=triangle 45,x=1.4cm,y=1.4cm] \begin{scope} \clip (0,0) -- (1,.2) arc (11.46:90:1.02) -- (0,0); \shade[inner color=blue,outer color=white] (0,0) circle (1); \end{scope} \draw (-1,0) -- (1,0) node[anchor=south west] {{$\ \xrightarrow{\text{\Tiny shear}}$}}; \draw (0,-1) -- (0,1) node[anchor=south] {}; \draw[style=dashed] (-1,-1) -- (1,1) node[anchor=west] {$e^+$}; \draw[style=dashed] (-1,1) -- (1,-1) node[anchor=west] {$e^-$}; \draw[style=dotted] (-1,-.2) -- node[very near start,sloped,below]{$a=Kb$} (1,.2); \node at (.5,.7) {$0\le a\le Kb$}; \end{tikzpicture} \begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.4cm,y=1.4cm] \begin{scope} \clip (0,0) -- (-1,.2) arc (0:-11.46:1.02) -- (0,0); \shade[inner color=blue,outer color=white] (0,0) circle (1); \end{scope} \draw (-1,0) -- (1,0) node[anchor=south west] {$\ \xrightarrow{\substack{\text{\Tiny return}\\\text{\Tiny map}}}$}; \draw (0,-1) -- (0,1) node[anchor=south] {}; \draw[style=dashed] (-1,-1) -- (1,1) node[anchor=west] {$e^+$}; \draw[style=dashed] (-1,1) -- (1,-1) node[anchor=west] {$e^-$}; \draw[style=dotted] (-1,0.2) -- node[very near start,sloped,below]{$\scriptstyle \quad\; a\le\;-Kb\le0$} node[very near end,sloped,below]{$a=-Kb$} (.9,-.2); \end{tikzpicture} \begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.4cm,y=1.4cm] \begin{scope} \clip (0,0) -- (-1,-.2) arc (-168.54:-90:1.02) -- (0,0); \shade[inner color=blue,outer color=white] (0,0) circle (1); \end{scope} \draw (-1,0) -- (1,0) node[anchor=south west] {}; \draw (0,-1) -- (0,1) node[anchor=south] {}; \draw[style=dashed] (-1,-1) -- (1,1) node[anchor=west] {$e^+$}; \draw[style=dashed] (-1,1) -- (1,-1) node[anchor=west] {$e^-$}; \draw[style=dotted] (-1,-.2) -- node[very near end,sloped,above]{$a=Kb$} (1,.2); \node at (-.7,-.7) {$0\ge a\ge Kb$}; \end{tikzpicture} \caption{The cones in the proof of Proposition~\ref{PRPNotAnosovIfq<0}}\label{FIGPRPNotAnosovIfq<0} \end{figure} Note that one can perform a positive surgery on an Anosov flow (and therefore obtain another Anosov flow) then undo it by performing a negative surgery and obtain again an Anosov flow. This is compatible with the statement of Proposition~\ref{PRPNotAnosovIfq<0}, as $q$ and $\epsilon$ are fixed in the negative surgery (and thus Proposition~\ref{PRPNotAnosovIfq<0} does not apply). Returning to the case of positive $q$, we note from the preceding: \begin{prop}\label{PROPFoliationsOrientability} The stable and unstable foliations of $(M_S,\alpha_A)$ as described in Theorem~\ref{THMMain} are orientable. \end{prop} \begin{proof} The strong stable foliation is contained in the positive cone of $Q^-$ and the strong unstable foliation in the positive cone of $Q^+$, so the stable foliation is orientable if and only if the positive cone of $Q^-$ is orientable (an orientation of the positive cone is a choice of a connected component of this cone). The stable and unstable foliations of the unit tangent bundle over a hyperbolic surface are orientable. Additionally, $Q^-(\tfrac{\partial}{\partial s},\tfrac{\partial}{\partial s})=0$ and $F^*\tfrac{\partial}{\partial s}=\tfrac{\partial}{\partial s}$, so the surgery preserves the orientation of $Q^-$, and $Q^-$ is orientable. This implies that $Q^+$ is orientable. \end{proof} \subsection{Impact on entropy} The nature of the surgery map implies: \begin{prop}\label{PRPsameLyapunov} If $q\ge0$, then $\hmu(X_{HT})\ge\hmu(X)$. \end{prop} \begin{proof} By the Pesin entropy formula it suffices to show that the positive Lyapunov exponent of $X_{HT}$ is no less than that of $X$. Volume-preserving Anosov 3-flows are ergodic~\cite[Theorem~20.4.1]{KatokHasselblatt}, so the positive Lyapunov exponent, being a flow-invariant bounded measurable function, is a.e. constant. The earlier observation that for the geodesic flow on a hyperbolic surface the expanding vector is of the form $e^te^+$ means that the Lyapunov exponent of (the normalized) Liouville measure is~1. Therefore, we will show that the positive Lyapunov exponent of $X_{HT}$ is at least~1. To that end we verify that the differential of its time-1 map expands unstable vectors by at least a factor of $e$ with respect to a suitable norm. For the geodesic flow the Sasaki metric induces a natural norm, and this norm is what is called an \emph{adapted} or \emph{Lyapunov} norm: for unstable vectors, it grows by exactly $e^t$ under the flow, and on each tangent space it is a product norm. Our argument involves only vectors in unstable cones, so we pass to a norm $\|\cdot\|_+$ that is (uniformly) equivalent when restricted to such vectors: the norm of the unstable component. Geometrically, this means that at each point we project tangent vectors to $E^+$ along $E^-\oplus\R X$ and take the length of this unstable projection as the norm of the vector. Thus, $\|Dg^t(v)\|_+=e^t\|v\|_+$ for $t\ge0$. The proof of hyperbolicity of $X_{HT}$ shows that the cone field defined by the Lyapunov--Lorentz functions is well-defined on the surgered manifold and invariant under $X_{HT}$. Thus, this adapted norm for the geodesic flow defines a (bounded, though discontinuous) norm $\|\cdot\|_+$ on unstable vectors for the flow $\varphi^t$ defined by$X_{HT}$. We now show that $\|D\varphi^1(v)\|_+\ge e\|v\|_+$ for any $v$ in an unstable cone. This is clear (with equality) when the underlying orbit segment does not meet the surgery annulus because the action is that of the geodesic flow. If there is an encounter with the surgery annulus at time $t\in(0,1]$, then $v'\dfn D\varphi^t(v)$ satisfies $\|v'\|_+=e^t\|v\|_+$, and we will check that $v''\dfn DF(v')$ satisfies $\|v''\|_+\ge\|v'\|$, which implies that $\|D\varphi^1(v)\|_+=\|D\varphi^{1-t}(v'')\|_+=e^{1-t}\|v''\|_+\ge e^{1-t}\|v'\|_+=e^{1-t}e^t\|v\|_+=e\|v\|_+$, as required. That $\|DF(v')\|_+\ge\|v'\|$ follows from the same argument as hyperbolicity of $X_{HT}$ as suggested by Figure~\ref{FIGShearGrowsUnstable}, which superimposes the tangent spaces at some $x$ and $F(x)$ in the surgery annulus (using the identification from the canonical isometries between these tangent spaces). $DF$ is a positive shear, and in the $H$-$V$-frame in the figure the addition of a multiple of the projection of $\frac{\partial}{\partial s}$ (which is close to $H$) by a positive shear results in an increase in the projection to $E^+$, which is spanned by $e^+=V+H$.