $\t:=\t_\phi$ and $\ov\t:=\t_{\ov\phi}$ such that for all $a\in\a_\sfG^+$ and $b\in\a_{\bar{\sfG}}^+$ one has \begin{equation}\label{piecewise} \begin{aligned} \slroot(a)&:=\slroot_1(\phi(a)) =\min\bigl\{\sroot(a):\sroot\in\t\bigr\}\\ \ov\tau(b)&:=\ovsloot_1(\ov\phi (b)) =\min\bigl\{\ov\sroot(b):\ov\sroot\in\ov\t\bigr\}. \end{aligned} \end{equation} In particular, for every $\g\in\grupo$ one has $\slroot_1(\lambda(\g))=\slroot(\lambda_\sfG(\iota\g))$, and similarly for $\ovrho$. Since $\phi$ and $\ov\phi$ are faithful, $\t$ and $\ov\t$ contain at least one root of each factor of, respectively, $\sfG$ and $\bar{\sfG}$. If $\vartheta\subset\t$ then we let \[ \slroot^{\vartheta}(v)=\min_{\sigma\in\vartheta}\sigma(v),\ v\in\a_{\sfG}. \] If $\sf H$ is a non-trivial product of simple factors of $\sfG$ then we let $\iota_\sf H:\grupo\to\sf H$ be the composition of $\iota$ with the projection of $\sfG$ onto $\sf H$. By Zariski-density of $\iota(\grupo)$, each representation $\iota_\sf H$ has Zariski-dense image (though unlikely to be discrete). We also~let \[ \t_{\sf H} =\t\cap\simple_{\sf H}. \] Each $\t_\sf H$ is non-empty. We analogously define $\ov\iota_{\bar{\sf H}}$, $\t^{\bar{\sf H}}$ and $\ovsloot^{\bar{\sf H}}$. We now let $\sf L$ be the largest product of simple factors, simultaneously of $\sfG$ and $\bar{\sfG}$, so that $\iota_\sf L$ is conjugated (up to finite index) to $\ov\iota_\sf L$. Let $\sf H$ and $\bar{\sf H}$ be the remaining factors of $\sfG$ and $\bar{\sfG}$ respectively, i.e. \[ \sfG=\sf L\times \sf H\quad\text{and}\quad\bar{\sfG}=\sf L\times\bar{\sf H}, \] and moreover, by definition of $\sf L$, the representation $\scr r:\grupo\to \sf L\times\bar{\sf H}\times\sf H$ \begin{equation}\label{repsZ1} \scr r:g\mapsto \bigl(\iota_{\sf L}(g),\;\ov\iota_{\bar{\sf H}}(g),\;\iota_\sf H(g)\bigr) \end{equation} has Zariski-dense image, see for example~\cite[Corollary~11.6]{pressure}. %Bridgeman--Canary--Labourie-S. We remark that we are not assuming that any of $\sf L$, $\bar{\sf H}$ or $\sf H$ is non-trivial (they can't, of course, be all trivial). If $(u,v,w)\in\a_{\sf L}\times\a_{\bar{\sf H}}\times\a_{\sf H}$ we naturally think of $(u,v)$ as an element of $\a_{\bar{\sfG}}$ and of $(u,w)$ as an element of $\a_{\sfG}$. We now write \begin{align*} \Theta &=\t_\sf L\cap\ov\t_\sf L,\\ \Theta_{\sf L} &=\t_\sf L\setminus\Theta,\\ \ov\Theta_{\sf L} & =\ov\t_{\sf L}\setminus\Theta. \end{align*} One has, for all $(u,v,w)\in\a_{\sf L}\times\a_{\bar{\sf H}}\times\a_{\sf H}$ that \begin{equation}\label{tauL} \begin{aligned} \slroot(u,w) &=\min\left\{\slroot^{\Theta_\sf L}(u),\slroot^{\Theta}(u),\slroot^{\t_\sf H}(w)\right\}\\ \ovsloot(u,v) &=\min\left\{\slroot^{\ov\Theta_\sf L}(u),\slroot^{\Theta}(u),\ovsloot^{\t_{\bar{\sf H}}}(v)\right\}. \end{aligned} \end{equation} By assumption, there exists $g\in\grupo$ such that $\rho(g)$ and $\ovrho(g)$ are proximal and $\slroot(\lambda_{\sfG}(\iota g))\neq\ovsloot(\lambda_{\bar{\sfG}}(\ov\iota g))$. Assume, without loss of generality, that \begin{equation}\label{eqA} \slroot(\lambda_{\sfG}(\iota g))<\ovsloot\left(\lambda_{\bar{\sfG}}(\ov\iota g)\right). \end{equation} By means of Eqs.~\eqref{tauL} we see that in this situation one has \[ \slroot^{\Theta_{\sf L}\cup\t_\sf H}(\lambda_\sfG(\iota g))=\slroot(\lambda_\sfG(\iota g))<\ovsloot(\lambda_{\bar{\sfG}}(\ov\iota g)), \] in particular the union $\Theta_\sf L\cup\t_\sf H$ must be non-empty. Moreover, this strict inequality yields the existence of a small closed cone with non-empty interior $\scr C_0\subset\calL_\rho\subset\a_{\sfG}^+$ about $\R_+\lambda_\sfG(\rho g) $ such that \begin{equation}\label{tauH=tau1} \slroot^{{\Theta_{\sf L}\cup\t_\sf H}}(a)=\slroot_1(a)\,\forall a\in\scr C_0. \end{equation} Consider now the representation $\scr r:\grupo\to\sf L\times\bar{\sf H}\times\sf H$ from~\eqref{repsZ1} and a closed cone with non-empty interior $\scr C \subset\calL_{\scr r(\grupo)}\subset\a_{\sf L}^+\times\a_{\bar{\sf H}}^+\times\a_{\sf H}^+$ whose natural projection onto $\a_\sfG^+=\a_{\sf L}^+\times\a_{\sf H}^+$ is $\scr C_0$. Lemma~\ref{semi} applied to the group $G=\sf L\times\bar{\sf H}\times\sf H$, the representation $\scr r$, the disjoint non-empty subsets $\vartheta=\Theta_\sf L\cup\t_\sf H$ and $\ov\vartheta=\ov\t_\sf L\cup\t_{\bar{\sf H}}$ and the cone $\scr C$, provides the desired conclusion. \end{proof} We conclude with the following corollary that we don't need but is of independent interest. \begin{coro}\label{corraiz} Let $\rho:\G\to\SL(d,\K)$ and $\ovrho:\G\to\SL(\ovd,\bar{\K})$ be $\R$-irreducible and $\{\slroot_1,\slroot_2\}$-Anosov and $\{\ovsloot_1,\ovsloot_2\}$-Anosov respectively. If $\K=\H$ assume moreover the Zariski closure of $\rho(\G)$ does not contain compact factors, and analogously for $\ovrho$. If $\rho$ and $\ovrho$ are not gap-isospectral then \[ \II_{\ovsloot_1}(\slroot_1)>{\hJ{ \ovsloot_1}}/{\hJ{ \slroot_1}}. \] \end{coro} \begin{proof} Since both representations are projective-Anosov every element has proximal image in both representations. Pro\-po\-si\-tion~\ref{nonA} implies then that, since they are not gap-isospectral, the group spanned by the pairs $\{(\slroot_1(\lambda(\g)),\ovsloot_1(\lambda(\ov\g))):\g\in\G\}$ is dense. Since $\slroot_1=2\peso_1-\peso_2$, one has $\slroot_1\in\a_{\{\slroot_1,\slroot_2\}}$ and thus we can compose the refraction cocycle of $\rho$ with $\tau_1$ to obtain a real valued cocycle with periods $\tau_1\circ\lambda$. The same holds for $\ovrho$ and so Proposition~\ref{ineq} applied to this pair of cocycles yields desired strict inequality. \end{proof} \subsection{Proof of Theorem~\ref{tutti.LCdiff}}\label{ProofA} Theorem~\ref{tutti.LCdiff} follows from Proposition~\ref{nonA} giving the desired non-arithmeticity of periods, Lemma~\ref{generalcase} identifying the set $ \Ext_{\rho,\ovrho}^\hol$ with the set of $\hol$-conical points of $(\rho,\ovrho)$ and Theorem~\ref{Hffconical} computing the Hausdorff dimension of the latter when the periods are non-arithmetic. The last equality is a direct consequence of Proposition~\ref{p.uppergeneral}.\qed \section{Theorem~\ref{t.Zcl}: Zariski closures of real-hyperconvex surface-group representations}\label{s.6} In this section we prove Theorem~\ref{t.Zcl} giving a preliminary classification of Zariski closures of irreducible real $(1,1,2)$-hyperconvex representations of surface groups. For most of the section we work with a pair of $(1,1,2)$-hyperconvex representations and eventually reduce the proof of Theorem~\ref{t.Zcl} to a situation like this; we will crucially use Theorem~\ref{t.C1}. \subsection{When \texorpdfstring{$\Xi$}{Xi} has oblique derivative}\label{s.6.1} We prove here a result of independent interest, albeit possibly known to experts. This subsection only requires Section~\ref{cont} and Section~\ref{s.Anosov} and will be needed not only for Theorem~\ref{t.Zcl} but also for Theorems~\ref{tutti} and~\ref{thm.tuttiC}. Either $\G$ has boundary homeomorphic to a circle, or it is a Kleinian group. In the first case we let \[ \rho,\ovrho:\G\to\Diff^{1+\nu}(\circle) \] be H\"older conjugated to action of $\G$ on its boundary; if instead $\G<\PSL(2,\C)$ is a~Kleinian group we let $\rho,\ovrho:\G\to\PSL(2,\C)$ be two convex co-compact representations that lie in the same connected component of the subset of the character variety $\fkX(\G,\PSL(2,\C))$ consisting of convex co-compact representations. We let $X$ be either the circle or $\bord\H^3$. To simplify notation we will denote the action of $\g\in\G$ on $X$ via $\rho$ by $\g$, the action via $\ovrho$ by $\ov{\!\g}$, and the limit sets of $\rho$ and $\ovrho$ by $\bord\G,\bar{\bord\G}\subset X$ respectively. In both situations there exists a H\"older-continuous map \[ \Xi:X\to X \] conjugating $\rho$ and $\ovrho$. Indeed while in the surface case this holds by definition, in the Kleinian case this is a theorem by Marden~\cite{Marden}, see also Anderson's survey~\cite[p.