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 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.

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The now classical convergence in distribution theorem for well normalized sums of stationary martingale increments has been extended to multi-indexed martingale increments. In the present article we make progress in the identification of the limit law.

In dimension one, as soon as the stationary martingale increments form an ergodic process, the limit law is normal, and it is still the case for multi-indexed martingale increments when one of the processes defined by one coordinate of the multidimensional time is ergodic. In the general case, the limit may be non normal.

In the present paper we establish links between the dynamical properties of the $\mathbb{Z}^d$-measure preserving action associated to the stationary random field (like the positivity of the entropy of some factors) and the existence of a non normal limit law. The identification of a natural factor on which the $\mathbb{Z}^d$-action is of product type is a crucial step in this approach.

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We are interested in the scattering problem for the cubic 3D nonlinear defocusing Schrödinger equation with variable coefficients. Previous scattering results for such problems address only the cases with constant coefficients or assume strong variants of the non-trapping condition, stating that all the trajectories of the Hamiltonian flow associated with the operator are escaping to infinity. In contrast, we consider the most general setting, where strong trapping, such as stable closed geodesics, may occur, but we introduce a compactly supported damping term localized in the trapping region, to explore how damping can mitigate the effects of trapping.

In addition to the challenges posed by the trapped trajectories, notably the loss of smoothing and of scale-invariant Strichartz estimates, difficulties arise from the damping itself, particularly since the energy is not, a priori, bounded. For $H^{1+\epsilon }$ initial data (chosen because the local-in-time theory is a priori no better than for 3D unbounded manifolds, where local well-posedness of strong $H^1$ solutions is unavailable) we establish global existence and scattering in $H^{s}$ for any $0 \le s <1$ in positive times, the inability to reach $H^1$ being related to the loss of smoothing due to trapping.

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We show that there is a rank $1$ transformation that is mildly mixing but does not have minimal self-joinings, answering a question of Thouvenot.

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Many results of smooth hypoellipticity are available for scalar equations. Much remains to be done for systems and/or at different levels of regularity and in particular for $L^1$-hypoellipticity. In this article we provide some examples and counter-examples.

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We study the distribution of the angles between Oseledets subspaces and their log-integrability, focusing on dimension $2$. For random i.i.d. products of matrices, we construct examples of probability measures on $\operatorname{GL}_2(\mathbb{R})$ with finite first moment where the Oseledets angle is not log-integrable. We also show that for probability measures with finite second moment the angle is always log-integrable. We then consider general measurable $\operatorname{GL}_2(\mathbb{R})$-cocycles over an arbitrary ergodic automorphism of a non-atomic Lebesgue space, proving that no integrability condition on the matrix distribution ensures log-integrability of the angle. In fact, the joint distribution of the Oseledets spaces can be chosen arbitrarily. A similar flexibility result for bounded cocycles holds under an unavoidable technical restriction.

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In 2000, Margulis proved that any group of homeomorphisms of the circle either preserves a probability measure on the circle or contains a free subgroup on two generators, which is reminiscent of the Tits alternative for linear groups. In this article, we prove an analogous statement for groups of locally monotonic homeomorphisms of a compact subset of $\mathbb{R}$. The proof relies on dynamical properties of random walks on the group, which may be of independent interest.

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Suppose we have two finitely supported, admissible, probability measures on a hyperbolic group $\Gamma $. In this article we prove that the corresponding two Green metrics satisfy a counting central limit theorem when we order the elements of $\Gamma $ according to one of the metrics. Our results also apply to various other metrics including length functions associated to Anosov representations and to group actions on hyperbolic metric spaces.

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We introduce a multi-scale generalization of the notion of scalar product on real and complex vector spaces and study the corresponding Voronoi tilings. This framework allows to analyze limiting metric structures involving several levels of convergence. In particular, we describe metric degenerations of polarized tori and Hausdorff limits of Voronoi tilings.

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Given a probability measure $\mu $ on a finitely generated group $\Gamma $, the Green function $G(x,y|r)$ encodes many properties of the random walk associated with $\mu $. Endowing $\Gamma $ with a word distance, we denote by $H_r(n)$ the sum of the Green function $G(e,x|r)$ along the sphere of radius $n$. This quantity appears naturally when studying asymptotic properties of branching random walks driven by $\mu $ on $\Gamma $. Our main result exhibits a relatively hyperbolic group with convergent Poincaré series generated by $H_r(n)$.

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We introduce the notion of a flat extension of a connection $\theta $ on a principal bundle. Roughly speaking, $\theta $ admits a flat extension if it arises as the pull-back of a component of a Maurer–Cartan form. For trivial bundles over closed oriented $3$-manifolds, we relate the existence of certain flat extensions to the vanishing of the Chern–Simons invariant associated with $\theta $. As an application, we recover the obstruction of Chern–Simons for the existence of a conformal immersion of a Riemannian $3$-manifold into Euclidean $4$-space. In addition, we obtain corresponding statements for a Lorentzian $3$-manifold, as well as a global obstruction for the existence of an equiaffine immersion into $\mathbb{R}^4$ of a $3$-manifold that is equipped with a torsion-free connection preserving a volume form.

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In a recent paper, Aldous, Blanc and Curien asked which distributions can be expressed as the distance between two independent random variables on some separable measured metric space. We show that every nonnegative discrete distribution whose support contains $0$ arises in this way, as well as a class of compactly supported distributions with density.

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The systole of a contact form $\alpha $ is defined as the shortest period of closed Reeb orbits of $\alpha $. Given a non-trivial $\mathbb{S}^1$-principal bundle over $\mathbb{S}^2$ with total space $M$, we prove a sharp systolic inequality for the class of tight contact forms on $M$ invariant under the $\mathbb{S}^1$-action. This inequality exhibits a behavior which depends on the Euler class of the bundle in a subtle way. As applications, we prove a sharp systolic inequality for rotationally symmetric Finsler metrics on $\mathbb{S}^2$, a systolic inequality for the shortest contractible closed Reeb orbit, and a particular case of a conjecture by Viterbo.

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We study quasi-isometric representations of finitely generated non-abelian free groups into some higher rank semi-simple Lie groups which are not Anosov, nor approximated by Anosov. We show in some cases that these can be perturbed to be non-quasi-isometric, or to have some instability properties with respect to their action on the flag space.

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Caffarelli’s contraction theorem bounds the derivative of the optimal transport map between a log-convex measure and a strongly log-concave measure. We show that an analogous phenomenon holds on the level of the trace: the trace of the derivative of the optimal transport map between a log-subharmonic measure and a strongly log-concave measure is bounded. We show that this trace bound has a number of consequences pertaining to volume-contracting transport maps, majorization and its monotonicity along Wasserstein geodesics, growth estimates of log-subharmonic functions, the Wehrl conjecture for Glauber states, and two-dimensional Coulomb gases. We also discuss volume-contraction properties for the Kim–Milman transport map.

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