\qedhere \begin{figure}[ht] \begin{tikzpicture}[line cap=round,line join=round,>=stealth,x=2.1cm,y=2.1cm] \clip(-0.79,-0.4) rectangle (3.52,1.9); \draw[red,thick](-0.77,-0.77) -- (3.52,3.52); \draw [blue] (-1,1) -- (0,0)node[midway,sloped,above,black]{\footnotesize{$e^-=V-H$}}; \draw[blue,thick](-0.77,0.77) -- (3.52,-3.52); \draw (0,-0.79) -- (0,3.51); \draw [domain=-0.77:3.52] plot(\x,{(-0-0*\x)/1}); \draw [dash pattern=on 1pt off 1pt,color=blue,domain=-0.77:3.52] plot(\x,{(-3--1*\x)/-1}); \draw [dash pattern=on 1pt off 1pt,color=blue,domain=-0.77:3.52] plot(\x,{(-2--1*\x)/-1}); \fill (1.5,1.5) circle (1.5pt); \fill (1,1) circle (1.5pt); \draw [->,dashed] (0,0) -- (1,1)node[midway,sloped,above,black]{\footnotesize{$e^+=V+H$}}; \draw (-.05,1.8) node[anchor=east] {\footnotesize{$V$}}; \draw (3.2,.2) node[anchor=north west] {\footnotesize{$H$}}; \draw [->,color=green!70!black] (1.2,.8) -- (2.2,.8)node[midway,above,black]{}; \draw [->] (0,0) -- (1.2,.8)node[midway,above,black]{\footnotesize{\quad $v'$}}; \draw [->,color=green!70!black,dashed] (1,1) -- (1.5,1.5)node[midway,sloped,above,black] {\Tiny $e^+$-increment}; \draw [->] (0,0) -- (2.2,.8)node[midway,sloped,below,black]{\footnotesize{$v''=DF(v')$}}; \end{tikzpicture} \caption{$DF$ increases the unstable component }\label{FIGShearGrowsUnstable} \end{figure} \end{proof} \begin{rema} Alternatively, let $z$ be a point on the surgery annulus such that its orbit under the flow $X_s$ crosses the surgery region infinitely often. In local coordinates, $z$ is the identification of $p=(t,s,w)$ with $F(p)=(t,s+f(w),w)$. Consider a vector $v=ae^+(F(p))+be^-(F(p))$ in the preserved cone at $F(p)$ given by $a>0$, $0\leq b\leq a$. Consider the first return, at time $t$. Let $q=\phi_t(F(p))=(0,s_t,w_t)$. The image of $v$ by the flow differential at $t$ before the identification between $q$ and $F(q)$ is $v'=ae^te^+(q)+be^{-t}e^-(q)$. To compute its image $v''=DF(v')$ after identification we need to consider the change of coordinate between the basis \begin{align*} \left(e_+(q),e_-(q),X(q)\right) &\text{ and } \left(\frac{\partial}{\partial s}(q),\frac{\partial}{\partial w}(q),X(q)\right). \\ \intertext{Let} \frac{\partial}{\partial s}(q)=a_0e^+(q)+b_0e^-(q)+c_0X(q) &\text{ and } \frac{\partial}{\partial w}(q)=a_1e^+(q)+b_1e^-(q)+c_1X(q). \end{align*} If the surgery is performed in a small annulus then $e_+(q)$ is arbitrarily close to $\frac{\partial}{\partial s}(q)+\frac{\partial}{\partial w}(q)$ and $e_-(q)$ is arbitrarily close to $-\frac{\partial}{\partial s}(q)+\frac{\partial}{\partial w}(q)$. We obtain \begin{align*} v'' &= a'e^+(q) + b' e^-(q) + c' X(q) \\ \intertext{with} a' &= ae^t + \frac{df}{dw} (w_t)\,a_0\frac{aa_0e^t-ba_1e^{-t}}{a_0b_1-a_1 b_0}. \end{align*} As $t$ is bounded below, we obtain $a'\geq a e^{t}$ for a surgery performed in a small annulus. This gives the desired inequality for the projected norm. \end{rema} \begin{rema} We emphasize that the entropy-increase is manifested for $X_{HT}$ and thus results from the surgery and not from the time-change that makes the flow contact. \end{rema} We are now ready to pursue the growth of periodic orbits. \begin{proof}[Proof of Theorem~\ref{THMLargerTopEnt}] Abramov's formula~\eqref{eqAbramov} with $ g\dfn\tfrac{c}{1\pm\dd h(X_{HT})}$ and $\mu_g$ the normalized volume defined by $\alpha_A$ gives \[ \hmu(X_h)=\hmu(X_{HT})\int\dfrac{c}{1\pm\dd h(X_{HT})}\alpha\wedge\dd\alpha=\hmu(X_{HT}). \] Combined with our previous result, this gives \begin{equation}\label{eqLiouvilleEntDefect} \hmu(\varphi^t)=\hmu(X_h)= \underbracket{\hmu(X_{HT})\ge\hmu(X)}_{\text{Proposition~\ref{PRPsameLyapunov}}}=\hmu(g^t). \end{equation} This in turn yields a comparison of topological entropies: \[ \rlap{$\underbracket{\phantom{\htop(g^t)=\hmu(g^t)}}_{\text{constant curvature}}$}\htop(g^t)=\rlap{$\overbracket{\phantom{\hmu(g^t)\le\hmu(\varphi^t)}}^{\eqref{eqLiouvilleEntDefect}}$}\hmu(g^t)\le\underbracket{\hmu(\varphi^t)<\htop(\varphi^t)}_{\text{Theorem~\ref{THMMain}$\MK$\eqref{itemBMMnotVolume}}}.\qedhere \] \end{proof} \begin{proof}[Proof of Theorem~\ref{THMLargerSharpGrowth}] By~\eqref{eqSharpCounting}, increased cohomological pressure suffices: \[ \overbracket{\hmu(g^t)\le P(g^t)\le\htop(g^t)}^{\text{\eqref{eqFanginequality}}}=\rlap{$\overbracket{\phantom{\hmu(g^t)\le\hmu(\varphi^t)}}^{\eqref{eqLiouvilleEntDefect}}$}\hmu(g^t)\llap{$\underbracket{\phantom{\htop(g^t)=\hmu(g^t)}}_{\text{constant curvature}}$}\le\underbracket{\hmu(\varphi^t)=1pt,node distance=2.0cm,auto] \node (A) {$\rmC\bbH^\Lambda_{\leq\,T}(\alpha_0)$}; \node (B) [below right of=A] {$\rmC\bbH^\Lambda(\alpha_0)$}; \node (C) [above right of=B] {$\rmC\bbH^\Lambda_{\leq\,T'}(\alpha_0)$}; \path[->] (A) edge node {$\phi_{T,\,T'}$} (C) edge node [swap] {$\phi_{T} $ } (B) (C) edge node {$\phi_{T'} $ } (B); \end{tikzpicture} \end{center} \item Let $\alpha=f\alpha_0$ be another non\-degenerate contact form. Assume $f>0$, and let $B$ be such that $1/B\leq f(m)\leq B$ for all $m\in M$. There exist $C=C(B)$ and morphisms $\psi_T\colon \rmC\bbH^\Lambda_{\leq\,T}(\alpha_0)\longrightarrow \rmC\bbH^\Lambda_{\leq\,CT}(\alpha)$ such that the following diagram commutes: \begin{center} \begin{tikzpicture}[shorten >=1pt, auto] \node (A) {$\rmC\bbH^\Lambda_{\leq T}(\alpha_0)$}; \node (B) [right =1.2cm of A] {$\rmC\bbH^\Lambda_{\leq\,CT}(\alpha)$}; \node (C) [below =1.1cm of A] {$\rmC\bbH^\Lambda_{\leq\,T'}(\alpha_0)$}; \node (D) [right =1.2cm of C] {$\rmC\bbH^\Lambda_{\leq\,CT'}(\alpha)$}; \path[->] (A) edge node [swap] {$\phi_{T,\,T'}(\alpha_0)$} (C) edge node {$\psi_{T} $ } (B) (C) edge node {$\psi_{T'} $ } (D) (B) edge node {$\phi_{CT,\,CT'}(\alpha)$} (D); \end{tikzpicture} \end{center} This defines a morphism of directed system. \end{enumerate} \end{theo} Contact homology was introduced by Eliashberg, Givental and Hofer~\cite{EGH00}. The filtration properties come from~\cite{ColinHonda08}. The description in terms of directed systems takes its inspiration from~\cite{McLean2010} and is presented in~\cite[Section~4]{Vaugon12}. Though commonly accepted, existence and invariance of contact homology remain unproven in general. This has been studied by many people using different techniques. This paper uses only proved results and follows the approaches of Dragnev and Pardon~\cite{Dragnev04,Pardon}. If $\alpha$ is hypertight and $\Lambda$ contains only primitive free homotopy classes, the properties of contact homology described in Theorem~\ref{THMFPCH} derive from~\cite{Dragnev04} (see~\cite[Section~2.3]{Vaugon12}). In the general case, Theorem~\ref{THMFPCH} can be derived from~\cite{Pardon}. Cylindrical contact homology for hypertight contact forms (and possibly nonprimitive homotopy classes) and the action filtration are described in~\cite[Section~1.8]{Pardon}. The case of a not hypertight contact form when there exists an hypertight contact form derives from the contact homology of contractible orbits~\cite[Section~1.8]{Pardon} and our invariant corresponds to $CH^\mathrm{\Lambda}_\bullet$. Note that when computed through a hypertight contact form, $CH^{\contr}_\bullet$ is trivial and $CH^\mathrm{\Lambda}_\bullet$ is the cylindrical contact homology. In the not hypertight case, our invariants can be interpreted geometrically using augmentations. This viewpoint is described in~\cite[Section~2.4 and Section~4]{Vaugon12}. Combining the two commutative diagrams from Theorem~\ref{THMFHC} and the invariance of contact homology we obtain the following inequality. %\pagebreak \begin{prop}\label{PROPGR} Let $\alpha_0$ and $\alpha=f\alpha_0$ be two non\-degenerate contact forms on $(M,\xi)$, where $M$ is a closed, $3$-dimensional manifold and $\xi$ is hypertight. Assume $f>0$, and let $B$ such that $1/B\leq f(m)\leq B$ for all $m\in M$. Then \[ N_L^\Lambda(\alpha)\geq \rank(\phi_L(\alpha)) \geq \rank(\phi_{L/C(B)}(\alpha_0)) \] for all $L>0$. \end{prop} If $\rmC\bbH^\Lambda(\alpha_0)$ is well-understood, one can get an easier estimate. \begin{coro}\label{CORGR} Let $\alpha_0$ and $\alpha=f\alpha_0$ be two non\-degenerate contact forms on $(M,\xi)$ where $M$ is a closed, $3$-dimensional manifold and $\xi$ is hypertight. Assume $f>0$, and let $B$ such that $1/B\leq f(m)\leq B$ for all $m\in M$. If \[ \rmC\bbH^\Lambda(\alpha_0)=\bigoplus_{R_{\alpha_0}\text{-Reeb-periodic orbit $\gamma$ in }\Lambda}\bbQ\gamma \] then, $ N_L^\Lambda(\alpha)\geq N_{L/C(B)}^\Lambda(\alpha_0)$ for all $L>0$. \end{coro} In fact, one can derive another invariant of contact structures from these properties of contact homology. Two non\-decreasing functions $f\colon\R_+\to \R_+$ and $g\colon\R_+\to \R_+$ have the same \emph{growth rate type} if there exists $C>0$ such that \[ f\left(\frac{x}{C}\right)\leq g(x)\leq f(Cx) \] for all~$x\in\R_+$ (for instance, a function grows exponentially is it is in the equivalence class of the exponential). The \emph{growth rate type of contact homology} is the growth rate of $T\mapsto \rank(\phi_T)$. Two non\-degenerate contact forms associated to the same contact structure have the same growth rate type (by Proposition~\ref{PROPGR}) and therefore, the growth rate type of contact homology is an invariant of the contact structure. The growth rate of contact homology was introduced in~\cite{BourgeoisColin05}. It ``describes'' the asymptotic behavior with respect to $T$ of the number of Reeb-periodic orbits with period smaller than $T$ that contribute to contact homology. For a more detailed presentation one can refer to~\cite{Vaugon12}. Colin and Honda's conjecture~\cite[Conjecture~2.10]{ColinHonda08} (see Section~\ref{SIntro}) for the contact structures from Theorem~\ref{THMMain}, and Theorem~\ref{THMFHC} for non\-degenerate contact forms follow from \begin{prop}\label{PROPEvenOrbits} Let $(M,\xi)$ be a compact contact 3-manifold and assume there exists a contact form $\alpha_0$ on $(M,\xi)$ whose Reeb flow is Anosov with orientable stable and unstable foliations. Then any $R_{\alpha_0}$-periodic orbit is even and hyperbolic. \end{prop} Indeed, by Proposition~\ref{PROPFoliationsOrientability}, one can apply Proposition~\ref{PROPEvenOrbits} to $(M_S,\alpha_A)$, which is hypertight as the Reeb flow is Anosov. Therefore, the differential in contact homology is trivial (Theorem~\ref{THMFPCH}$\MK$\eqref{itemContHomDiffOddEven}) and for any set $\Lambda$ of free homotopy classes, \[ \rmC\bbH_{\cyl}^\Lambda(\alpha_A)=\bigoplus_{R_{\alpha_A}\text{-Reeb-periodic orbit $\gamma$ in }\Lambda}\bbQ\gamma. \] Let $\alpha=f\alpha_A$ be non\-degenerate with $f>0$ and let $B$ be such that $1/B\leq f(m)\leq B$ for all $m\in M$. Applying Corollary~\ref{CORGR} for $\Lambda=\{\rho\}$, we get $ N_L^\Lambda(\alpha)\geq N_{L/C(B)}^\Lambda(\alpha_A)$ for any $L>0$. Using the Barthelmé--Fenley estimates from~\cite[Theorem~F]{BarthelmeFenley2} we obtain the desired logarithmic growth. This finishes the proof of Theorem~\ref{THMFHC} in the nondegenerate case. Additionally, the number of periodic orbits of an Anosov flow in primitive homotopy classes grows exponentially with the period. Applying Corollary~\ref{CORGR} for $\Lambda$ the set of all primitive free homotopy classes in $M_S$ proves the Colin--Honda conjecture for contact structures from Theorem~\ref{THMMain} and non\-degenerate contact forms. \begin{proof}[Proof of Proposition~\ref{PROPEvenOrbits}] By definition of stable and unstable foliations, $D\phi^T_{\vert\xi}(p)$ has real eigenvalues $\mu$ and $\tfrac{1}{\mu}$ and the associated eigenspaces are $E^+$ and $E^-$. As the strong stable foliation is orientable, the eigenvalues are positive. Thus $\gamma$ is even and hyperbolic. \end{proof} \section{Orbit growth in a free homotopy class for degenerate contact forms}\label{SECOrbitGrowthFHC} We now prove Theorem~\ref{THMFHC} for degenerate contact form (the non\-degenerate case is explained in the previous section). The proof derives from the proof of~\cite[Theorem~1]{Alves3}. Yet Alves' goal was to obtain one orbit with bounded period in some free homotopy class and not control the number of orbits in this class, and the following result is not explicit in~\cite{Alves3}. \begin{coro}\label{CORExistencePO} Let $(M,\xi)$ be a closed manifold and $\alpha_0$ an Anosov contact form on $(M,\xi)$. Let $\rho$ be a primitive free homotopy class of $M$ such that \[ \rmC\bbH_{\cyl}^\rho(\alpha_0)=\bigoplus_{R_{\alpha_0}\text{-periodic orbit $\gamma$ in }\rho}\bbQ\gamma\neq\{0\}. \] Then, for any contact form $\alpha=f_\alpha\alpha_0$ on $(M,\xi)$ and for any $R_{\alpha_0}$-periodic orbit of period $T$, there exists an $R_\alpha$-periodic orbit in $\rho$ of period $T'$ with $e\dfn\min|f_\alpha|\linebreak\leq T'/T\leq E\dfn\max|f_\alpha|$. \end{coro} \begin{proof}[Proof of Corollary~\ref{CORExistencePO}] Fix $0<\epsilon

0$. We follow Alves' proof of~\cite[Theorem~1]{Alves3} and consider $\alpha=f_\alpha\alpha_0$ on $(M,\xi)$ non\-degenerate. For any $R>0$, Alves constructs (Step 1) a symplectic cobordism $\R\times M_S$ between $(E+\epsilon)\alpha_0$ and $(e-\epsilon)\alpha_0$ which corresponds to the symplectization of $\alpha$ on $[-R,R]\times M_S$, and a map \[ \Psi_R\colon \rmC\bbH_{\cyl}^\rho((E+\epsilon)\alpha_0)\longrightarrow \rmC\bbH_{\cyl}^\rho\left((e-\epsilon)\alpha_0\right) \] by counting holomorphic cylinders in the symplectic cobordism. As $\rmC\bbH_{\cyl}^\rho(C\alpha_0)$ is canonically isomorphic to $\rmC\bbH_{\cyl}^\rho(\alpha_0)$ for any $C>0$, $\Psi_R$ induces an endomorphism of $\rmC\bbH_{\cyl}^\rho(\alpha_0)$ and Alves proves that this endomorphism is, in fact, the identity. Let $\gamma$ be a $R_{\alpha_0}$-periodic orbit of period $T$. For any $C>0$, it induces a $R_{C\alpha_0}$-periodic orbit $\gamma_C$ of period $CT$. As \[ \rmC\bbH_{\cyl}^\rho(\alpha_0)=\bigoplus_{R_{\alpha_0}\text{-Reeb-periodic orbit $\gamma$ in }\rho}\bbQ\gamma, \] $\Psi_R(\gamma_{E+\epsilon})=\gamma_{e-\epsilon}$ and therefore, there exists a holomorphic cylinder between $\gamma_{E+\epsilon}$ and $\gamma_{e-\epsilon}$. Now as $R$ tends to infinity (Step 2), SFT compactness (see~\cite{Alves3}) shows that our family of cylinders breaks and an $R_\alpha$-periodic orbit $\gamma_\epsilon$ of period $T_\epsilon$ appears in an intermediate level. By construction, $(e-\epsilon)T\leq T_\epsilon\leq (E+\epsilon)T$. Now, let\linebreak$\epsilon$ tend to $0$ and use the Arzelà--Ascoli Theorem to obtain an $R_\alpha$-periodic orbit $\gamma'$ with period $T'$ such that $eT\leq T'\leq ET$. If $\alpha$ is degenerate (Step 4), there exists a sequence $(\alpha_n)_{n\,\in\,\bbN}$ of non\-degenerate contact forms converging to $\alpha$ and the Arzelà-Ascoli Theorem can again be applied to obtain the desired periodic orbit. \end{proof} \begin{proof}[Proof of Theorem~\ref{THMFHC} for degenerate contact forms] As $M_S$ is hyperbolic, there are positive real numbers $a_1, b_1,a_2,b_2>0$ such that \[ \frac{1}{a_2}\ln(T)-c_2\leq N_T^\rho(\alpha_A)\leq a_1\ln(T)+c_1 \] for all $T>0$~\cite[Theorem~F]{BarthelmeFenley2}. Let $(\gamma_n)_{n\,\in\,\bbN}$ be a sequence of $R_{\alpha_A}$-periodic orbits in $\rho$ of periods $(T_n)_{n\,\in\,\bbN}$ such that \begin{itemize} \item $\gamma_0$ is a $R_{\alpha_A}$-periodic orbit in $\rho$ with minimal period; \item for all $n\geq 0$, $\gamma_{n+1}$ is a $R_{\alpha_A}$-periodic orbit in $\rho$ with period $T_{n+1}>\tfrac{E}{e}T_n$ and such that there exists no periodic orbit in $\rho$ with period in $(\tfrac{E}{e}T_n,T_{n+1})$. \end{itemize} By Corollary~\ref{CORExistencePO}, for any $n\geq 0$, there exists a $R_\alpha$-periodic orbit $\gamma'_n$ of period $T'_n$ such that $eT_n\leq T'_n\leq ET_n$. Therefore, $T'_n\leq ET_n 1$ such that $T_{n+1}\leq (c_3 T_n)^{a_3}$ for all $n\geq 0$. Thus, there exists $c_4> 0$ such that \[ \ln(T_{n+1})\leq c_4 a_3^{n+1} \] for all $n\geq 0$ and there exists $c_5\in\R$ such that \[ \ln(\ln(T_{n+1}))\leq \ln(a_3)(n+1)+c_5 \] for all $n\geq 0$. Now, if $eT_{n-1}\leq T'_{n-1}\leq T\leq T'_n\leq ET_n$, then \[ N_T^\rho(\alpha)\geq n \geq \frac{1}{\ln(a_3)}\ln(\ln(T_{n}))-c_5\geq \frac{1}{\ln(a_3)}\ln(\ln(T))-c_6 \] for some $c_6\in\R$. This proves Theorem~\ref{THMFHC}. \end{proof} \begin{rema} If $a_1a_2=1$, one can get better estimates and obtain the same growth as in the non\-degenerate case. \end{rema} \section{Exponential orbit growth after surgery on a simple closed geodesic}\label{SECOrbitGrowth_simple_geodesic} We now prove Theorem~\ref{th_courbe_simple_entropie} using the following result by Alves about the \emph{exponential homotopical growth} of cylindrical contact homology. \begin{defi}[Exponential homotopical growth~\cite{Alves1}] Let $(M,\xi)$ be a closed contact manifold and $\alpha_0$ a hypertight contact form on $(M,\xi)$. For $T>0$, let $N_T^{\cyl}(\alpha_0)$ be the number of free homotopy classes $\rho$ of $M$ such that \begin{itemize} \item all the $R_{\alpha_0}$-periodic orbits in $\rho$ are simply-covered, non\-degenerate and have period smaller than $T$; \item $\rmC\bbH_{\cyl}^\rho(\alpha_0)\neq 0$. \end{itemize} We say that the cylindrical contact homology of $(M,\alpha_0)$ has \emph{exponential homotopical growth} if there exist $T_0\geq 0$, $a>0$ and $b\in\R$ such that, for all $T\geq T_0$, \[ N_T^{\cyl}(\alpha_0)\geq e^{aT+b}. \] \end{defi} \begin{theo}[{Alves~\cite[Theorem~2]{Alves1}}]\label{th_alves} Let $\alpha_0$ be a hypertight contact form on a closed contact manifold $(M,\xi)$ and assume that the cylindrical contact homology has exponential homotopical growth. Then every Reeb flow on $(M,\xi)$ has positive topological entropy. \end{theo} If $\rho$ is a free