~32]{SurveyAnderson}: the equivariant limit map $\Xi:\bord\G\to\bord\G$ conjugating the actions $\rho$ and $\ovrho$ on their respective limit sets extends to a $\G$-equivariant, H\"older continuous homeomorphism of the whole Riemann sphere $\bord\H^3$. We study differentiability points of $\Xi$ with oblique derivative. We let $d$ be either a visual distance on $X$ (in the complex case) or a distance inducing the chosen $\class^1$ structure on the circle $\circle$. \begin{defi}\label{assuA} An action $\rho$ admits a \emph{Lipschitz-compatible cover} if there exists a finite open cover $\cal B$ of $X$ and a map $\G\to\cal B$, $\gamma\mapsto\cal B_\infty(\g)$ such that: \begin{enumerate}\romanenumi \item\label{defi7.1.1} for any $a,b\in \G$ so that $|ab|=|a|+|b|$ one has \begin{enumerate} \item\label{defi7.1.1.a} $b\cal B_{\infty}(ab)\subset \cal B_\infty(a)$, \item\label{defi7.1.1.b} $\cal B_{\infty}(ab)\subset\cal B_{\infty}(b)$; \end{enumerate} \item\label{defi7.1.2} there exist $\lambda>0$, $C$ and $L\in\N$ such that if $|\g|\geq L$ and $x,y\in\cal B_\infty(\g)$ then \[ d(\g x,\g y)\leq Ce^{-|\g|\lambda}d(x,y); \] \item\label{defi7.1.3} there exist constants $r_1,r_2$ and a function $\slroot:\G\to\RR$ with $\slroot(\g)\geq\lambda|\g|$ such that for every $\g\in\G$ and every $x\in\cone_\infty(\g)$, \[ B\left(x,r_1e^{-\slroot(\g)}\right)\subset\g\cal B_\infty(\g)\subset B\left(x, r_2e^{-\slroot(\g)}\right). \] \end{enumerate} \end{defi} The goal of the subsection is to prove the following result, similar arguments can be found in Guizhen~\cite{cui} in the context of conjugacies of expanding circle maps. \begin{prop}\label{c.indepPer} Let $\rho,\ovrho$ be as above and assume both admit a Lipschitz compatible cover. If there exists $p\in\bord\G$ such that $\Xi$ has a finite non-vanishing derivative (complex derivative in the Kleinian case) at $p$ then $\Xi|\bord\G$ is bi-Lipschitz. \end{prop} We work under the assumptions of Proposition~\ref{c.indepPer} and begin its proof with the following lemma. For $\g\in\G$ we denote its derivative at $x\in X$ by $\g'(x)\in\K$ defined, according our two situations, by: \begin{itemize} \item $X=\bbS^1$: the derivative $\tilde\g'(\tilde x)$ of a lift of $\g$ to the universal cover $\R$ of $\bbS^1$, and a lift $\tilde x\in\R$ of $x$, the number $\tilde \g'(\tilde x)$ is independent of these choices; \item $X=\bord\H^3$: we fix an arbitrary point $\infty\notin\bord\G$, identify $X-\{\infty\}$ with $\C$ via the stereographic projection and let $\g'(x)$ be the standard complex derivative. \end{itemize} \begin{lemm}\label{controlC} Let $\rho:\G\to\Diff^{1+\nu}(X)$ admit a Lipschitz compatible cover. There exists a constant $\kappa>0$ and $N\in\N$ such that for all $\g\in\G$ with $|\g|\geq N$ and $x,y\in\cal B_\infty(\g)$ one has \[ \bigl\lvert\log\left|\g'(x)\right|-\log\left|\g'(y)\right|\bigr|\leq \kappa d(x,y)^\nu. \] \end{lemm} \begin{proof} We consider $L$ from Definition~\ref{assuA}, so that for every $\eta\in\G$ with $|\eta|\geq L$ and $x,y\in\cal B_\infty(\eta)$ one has \begin{equation}\label{lip} d(\eta x,\eta y)\leq Ce^{-|\eta|\lambda}d(x,y). \end{equation} Since the action is $\class^{1+\nu}$ we can find a positive $K$ such that for every $\beta$ with $|\beta|\leq L$ and $u,w\in X$ one has \begin{equation}\label{derivativeGen} \bigl\lvert\log\left|\beta'(u)\right|-\log\left|\beta'(w)\right|\bigr|\leq Kd(u,w)^\nu. \end{equation} We let then $K'=\max\{K,KC^\nu\}$. We begin by showing, by induction on $k$, that if $|\g|=kL$ then for all $x,y\in\cal B_\infty(\g)$, one has \begin{equation}\label{induc} \bigl\lvert\log\left|\g'(x)\right|-\log\left|\g'(y)\right|\bigr|\leq K'\left(\sum_{i=0}^{k-1}e^{-\nu\lambda L i}\right)d(x,y)^\nu. \end{equation} Eq.~\eqref{derivativeGen} gives the base case, so assume that the result holds up to $k-1$. We write $\g=\beta\eta$ with $|\beta|=L$, $|\eta|=(k-1)L$. By Definition~\ref{assuA} (ib) we have \begin{equation}\label{conocont} \cal B_\infty(\g)\subseteq\cal B_\infty(\eta). \end{equation} Applying the chain rule gives that for every $u\in X$ one has \[ \log\left|\g'(u)\right|=\log\left|(\beta)'(\eta u)\right|+\log\left|(\eta)'(u)\right| \] and thus, when $x,y\in\cal B_\infty(\g)$, \begin{multline*} \bigl\lvert\log\left|\g'(x)\right|-\log\left|\g'(y)\right|\bigr\rvert\\ \begin{aligned} & \leq \bigl\lvert\log\left|\beta'(\eta x)\right|-\log\left|\beta'(\eta y)\right|\bigr\rvert+\bigl|\log\left|\eta'(x)\right| -\log\left|\eta'(y)\right|\bigr|\\ & \leq K d(\eta x,\eta y)^\nu + K'\left(\sum_{i=0}^{k-2}e^{-\nu\lambda L i}\right)d(x,y)^\nu\quad (\text{by~\eqref{derivativeGen} and induction})\\ & \leq KC^\nu e^{-|\eta|\nu\lambda}d(x,y)^\nu+ K'\left(\sum_{i=0}^{k-2}e^{-\nu\lambda L i}\right)d(x,y)^\nu\quad (\text{by~\eqref{conocont} and~\eqref{lip}}). \end{aligned} \end{multline*} This shows Eq.~\eqref{induc} which implies that for $\kappa_0=K'/(1-e^{-\nu\lambda L})$, every $\g\in\G$ whose word-length is an integer multiple of $L$, and $x,y\in\cal B_\infty(\g)$ one has \[ \bigl\lvert\log\left|\g'(x)\right|-\log\left|\g'(y)\right|\bigr|\leq \kappa_0 d(x,y)^\nu. \] To conclude the lemma we consider an arbitrary $\g$ with $|\g|=mL+t$ and $t< L$. We write $\g=\beta \eta$ with $|\beta|=mL$. By Definition~\ref{assuA}$\MK$\eqref{defi7.1.1.a} it holds \begin{equation}\label{e.6.6} \eta \cal B_\infty(\g)\subset\cal B_\infty(\beta). \end{equation} Applying the chain rule gives then \begin{align*} \bigl\lvert\log\left|\g'(x)\right|-\log\left|\g'(y)\right|\bigr|& \leq \bigl\lvert\log\left|\beta'(\eta x)\right|-\log\left|\beta'(\eta y)\right|\bigr| +\bigl\lvert\log\left|\eta'(x)\right| -\log\left|\eta'(y)\right|\bigr|\\ & \leq \kappa_0d(\eta x,\eta y)^\nu + Kd(x,y)^\nu \quad (\text{by~\eqref{derivativeGen} and~\eqref{e.6.6}})\\ & \leq \left(\kappa_0C^\nu e^{-mL\lambda}+K\right) d(x,y)^\nu \;\:(\text{by~\eqref{lip}}) \end{align*} so taking $\kappa=K+\kappa_0C^\nu e^{-L\lambda}$ we conclude the proof of Lemma~\ref{controlC}. \end{proof} \begin{proof}[Proof of Proposition~\ref{c.indepPer}] Let $p\in\bord\G$ be such that $\Xi$ has a finite non-vanishing derivative. Fix a geodesic ray $(\alpha_n)_{0}^\infty$ through the identity with $\alpha_{n}\to p$. By definition for all $n$ one has $p\in\alpha_n\cone_\infty(\alpha_n)$. Without loss of generality we may also assume that \[ p=0=\Xi(0) \] and we may write the derivative as the incremental limit \[ \Xi'(0)=\lim_{y\to 0}\frac{\Xi(y)}{y}\in\K-\{0\}. \] For each $n$ we let $s_n=r_1e^{-\slroot(\alpha_n)}$, so that by Definition~\ref{assuA}$\MK$\eqref{defi7.1.3}, \[ B(0,s_n)\subset\alpha_n\cal B_\infty (\alpha_n). \] \pagebreak{} We consider the scaling map \[ g_n:B(0,1)\to \alpha_n\cal B_\infty(\alpha_n) \] defined by $g_n(z)=s_n z$. Let $a_n$ be an arbitrary point at distance $s_n$ from $0$ and let $\tilde s_n=\Xi(a_n)$. Observe that since $\Xi$ is differentiable at zero, for $n$ big enough the image $\Xi(B(0,s_n))$ is coarsely a ball around zero of size comparable to that of $\ov\alpha_n\cone_\infty(\ov\alpha_n)$, and in particular we can assume, since the cover $\{\cal B_\infty(\ov{\!\g})\}$ is Lipschitz compatible (Definition~\ref{assuA}$\MK$\eqref{defi7.1.3}, that $\Xi(B(0,s_n))$ is contained in $\ov\alpha_n\cal B_\infty(\ov\alpha_n)$. Furthermore we deduce that there exist positive constants $d,D$ such that for every $n$ \[ d<\frac{\ov r_2e^{-\ovsloot(\ov\alpha_n)}}{|\tilde s_n|}0$. However, the function \[ (x,y)\mapsto \frac{d(f(x),f(y))}{d(x,y)} \] is continuous and non-vanishing on $\{(x,y):d(x,y)\geq\eps\}$. Since the latter space is compact we obtain the bi-Lipschitz property for points which are far apart, as desired. \end{proof} The following Lemma guarantees we can later apply the results of this section to the situation of our interest. \begin{lemm}\label{usar} \leavevmode \begin{itemize} \item Assume $\bord\G$ is homeomorphic to a circle and let $\rho:\G\to\SL(d,\R)$ be $(1,1,2)$-hyperconvex. Then the induced action of $\rho(\G)$ on the $\class^{1+\nu}$ circle $\xi(\bord\G)$ admits a Lipschitz compatible covering. \item If $\G$ is a convex co-compact Kleinian group then the action of $\G$ on $\bord_\infty\H^3$ admits a Lipschitz compatible cover. \end{itemize} \end{lemm} \begin{proof} Recall from Section~\ref{cont} that we have fixed a word metric on $\G$ and we denote by $\cone_\infty(\g)\subset\partial\G\subset X$ the set of endpoints of geodesic rays contained in the cone type $\cone(\g)$. Let $\delta_\rho$ be the fundamental constant of $\rho$ from Definition~\ref{FConstant}, and let $\cal B_{\infty}(\g)=X_\infty(\gamma)$ be the $\delta_\rho/2$-neighbourhood of $\cone_\infty(\g)$ inside $X$. This is the thickened cone type at infinity considered in~\cite[Section~5]{PSW1} (see also the proof of Proposition~\ref{conetypesBalls}). It is a proper subset of $X$ by Corollary~\ref{c.fundamentalConstant}. The cover $\cal B$ is finite since there are only finitely many cone types~\cite[p.