homotopy class containing only one $R_{\alpha_0}$-periodic orbit and if this orbit is simply-covered and non\-degenerate, it is a direct consequence of the definition of contact homology that $\rmC\bbH_{\cyl}^\rho(\alpha_0)=\bbQ$. Therefore, to prove Theorem~\ref{th_courbe_simple_entropie}, it suffices to prove the following propositions. \begin{prop}\label{prop_ht} The contact form $\alpha_A$ is hypertight in $M_S$. \end{prop} \begin{prop}\label{prop_N'} Let $(M_S,\alpha_A)$ be a contact manifold obtained after a contact surgery along a Legendrian projecting to a simple closed geodesic and $N'_T(\alpha_A)$ the number of free homotopy classes $\rho$ that contain only one $R_{\alpha_A}$-periodic orbit and this orbit is simply-covered, non\-degenerate and of period smaller than $T$. Then there exist $T_0\geq 0$, $a>0$ and $b\in\R$ such that, for all $T\geq T_0$, $N'_T(\alpha_A)\geq e^{aT+b}$. \end{prop} Indeed, the exponential growth of $N'_T(\alpha_A)$ with respect to $T$ induces the exponential homotopical growth of $(M_S,\alpha_A)$ and we can apply Theorem~\ref{th_alves}. We now turn to the proofs of Proposition~\ref{prop_ht} and Proposition~\ref{prop_N'}. In $S\Sigma$, $\bbT = \pi^{-1}(c)$ is a torus, and our surgery preserves this torus. Let $\bbT_S$ denote the associated torus in $M_S$. Van Kampen's Theorem tells us that $\bbT_S$ is $\pi_1$-injective. To prove Proposition~\ref{prop_N'}, we want to find free homotopy classes with only one periodic Reeb orbit. We will consider free homotopy classes containing a periodic orbit disjoint from $\bbT_S$ and prove there are enough of such classes. First, we describe Reeb-periodic orbits and study the properties of free homotopies between them. \begin{enonce}{Claim}\label{po_type} There are three types of $R_{\alpha_A}$-periodic orbits: \begin{enumerate} \item periodic orbits contained in $\bbT_S$, the only periodic orbits of this kind are $\geodesic$, $-\geodesic$ ($\geodesic$ with the reverse orientation) and their covers, \item periodic orbits disjoint from $\bbT_S$, these orbits correspond to closed geodesics in $\Sigma$ disjoint from $\pi(\geodesic)$ (this includes multiply-covered geodesics), \item periodic orbits intersecting $\bbT_S$ transversely. \end{enumerate} \end{enonce} Therefore, a free homotopy between two $R_{\alpha_A}$-periodic orbits can always be perturbed to be transverse to $\bbT_S$. \begin{prop}\label{prop_H} Let $\delta_0,\delta_1$ be two smooth loops in $M_S$ and $H\colon [0,1]\times S^1\to M_S$ be a free homotopy between $\delta_0$ and $\delta_1$ transverse to $\bbT_S$. $N\dfn H^{-1}(\bbT_S)$ is a smooth manifold of dimension~$1$ properly embedded in $[0,1]\times S^1$. Therefore, \begin{enumerate} \item\label{p1} one can modify $H$ so that $N$ does not contain contractible circles, \item if $\delta_0$ is a $R_{\alpha_A}$-periodic orbit transverse to $\bbT_S$, $N$ does not contain a segment with both end-points on $\{0\}\times S^1$. \end{enumerate} \end{prop} \begin{proof} Consider an innermost contractible circle $c_0$ in $N\subset [0,1]\times S^1$, $c_0$ bounds a disk $D_0$ in $[0,1]\times S^1$. The image of $c_0$ is contractible in $\bbT_S$ as $\bbT_S$ is $\pi_1$-injective. Therefore, there exists a continuous $G\colon D_0\to \bbT_S$ such that $H_{|c_0}=G_{|c_0}$ and one can replace $H_{|D_0}$ by $G$ to obtain a new homotopy (still denoted by $H$) between $\delta_0$ and $\delta_1$. Now, consider a neighborhood $[-\nu,\nu]\times \bbT_S$ of $\bbT_S$ in $M_S$ (with $\bbT_S\simeq \{0\}\times\bbT_S$) and a disk $D_1$ containing $D_0$ such that $H(D_1)\subset [0,\nu]\times \bbT_S$ and $H(D_1\setminus D_0)\subset (0,\nu]\times \bbT_S$. One can perturb $H$ in $\intt(D_1)$ so that $H(D_1)\subset (0,\nu]\times \bbT_S$. Performing this inductively on the contractible circles proves~\eqref{p1}. We now assume $\delta_0$ is an $R_{\alpha_A}$-periodic orbit transverse to $\bbT_S$. By contradiction, consider an innermost segment $c_0$ in $N$ with end-points on $\{0\}\times S^1$. The end-points of $c_0$ correspond to consecutive intersection points of $\delta_0$ with $\bbT_S$. Let $c_1$ be the segment in $\{0\}\times S^1$ joining these two end-points and homotopic (relative to end-points) to $c_0$. By construction, there exists a homotopy $(\eta_t)_{t\,\in\,[0,1]}\colon [0,1]\to M_S$ (relative to end-points) between $\eta_0=H(c_0)$ et $\eta_1=H(c_1)$ such that $\eta_t(s)\in \bbT_S$ if and only if $t=1$ or $s=0,1$. Let $M'$ be the manifold with boundary obtained by cutting $M_S$ along $\bbT_S$. Note that $M'$ can also be obtained by cutting $S\Sigma$ along $\bbT$. The projection $M'\to M_S$ is injective in the interior of $M'$, therefore $\eta_t(s)$ is well-defined in $M'$ if $t\neq 0$ and $s\neq 0,1$. Thus, there exists a homotopy $\eta'_t$ in $M'$ lifting $\eta_t$. This homotopy induces a homotopy in $S\Sigma$ and, as a result, a homotopy in $\Sigma$ between a geodesic arc contained in $\pi(\geodesic)$ and a geodesic arc with end-points on $\pi(\geodesic)$. As $\Sigma$ is hyperbolic, this can only happen if our second geodesic arc is also contained in $\pi(\geodesic)$, a contradiction. \end{proof} \begin{proof}[Proof of Proposition~\ref{prop_ht}] By contradiction, assume there exists a free homotopy $H$ between $\delta$, a $R_{\alpha_A}$-periodic orbit, and a point $p\notin \bbT_S$. As $\bbT_S$ is $\pi_1$-injective, $\delta$ cannot be contained in $\bbT_S$. Without loss of generality we may assume that $H$ is transverse to $\bbT_S$ and apply Proposition~\ref{prop_H}. If $\delta$ is disjoint from $\bbT_S$, then $N\subset [0,1]\times S^1$ (see Proposition~\ref{prop_H}) can only contain circles parallel to the boundary. We will now prove that we can modify $H$ so that $N$ is empty. Let $c_0$ be the circle in $N$ closest to $\{0\}\times S^1$ and let $C$ be the closure of the connected component of $([0,1]\times S^1)\setminus c_0$ containing $\{1\}\times S^1$. Then $H(c_0)$ is an immersed circle contractible in $\bbT_S$ and there exists a continuous map $G\colon C\to \bbT_S$ such that $G_{|c_0}=H$ and $G_{\{1\}\,\times\,S^1}$ is constant. We replace $H_{|C}$ with $G$ to obtain a new homotopy $H$. Now, consider a neighborhood $[-\nu,\nu]\times \bbT_S$ of $\bbT_S$ in $M_S$ and a neighborhood $C_1$ of $C$ such that $H(C_1)$ is contained in $[0,\nu]\times \bbT_S$. We can perturb $H$ so that $H(C_1)$ is contained in $(0,\nu]\times \bbT_S$. Therefore we may assume that $N$ is empty and $H$ is an homotopy in $M_S\setminus \bbT_S$. It induces an homotopy in $S\Sigma$, a contradiction as the periodic orbits are not contractible in $S\Sigma$. Finally, we consider the case of $\delta$ transverse to $\bbT_S$. In this case, $N$ has boundary points on $\{0\}\times \bbT_S$ but not on $\{1\}\times \bbT_S$. This contradicts Proposition~\ref{prop_H}. \end{proof} \begin{prop} If $\delta$ is a $R_{\alpha_A}$-periodic orbit disjoint from $\bbT_S$, then the free homotopy class of $\delta$ contains exactly one $R_{\alpha_A}$-periodic orbit. \end{prop} \begin{proof} By contradiction, consider a free homotopy $H$ from $\delta$ to $\delta_1$, a distinct $R_{\alpha_A}$-periodic orbit. Without loss of generality, we may assume that $H$ is transverse to $\bbT_S$ (apply Proposition~\ref{prop_H}). If $\delta_1$ is disjoint from $\bbT_S$, then $N$ can only contain circles parallel to the boundary. If $N$ is empty, $H$ induces a homotopy in $S\Sigma$ and therefore in $\Sigma$. Yet, two closed geodesics on a hyperbolic surface are not homotopic. This proves that $N\neq\emptyset$. Let $c_0$ be the circle in $N$ closest to $\{0\}\times S^1$ and $M'$ be the manifold with boundary obtained by cutting $M_S$ along $\bbT_S$. The homotopy $H$ induces a homotopy $G$ between $\delta$ and $H(c_0)\subset \bbT_S$. The homotopy $G$ lifts to $M'$ and therefore induces a free homotopy in $S\Sigma$ and, as a result, a free homotopy in $\Sigma$ between a closed geodesics and a loop contained in the geodesic $\pi(\geodesic)$. This can happen only if our first geodesic is a cover of $\pi(\geodesic)$. Yet this implies $\delta\subset\bbT_S$, a contradiction. If $\delta_1$ is transverse to $\bbT_S$, the manifold $N$ is not empty and has end-points on $\{1\}\times S^1$ but cannot have end-points on $\{0\}\times S^1$. This contradicts Proposition~\ref{prop_H}. Finally, the case $\delta_1$ contained in $\bbT_S$ is similar to the case $\delta_1$ disjoint from $\bbT_S$. In this case, $N$ contains only circles parallel to the boundary and $\{1\}\times S^1$ is in $N$. \end{proof} \begin{proof}[Proof of Proposition~\ref{prop_N'}] If $\pi(\geodesic)$ is nonseparating, by cutting $\Sigma$ along $\pi(\geodesic)$ we obtain a surface of genus at least $1$ with two boundary components. Let $\ell_1$ and $\ell_2$ be two loops in $\Sigma\setminus \geodesic$, homotopically independent and with the same base-point. Then, any nontrivial word in $\ell_1$ and $\ell_2$ defines a nontrivial free homotopy class for $\Sigma$ and there exists a closed geodesic on $\Sigma$ representing this class. This $R_{\alpha_A}$-periodic orbit is always nondegenerate. Additionally, we may assume that the orbits associated to $\ell_1$ and $\ell_2$ are simply-covered. If a word is not the repetition a smaller word, the associated orbit is therefore simply covered. As $\ell_1$ and $\ell_2$ are independent, all these geodesics are disjoint and their number grows exponentially with the period. Finally, these geodesics do not intersect $\geodesic$ as geodesics always minimize the intersection number. If $\pi(\geodesic)$ is separating, then by cutting $\Sigma$ along $\pi(\geodesic)$, we obtain two surfaces of genus at least $1$ with one boundary component. The proof is similar. \end{proof} \section{Surgery on a periodic Reeb flow}\label{SMoreContactFlows} We now consider the coexistence of diverse Reeb flows and prove Theorem~\ref{th_courbe_simple_polynom}. \subsection{Dynamical properties of the periodic Reeb flow after surgery} We next apply the general construction of contact surgery along a Legendrian curve described in Section~\ref{SScontact} to the contact structure with contact form $\beta$ and periodic Reeb flow described in Section~\ref{SIMoreContactFlows}. Select a closed geodesic on a hyperbolic surface $\Sigma$ and consider its lift $\geodesic\colon S^1=\R/\bbZ\to M$ to the unit tangent bundle $M$ of $\Sigma$. Consider the Legendrian knot $\gamma$ in $(M,\ker(\beta))$ obtained by rotating the unit vector field along $\geodesic$ by the angle $\theta=\pi/2$. Note that the Legendrian knot $\gamma$ is the same as in Section~\ref{SSAnosov} (and is tangent to $H$). To obtain standard coordinates in a neighborhood of $\gamma$ we first consider an annulus $A$ in $S\Sigma$ transverse to the fibers with coordinates $(s,w)\in S^1\times (-2\epsilon,2\epsilon)$ such that $\beta_{|A}=w\dd s$ and then flow along the Reeb vector field $R_\beta$ to obtain coordinates $(t,s,w)\in S^1\times A=N$ such that $\beta=\dd t + w\dd s$ (to remain coherent with previous conventions our circles have different lengths, more precisely $t\in \R/2\pi \bbZ$ and $s\in \R/\bbZ$)\rlap.\footnote{These coordinates along $\gamma$ are different from the coordinates defined for the surgery associated to the contact form $\alpha$ as, for instance, the surgery annulus is different. It is possible to derive a contact form from $\beta$ on the surgered manifold using the coordinates and surgery associated to $\alpha$: write $\beta$ in local coordinates, compute $F^*\beta$ and interpolate using bump and cut-off functions. Unfortunately, this construction yields a complicated Reeb vector field. Note that the contact structure obtained this way is isotopic to $\ker(\beta_S)$. This can be proved as follows. First the two surgeries result in the same manifold. Moreover, a surgery can be described as the gluing of a solid torus on an excavated manifold. Therefore we just need to prove that the contact structures on the glued tori are the same. This derives from the classification of contact structures on $\bbD^3$ by Eliashberg. See~\cite{MakarLimanov} for an application to the torus.