~455]{BH}. Definition~\ref{assuA}$\MK$\eqref{defi7.1.1} holds since the same property holds for $\cone_\infty(\g)$, property~\eqref{defi7.1.2} is a~consequence of Proposition~\ref{Lipschitz-compatibleB}. Finally, Property~\eqref{defi7.1.3} was proven in~\cite[Corollary~5.10]{PSW1} choosing $\slroot(\g):=\slroot_1(\cartan\rho(\g))$ (see also the proof of Proposition~\ref{conetypesBalls}). Observe that in the real case by considering $X=\circle$ we are implicitly considering only the intersection with the limit set, while in the Kleinian group case it is not necessary to intersect with the limit set since the $\G$-action on the whole $X$ is conformal. \end{proof} We now establish the following corollary that will be used in the sequel. \begin{coro}\label{diffpointreal} Assume $\bord\G$ is homeomorphic to a circle. Let $\rho:\piS\to\PGL(d,\R)$ and $\ovrho:\piS\to\PGL(\ovd,\R)$ be $(1,1,2)$-hyperconvex, consider the map between $\class^{1+\nu}$ circles \[ \Xi=\ovxi\circ\xi^{-1}:\xi(\bord\piS)\to\ovxi(\bord\piS). \] If $\Xi$ has a differentiability point with finite non-vanishing derivative then $\rho$ and $\ovrho$ are gap-isospectral. \end{coro} \begin{proof} By Lemma~\ref{usar} we can apply Proposition~\ref{c.indepPer} to obtain that $\Xi$ is bi-Lipschitz. The following standard lemma from linear algebra (see for example~\cite{convexes1} and~\cite[Lemma~3.4]{exponential}) gives the period computation completing the proof. \end{proof} \begin{lemm}\label{lambda2} Let $g\in\PGL(d,\R)$ be proximal with attracting point $g_+\in\P(\R^d)$ and repelling hyperplane $g_-\in\P((\R^d)^*)$. Let $V_{\lambda_2(g)}$ be the sum of the characteristic spaces of $g$ whose associated eigenvalue is of modulus $\exp\lambda_2(g)$, Then for every $v\notin\P(g_-)$, with non-zero component in $V_{\lambda_2(g)}$, one has \[ \lim_{n\to\infty}\frac{\log d_\P(g^n(v),g_+)}n= -\slroot_1(\lambda(g)). \] \end{lemm} \subsection{Limit curves in non-maximal flags} We proceed with another intermediate step for the proof of Theorem~\ref{t.Zcl} describing differentiability points of boundary maps in partial flag manifolds $\calF_{\{\sroot,\bb\}}$ for $\{\sroot,\bb\}$-Anosov representations. This step follows from the combination of Theorem~\ref{t.C1} and Corollary~\ref{diffpointreal}. Let $\sfG$ be real-algebraic and semi-simple. Let $\{\sroot,\bb\}\subset\simple$ be two distinct simple roots. The partial flag space $\calF_{\{\sroot,\bb\}}$ carries two transverse foliations that are the level sets of the natural projections $\calF_{\{\sroot,\bb\}}\to\calF_{\{\sroot\}}$ and $\calF_{\{\sroot,\bb\}}\to\calF_{\{\bb\}}$. We will refer to these as the \emph{canonical foliations} of $\calF_{\{\sroot,\bb\}}$. \begin{coro}\label{nonC1} Let $\sfG$ be real-algebraic and semi-simple and let $\{\sroot,\bb\}\subset\simple$ distinct. Let $\rho:\piS\to\sfG$ be Zariski-dense and $\{\sroot,\bb\}$-Anosov. If both curves $\xi^\sroot(\bord\piS)$ and $\xi^{\bb}(\bord\piS)$ are $\class^{1+\alpha}$ then every differentiability point of $\xi^{\{\sroot,\bb\}}(\bord\piS)$ is tangent to one of the canonical foliations of $\calF_{\{\sroot,\bb\}}$. \end{coro} \begin{proof} By Benoist's Theorem~\ref{densidad} the limit cone of $\rho$ has non-empty interior, in particular there exists $\g\in\piS$ such that \begin{equation}\label{isospe} \sroot(\lambda(\g))\neq\bb(\lambda(\g)). \end{equation} Consider the Tits representations $\Fund_\sroot$ and $\Fund_\bb$ associated to $\sroot$ and $\bb$. Since $\rho(\piS)$ is Zariski-dense, both representation $\Fund_\sroot\rho$ and $\Fund_\bb\rho$ are irreducible and since $\rho$ is $\{\sroot, \bb\}$-Anosov both representation $\Fund_\sroot\rho$ and $\Fund_\bb\rho$ are projective Anosov. Recall that by definition of $\Fund_\sroot$, for every $g\in\sfG$ one has \[ \slroot_1(\lambda(\Fund_\sroot(g)))=\sroot(\lambda(g)), \] so by Eq.~\eqref{isospe} the representations $\Fund_\sroot\rho$ and $\Fund_\bb\rho$ are not gap-isospectral. Since the maps $\zeta_\sroot$ and $\zeta_\bb$ are analytic, both projective curves $\zeta_\sroot\xi^{\sroot}(\bord\piS)$ and $\zeta_\bb\xi^{\bb}(\bord\piS)$ are $\class^{1+\alpha}$ and thus by Zhang--Zimmer's Theorem~\ref{t.C1} the representations $\Fund_\sroot\rho$ and $\Fund_\bb\rho$ are $(1,1,2)$-hyperconvex. The natural embedding $\calF_{\{\sroot,\bb\}}\to\P(V_\sroot)\times\P(V_\bb)$ sends $\xi^{\{\sroot,\bb\}}$ to the graph of the map $\Xi$ from Corollary~\ref{diffpointreal} and thus the corollary implies the result. \end{proof} \subsection{Proof of Theorem~\ref{t.Zcl}}\label{s.pB} \label{s.Zd} The goal of the section is to prove Theorem~\ref{t.Zcl}, stating that the Zariski closure $\sfG$ of the image of an irreducible $(1,1,2)$-hyperconvex representation $\rho:\piS\to\PGL(d,\R)$ is simple and the highest weight of the induced representation $\Fund:\sfG\to\PGL(d,\R)$ is a multiple of a fundamental weight associated to a root whose root-space is one-dimensional. It is known that an irreducible subgroup $\sfG<\PGL(d,\R)$ containing a proximal element is semi-simple without compact factors (see~\cite[Lemma~8.6]{entropia} for an explicit argument following a suggestion by Quint). We consider the induced representation $\rho_0:\G\to\sfG$ and denote by $\Fund:\sfG\to\PGL(d,\R)$ the linear representation so that $\rho=\Fund\rho_0$. Let $\chi=\chi_\Fund\in\a^*$ be the highest weight of $\Fund$. As in Definition~\ref{trep} we consider \[ \t=\t_\Fund=\left\{\sroot\in\Delta\,:\, \chi-\sroot \text{ is a weight of }\Fund \right\} =\left\{\sroot\in\simple\,:\,\<\chi,\sroot\>\neq0\right\}. \] It is enough to show that $\t$ is reduced to a single root $\{\sroot_0\};$ indeed, if this is the case, upon writing $\chi$ in the basis of fundamental weights $\{\peso_\sroot:\sroot\in\simple\}$ (recall their defining Eq.~\eqref{pesoFund}) one has \[ \chi=\sum_{\sroot\in\simple}\<\chi,\sroot\>\peso_\sroot=\<\chi,\sroot_0\>\peso_{\sroot_0}, \] Moreover this gives: \begin{itemize} \item $\sfG$ is simple by Lemma~\ref{simple}; \item the weights on the first level consist solely on $\chi-\sroot_0$ and its associated weight space is $\phi(\ge_{-\sroot_0})V_{\chi_\Fund}$. By definition, if $a\in\inte\a^+$ then $\Fund(\exp(a))$ is proximal with associated eigenline $V^\chi$ and the generalized eigenspace associated with the second eigenvalue, in modulus, is $\phi(\ge_{-\sroot_0})V_{\chi_\Fund}$. Since $\rho(\G)$ is $\{\slroot_1,\slroot_2\}$-Anosov, we know that for all $\g\in\G$, $\rho(\g)$ and $\wedge^2\rho(\g)$ are proximal, thus the generalized eigenspace associated to the second eigenvalue of $\rho(\g)$ is $1$-dimensional. Combining both facts we conclude that $\phi(\ge_{-\sroot_0})V_{\chi_\Fund}$ is one-dimensional, but by Lemma~\ref{not} no element of $\ge_{-\sroot_0}$ acts trivially on $V_{\chi_\Fund}$ so $\ge_{-\sroot_0}$ is $1$-dimensional, as desired. \end{itemize} We proceed now to show that in the present situation $\t$ consists of only one element. By definition of $\t$ one has, for every $g\in\sfG$, that \[ \slroot_1\bigl(\lambda(\Fund(g))\bigr)=\min_{\sroot\in\theta}\bigl\{\sroot(\lambda_\sfG(g))\bigr\}. \] Consequently, the limit cone $\calL_{\rho_0}\subset\a^+_\sfG$ does not intersect the walls of elements in $\t$ and, since $\rho_0:\G\to\sfG$ is a quasi-isometry, Remark~\ref{wallsAnosov} implies that the representation $\rho_0$ is $\t$-Anosov. Recall from Eq.