} Note that $N$ can be interpreted as the suspension of the annulus $A$ by the identity map. Our non\-trivial surgery is defined by a twist (shear) $F$ along $A$. We denote by $M_S$ the manifold $S\Sigma$ after surgery and by $N_S\subset M_S$ the manifold (with boundary) $N$ after surgery. Let $\beta_S$ be the contact form on $M_S$ as described in Section~\ref{SScontact}. Note that $\beta$ and $\beta_S$ coincide outside $N$ and $N_S$ respectively. The manifold $N_S$ is the suspension of the annulus $A$ by the shear map $F$. Moreover, the map $p_S\colon N_S\to(-2\epsilon,2\epsilon)$ given by the $w$-coordinate is well-defined and is a trivial torus-bundle. For $w\in(-2\epsilon,2\epsilon)$, the torus $p_S^{-1}(w)$ is foliated by closed Reeb orbits if and only if \[ f\left(w\right)=2\pi \frac{p_w}{q_w}\in2\pi\bbQ \] where $p_w$ and $q_w$ are coprime. In this situation the orbits of $\tfrac{\partial}{\partial t}$ on $p_S^{-1}(w)$ are periodic of period $q_w$. The Reeb vector field is a renormalization of $\tfrac{\partial}{\partial t}$ (see~\eqref{eqXh}). Finally, let $\bbT = S^1\times S^1\times\{0\}$ in $N$ and $\bbT_S$ be its image in $M_S$. By van Kampen's theorem, this torus is incompressible. Therefore the contact form $\beta_S$ is hypertight. Note that if $f(w)\in2\pi\bbQ$ and $f(w')\in2\pi\bbQ$ but $f(w)\neq f(w')$, the associated periodic orbits are not freely homotopic. \subsection{Proof of Theorem~\ref{th_courbe_simple_polynom}} The contact form $\beta_S$ is degenerate and the renormalization from the surgery makes the direct study a bit harder. So, to estimate the growth rate of its contact homology, we will standardize and perturb our contact form. For any $w\in(-2\epsilon,2\epsilon)$, the vector fields $\tfrac{\partial}{\partial t}+\tfrac{f(w)}{2\pi}\frac{\partial}{\partial s}$ and $\tfrac{\partial}{\partial s}$ generate circles in the torus $p_S^{-1}(w)$. These circles induce a trivialisation of $N_S$. Let $(\tau,\sigma, w)$ be the coordinates on $N_S$ associated to this trivialisation. Without loss of generality, we may assume that the map $f$ defining the twist (shear) $F$ is constant on $(-2\epsilon,-\epsilon)\cup(\epsilon,2\epsilon)$, that $f'(w)\neq 0$ for any $w\in(-\epsilon, \epsilon)$ and that $f$ is invariant under reflection with respect to the point $(0,q/2)$. Therefore, for $w$ in $[-2\epsilon,-\epsilon]$, \[ \beta_S=\dd\tau+w\dd\sigma \] and for $w$ in $[\epsilon,2\epsilon]$, \[ \beta_S=\left(1+\frac{qw}{2\pi}\right)\dd\tau+w\dd\sigma. \] \begin{lemm} There exist smooth maps $h_0, k_0\colon (-2\epsilon, 2\epsilon)\to\R$ such that \[ \beta_0=h_0(w)\dd\tau+k_0(w)\dd\sigma \] is a contact form such that $\beta_0=\beta_S$ for $w$ close to $\pm2\epsilon$ and $R_{\beta_0}$ and $R_{\beta_S}$ are positively collinear on $N_S$. Therefore, $\beta_0$ and $\beta_S$ are isotopic (through contact forms). \end{lemm} \begin{proof} Let $h_0$ and $k_0$ be the maps defined by $k_0(w)=w$ and \[ h_0(w)=1+\int_{-2\epsilon}^w f(u)/2\pi\dd u \] for $w\in (-2\epsilon,2\epsilon)$. As $\int_{-\epsilon}^{\epsilon} f(u)d u = q\epsilon$, $\beta_0=\beta_S$ for $w\in(\epsilon,2\epsilon)$. Moreover, the contact condition is \[ 1+\int_{-2\epsilon}^w f(u)/2\pi\dd u -w f(w)/2\pi>0 \] and this condition is always satisfied for $\epsilon$ small enough. Additionally, the Reeb vector field is positively collinear to $(k'_0(w),-h'_0(w))=(1,-f(w)/2\pi)$. Finally, as $R_{\beta_0}$ and $R_{\beta_S}$ are positively collinear, $(u\beta_S+(1-u)\beta_0)_{u\,\in\,[0,\,1]}$ is a contact isotopy. \end{proof} The contact form $\beta_0$ is degenerate. To estimate the growth rate of its contact homology, we have to perturb it. Our perturbation draws its inspiration from Morse--Bott techniques. To describe our perturbation, we need to fix some notations. The manifold $S\Sigma\setminus p^{-1}((-\epsilon,\epsilon))$ is a trivial circle bundle. Let $S'$ be a surface (with boundary) transverse to the fibers and intersecting each fiber once: $S'$ provides us with a trivialisation $S'\times S^1$ of $S\Sigma\setminus p^{-1}((-\epsilon,\epsilon))$. The surface $S'$ has two boundary components. Let $\phi\colon S'\to \R$ be a Morse function such that $\phi=0$ on the boundary of $S'$ and, if $q>0$ (resp. $q<0$), the connected component of $\partial S'$ corresponding to $w=-\epsilon$ is a maximum (resp. a minimum) and the connected component corresponding to\linebreak $w=\epsilon$ a minimum (resp. a maximum). For any $w$ such that $f(w)=2\pi p(w)/q(w)\in 2\pi\bbQ$, we denote by $P(w)$ the period of the $R_{\beta_0}$-periodic orbits foliating $p_S^{-1}(w)$. Note that there exists $C_P>0$ such that $q(w)/C_P\leq P(w)\leq C_P q(w)$, this implies that the number of torus with $w\in(-\epsilon,\epsilon)$ foliated by Reeb-periodic orbits with period smaller than $L$ grows quadratically in $L$. For a contact form $\alpha$, let $\sigma(\alpha)$ denote the action spectrum: the set of periods of the periodic orbits of $R_\alpha$. \begin{prop}\label{prop_perturbation} Let $T>0$, $T\notin \sigma(\beta_0)$. There exists $\beta'=l\beta_0$ with $l\colon M_S\to\R_+$ arbitrarily close to 1 such that \begin{itemize} \item $\beta'$ is hypertight and nondegenerate \item the periodic orbits of $R_{\beta'}$ with period $\leq T$ are exactly: \begin{enumerate} \item the fibers associated to the critical points of $\phi$ and their multiple of multiplicity $\leq \lfloor \tfrac{T}{2\pi} \rfloor$ \item for all $w\in(-\epsilon,\epsilon)$ such that $P(w) 0$ such that for any $w\in(-\epsilon,-\epsilon\linebreak+\nu]\cup[\epsilon-\nu,\epsilon)$, if $f(w)=2\pi p(w)/q(w)\in 2\pi\bbQ$ then $q(w)>C_PT$. Let \[ N_S'=p_S^{-1}((-\epsilon,-\epsilon+\nu]\cup[\epsilon-\nu,\epsilon)). \] Let $S''$ be a smooth surface in $M_S$ with boundary obtained by adding to $S'$ two annuli in $N_S$, transverse to $R_{\beta_0}$ and projecting to $[-\epsilon,-\epsilon+\nu]\cup[\epsilon-\nu,\epsilon]$. We can therefore endow $S''\setminus S'$ with coordinates $(s',w')$ such that $w'$ lifts $w$. We now perturb $\phi$ and extend it to $S''$ so that $\phi(s',w')=\phi(w')$ on $S''\setminus S'$, $\phi'(w')\neq 0$ for all $w'\in [-\epsilon,-\epsilon+\nu)\cup(\epsilon-\nu,\epsilon]$, $\phi$ is flat (all its derivative are equal to $0$) for $w=\pm(\epsilon-\nu)$ and the critical points of $\phi$ are unaltered. Finally, we extend $\phi$ to $M_S$ to obtain a smooth function, $R_{\beta_0}$-invariant and such that $\phi\equiv 0$ in $N_S\setminus N_S'$. Let $\beta_\lambda=(1+\lambda \phi)\beta_0$. This is a standard Morse--Bott perturbation (see~\cite[Lemma~2.3]{Bourgeois02}) in $M_S\setminus p_S^{-1}((-\epsilon,\epsilon))$, therefore, for $\lambda\ll1$, the periodic orbits in this area correspond to the critical points of $k$. In the coordinates $(\tau,\sigma,w)$, we have \[ \beta_\lambda=(1+\lambda \phi(w))(h_0(w)\dd \tau+k_0(w)\dd \sigma). \] Therefore, in these coordinates, the Reeb vector field is positively collinear to \[ \left((1+\lambda \phi(w))k_0'(w)+\lambda \phi'(w)k_0(w)\right) \frac{\partial}{\partial \tau}-\left((1+\lambda \phi(w))h_0'(w)+\lambda \phi'(w)h_0(w)\right) \frac{\partial}{\partial \sigma}. \] The $\sigma$-coordinate is nonzero as $\phi$ and $h$ have the same monotonicity. For $\lambda\ll1$, the $\sigma$-coordinate is close to $-h'_0(w)$, the $\tau$-coordinate to $k'_0(w)$ and $R_{\beta_\lambda}$ is close to $R_{\beta_0}$. Therefore, for $\lambda\ll1$, if there is a $R_{\beta_\lambda}$-periodic orbit in $N'_S$, this orbit has slope $2\pi p'(w)/q'(w)\in2\pi\bbQ$ with $q'(w)>C_PT$. Thus there are no periodic orbit with period smaller than $T$ in $N'_S$ and the periodic orbits with period bigger than $T$ are not in the free homotopy classes of orbits with period smaller than $T$ as described in Proposition~\ref{prop_perturbation}. In $p_S^{-1}([-\epsilon+\nu,\epsilon-\nu])$, the periodic orbits with period $\leq T$ are contained in tori $p_S^{-1}(w)$ such that $P(w)\leq T$. These tori are foliated by periodic orbits. Morse--Bott techniques apply here and give the second type of periodic Reeb orbits: for any such $w$ we perturb $\beta$ in a neighborhood of $p_S^{-1}(w)$ with a function derived from a Morse function $\phi_w$ defined on $p_S^{-1}(w)/\text{Reeb flow} = S^1$ and the periodic orbits after perturbation correspond to the critical points of $\phi_w$. For a given $w$ we obtain two orbits (one associated to the maximum of $\phi_w$ and one associated to the minimum of $\phi_w$), their covers and some orbits with period bigger than $T$ and in the free homotopy class of arbitrarily large covers of our two simple orbits. This perturbation derives from~\cite[Lemma~2.3]{Bourgeois02} and is described for tori in~\cite[Section~3.1]{Vaugon12}. Lastly, standard perturbation techniques prove there exists an arbitrarily small perturbation of $\beta_\lambda$ with the following properties: \begin{itemize} \item it gives rise to a nondegenerate contact form, \item it does not change the periodic orbits with period smaller than $T$, \item it does not create periodic orbits of period bigger than $T$ in the free homotopy classes of orbits of period smaller than $T$.\qedhere \end{itemize} \end{proof} \begin{proof}[Proof of Proposition~\ref{PROPCH}] Let $\delta\in p_S^{-1}(w)$ be a $R_{\beta'}$-periodic orbit of period $\leq T$ of the second type in Proposition~\ref{prop_perturbation}. Then the $R_{\beta_0}$-periodic orbits in the class $[\delta]$ are exactly the orbits in $p_S^{-1}(w)$ (and all these orbits have the same period). As $\delta$ is simply-covered, Dragnev's~\cite{Dragnev04} results can be applied. Additionally, standard perturbations do not create contractible periodic Reeb orbits. Therefore, the differential for contact homology can be described using ``cascades'' from Bourgeois' work~\cite{Bourgeois02}. The case of a unique torus of orbits is explained in~\cite[Section~9.4]{Bourgeois02}. The cascades used to describe the differential in this degenerate setting mix holomorphic cylinders between orbits and gradient lines for $\phi_w$ in $p_S^{-1}(w)/\text{Reeb flow} = S^1$ (for some generic metric). As all periodic orbits in this class have the same period, there is no homolorphic cylinder in the cascade and the differential coincides with the Morse--Witten differential for $\phi_w$ (ie the differential associated to Morse homology). Therefore, cylindrical contact homology in the free homotopy class $\rho$ is $2$-dimensional. The cascades of Morse--Bott homology are explicitly described in~\cite{BourgeoisOancea} (in a slightly different setting). \end{proof} \begin{proof}[Proof of Proposition~\ref{th_courbe_simple_polynom}] Let $\beta'=f\beta$ be a nondegenerate hypertight contact form and $B$ be such that $1/B =1pt,node distance=3cm, auto] \node (A) {$\rmC\bbH_{\leq\,T_i/C(B)}(\beta')$}; \node (B) [right =1.1cm of A] {$\rmC\bbH_{\leq\,T_i}(\beta_i)$}; \node (C) [below =1.2cm of A] {$\rmC\bbH(\beta')$}; \node (D) [below =1.2cm of B] {$\rmC\bbH(\beta_i)$}; \path[->] (A) edge node [swap] {$\phi_{T_i/C(B)}(\beta')$} (C) edge node {} (B) (C) edge node {} (D) (B) edge node {$\phi_{T_i}(\beta_i)$} (D); \end{tikzpicture} \end{center} thus \[ rk\left(\phi_{T_i/C(B)}(\beta')\right)\leq rk\left(\phi_{T_i}(\beta_i)\right)\leq \dim\left(\rmC\bbH_{T_i}(\beta_i)\right) \leq a_1 \left(T_i^2\right). \] A symmetric commutative diagram implies \[ rk\left(\phi_{T_i/C(B)}(\beta_i)\right)\leq rk\left(\phi_{T_i}(\beta')\right) \] Propositions~\ref{prop_perturbation} and~\ref{PROPCH} prove that $\phi_{T_i/C(B)}$ is injective on the class of simply-covered periodic orbits of the second type (as defined in Proposition~\ref{prop_perturbation}). Therefore, we have $ rk(\phi_{T_i/C(B)}(\beta_i)) \geq a_0 T_i^2$ and the growth rate of contact homology is quadratic. \end{proof} \bibliography{AHL_Foulon} \end{document}