~\eqref{maps} that we have a $\Fund$-equivariant analytic embedding $\zeta_\t:\sfG/\sf P_{\theta}\to \P(\R^{d})$. One has moreover that $\xi^1_\rho=\zeta_\t\circ\xi^\t_{\rho_0}$. Since $\xi^2_\rho$ is H\"older-continuous we conclude that the boundary map $\xi^\theta$ has $\class^{1+\alpha}$-image. Composing with the projections $\calF_\t\to\calF_{\t'}$ one sees that, for any $\t'\subset\t$ the curve $\xi_{\rho_0}^{\t'}(\bord\G)$ is a also a $\class^{1+\alpha}$ circle. Assume now there exists two distinct roots $\sroot,\bb$ in $\t$ and consider the subset $\t'=\{\sroot,\bb\}$. By the previous paragraph the curve $\xi^{\t'}(\bord\G)$ is $\class^{1+\alpha}$. Corollary~\ref{nonC1} gives then that $\xi^{\t'}(\bord\G)$ is necessarily contained in one of the leaves of the canonical foliations of $\calF_{\t'}$, thus giving that one of the maps $\xi^{\sroot}$ or $\xi^\bb$ is constant, achieving a~contradiction. \qed \section{Non-differentiability and \texorpdfstring{$1$}{1}-conicality: the proof of Theorem~\ref{tutti}}\label{s.Hffproof} \subsection{Non-differentiability and \texorpdfstring{$1$}{1}-conicality} By means of Section~\ref{s.6.1} we can improve Lemma~\ref{generalcase} when we deal with a pair of real hyperconvex representations of surface groups, this is the missing ingredient for Theorem~\ref{tutti}: \begin{coro}\label{nondiffbconicalR} Assume $\bord\G$ is homeomorphic to a circle. Let $\rho,\ovrho$ two $(1,1,2)$-hyperconvex representations over $\R$ of $\G$ that are not gap-isospectral. Then, the set of non-differentiability points of $\Xi$ coincides with the set of $1$-conical points. \end{coro} \begin{proof} We choose a $\class^1$ identification of the $\class^1$ torus $\xi(\bord\G)\times\ovxi(\bord\G)\subset\P(\R^d)\times \P(\R^{\ovd})$ with the quotient of the square $[-1,1]\times[-1,1]$ preserving the product structure, and such that the point $(x,\Xi(x))$ corresponds to $(0,0)$. In these coordinates the graph of $\Xi$ is a monotone curve $[-1,1]\to[-1,1]$ passing through the origin. Since the chosen identification is $\class^1$, it is in particular $K$-bi-Lipschitz for some $K$, so we can write (coarsely in a small neighbourhood of $x$) $d(\xi(y),\xi(x))=|y|$ and $d(\ovxi(y),\ovxi(x))=|\Xi(y)|$. From Lemma~\ref{generalcase} we know that $x$ is $1$-conical if and only if either $\lim_{y\to x}\frac{|\Xi(y)|}{|y|}$ exists and is far from $0$ and $\infty$, either it does not exist. The proposition is settled if we show that the first situation cannot happen, so let's assume it does. However, since $\Xi$ is monotone we can remove the $|\,|$ and we get that $x$ is a differentiability point of $\Xi$ with oblique derivative. Corollary~\ref{diffpointreal} implies then that for all $\g\in\G$ one has $\slroot_1(\lambda(\g))=\ovsloot_1(\lambda(\ov\g))$, contradicting our assumption. \end{proof} \subsection{Proof of Theorem~\ref{tutti} and an analogous for Kleinian groups} We begin with the proof of Theorem~\ref{tutti} by recalling the following result from \cite{BeyP2}. \begin{coro}[{\cite{BeyP2}}]\label{l.weaklyirred} Assume $\bord\G$ is homeomorphic to a circle and let $\rho:\G\to \PGL(d,\R)$ be $(1,1,2)$-hyperconvex. Then there exists an irreducible $(1,1,2)$-hyperconvex representation $\rho_0:\G\to \PGL(m,\R)$ such that, for every $\g\in\G$ one~has \[ \slroot_1(\lambda(\g))=\slroot_1(\lambda(\rho_0\g)). \] \end{coro} We now prove Theorem~\ref{tutti}. Since there exists $\g\in\G$ with $\slroot_1(\lambda(\g))\neq\ov{\slroot}_1(\lambda(\ov\g))$, Corollary~\ref{l.weaklyirred} allows us to apply Proposition~\ref{nonA} to obtain the density assumption in Theorem~\ref{Hffconical}, so one has \[ \Hff\Xi(\{1\text{-conical points}\})=\hC{\max\{\slroot,\ov{\slroot}\}}. \] Corollary~\ref{nondiffbconicalR} states that the set of $1$-conical points coincides with the set of non-differentiability points of $\Xi$. The inequality $\hC{\infty,1} <1$ then follows from Theorem~\ref{Hffconical}. This completes the proof of Theorem~\ref{tutti}. \subsection{Proof of Corollary~\ref{cor.B}} We conclude the paper proving Corollary~\ref{cor.B}. Recall from Section~\ref{representaciones} that for every simple root $\sroot$ of $\sfG$ we chose a Tits representation $\Phi_\sroot:\sfG\to\PSL(V_\sroot)$. \begin{coro}\label{c.Hk} Assume $\bord\G$ is homeomorphic to a circle and let $\sfG$ be a simple Lie group. Let $\rho:\G\to\sfG$ have Zariski-dense image. If for $\sroot,\bb\in\simple$ the representations $\Phi_\sroot\circ\rho$ and $\Phi_\bb\circ \rho$ are $(1,1,2)$-hyperconvex, then \begin{enumerate}\romanenumi \item\label{coro8.3.1} the image of the limit curve $\xi^{\{\sroot,\bb\}}:\bord\G\to\calF_{\{\sroot,\bb\}}$ is Lipschitz and the Hausdorff dimension of the points where it is non-differentiable is $\hC{\max\{\sroot,\bb\}}$. \item\label{coro8.3.2} If the opposition involution $\ii$ on $\ge$ is non-trivial and $\bb=\ii\sroot$ then \[ \hC{\max\{\sroot,\bb\}}=\hC{(\sroot+\bb)/2}. \] \end{enumerate} \end{coro} \begin{proofc} \begin{proof}[{\eqref{coro8.3.1}}] Since the map $\Phi_\sroot:\calF_{\sroot}\to\P(V_{\sroot})$ is analytic, and $\Phi_\sroot\circ\xi^\sroot(\bord\G)$ is a $\class^1$-submanifold (Theorem~\ref{t.C1}), $\xi^\sroot(\bord\G)$ is a $\class^1$ submanifold as well. The curve $\xi^{\{\sroot,\bb\}}:=\calF_{\{\sroot,\bb\}}\cap (\xi^{\sroot}(\bord\G)\times\xi^{\bb}(\bord\G))$ is the graph of the homeomorphism $\Xi$ and is thus a Lipschitz curve. The second claim is then a direct consequence of Theorem~\ref{tutti}.\let\qed\relax \end{proof} \begin{proof}[{\eqref{coro8.3.2}}] Assume the opposition involution $\ii$ of $\ge$ is non-trivial and that $\bb=\ii\sroot$. Using notation from Section~\ref{s.lower} with $\sroot=\slroot$ and $\bb=\ii\sroot=\ov{\slroot}$ we let $V^*=\spa\{\sroot,\bb\}$, $V=\a_\t/\Ann(V^*)$, $\Pi:\a_\t\to V$ the quotient projection, $\|\,\|_\infty=\max\{|\sroot|,|\bb|\}$, $\|\,\|^1$ its dual norm on $V^*$ and ${\varphi^\infty_1}\in\calQ_\cc$ the only form minimizing $\|\,\|^1$. Since $\ii\sroot=\bb$, the space $V^*$ is preserved by $\ii$ and the fact that $\lambda(g^{-1})=\ii\lambda(g)$ (for all $g\in \sfG$) implies that $\calQ_\cc$ is $\ii$-invariant. Moreover, the norm $\|\,\|^1$ is $\ii$-invariant and by definition of ${\varphi^\infty_1}$ one has $\ii{\varphi^\infty_1}={\varphi^\infty_1}$. However, $(\sroot+\bb)/2$ is also $\ii$-invariant and $\hC{(\sroot+\bb)/2}(\sroot+\bb)/2\in\calQ_\cc$ whence \[ {\varphi^\infty_1}=\hC{(\sroot+\bb)/2}(\sroot+\bb)/2. \] %\end{proof} We now consider a refinement of Corollary~\ref{quintgrowththeta}. Indeed, examining its proof one readily sees that the statement holds replacing $\a_\t$ with $V$ and $\cartan_\t$ with $\Pi\cartan_\t$, thus one has $\hC{\max\{\sroot,\bb\}}=\|{\varphi^\infty_1}\|^1$. \end{proof} \let\qed\relax \end{proofc} \subsection{The \texorpdfstring{$\PSL(2,\C)$}{PSL(2, C)}-case}\label{klein} If $\rho,\ovrho:\G\to\PSL(2,\C)$ are convex co-compact representations that are connected by convex co-compact representations, it was proven by Marden~\cite{Marden} that the natural map $\Xi:\Lambda_\rho\to\Lambda_{\ovrho}$ conjugating the respective actions extends to a H\"older homeomorphism $\Xi:\C\P^1\to\C\P^1$ that is $(\rho,\ovrho)$-equivariant. We consider in this case the complex derivative of such an extension $\Xi$ and say that $\Xi$ is $\C$-\emph{differentiable} at a given $x\in\Lambda_\rho$ if, conformally identifying $\bord\H^3-\{\text{point}\}$ to $\C$, the limit \[ \Xi'(x):=\lim_{y\to x} \frac{\Xi(x)-\Xi(y)}{x-y} \] exists or is infinite. We let now $\Ndiff_{\rho,\ovrho}$ be the set of points $x\in\Lambda_\rho$ where the extended conjugating map $\Xi$ is not $\C$-differentiable. The proof of the following works verbatim as in Corollary~\ref{nondiffbconicalR}. \begin{prop}\label{nondiffbconicalC} Let $\rho,\ovrho:\G\to\PSL(2,\C)$ be non-gap-isospectral and in the same connected component of \[ \bigl\{\varrho:\G\to\PSL(2,\C):\,\varrho\text{\ is\ convex\ co-compact}\bigr\}. \] Then, the set of non-$\C$-differentiability points of $\Xi$ coincides with the set of $1$-conical points. \end{prop} Density of the group generated by the pairs $\{(\lambda(\g),\lambda(\bar\g)):\g\in\G\}$ follows readily from Benoist~\cite{benoist2} (see Theorem~\ref{densidad}), from this point on the exact same proof of Theorem~\ref{tutti} gives the following. \begin{theo}\label{thm.tuttiC} Let $\rho,\ovrho:\G\to\PSL(2,\C)$ be non-gap-isospectral convex co-compact representations that are connected by convex co-compact representations. Assume without loss of generality that $\hJ{\ovsloot}\geq \hJ{\slroot}$. If $\II_{\slroot}(\ovsloot)>1$, then \[ \Hff\left(\Ndiff_{\rho,\ovrho}\right)=\hC{\infty,1}. \] \end{theo} \bibliography{20251124d_pozzetti} \end{document}