\documentclass{beamer} \usetheme{Malmoe} %\usetheme{Munich} \usecolortheme{seagull} \setbeamercovered{transparent} \usepackage{amsmath,amsfonts,amsthm, amssymb} \usepackage{dsfont, epsfig, graphics, diagrams, amssymb, amsfonts, euscript ,stmaryrd,float,graphicx, color} \usepackage{mathrsfs} %\usepackage[applemac]{inputenc} %\usepackage{pstricks,pst-node,pst-text} \usepackage[german,francais]{babel} \usepackage[latin1]{inputenc} \usepackage{graphicx} \graphicspath{{./images/}} \newcommand{\RR}{{\mathbb R}} \newcommand{\DD}{{\mathbb D}} \newcommand{\CC}{{\mathbb C}} \newcommand{\NN}{{\mathbb N}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\QQ}{{\mathds Q}} \newcommand{\EE}{{\mathds E}} \newcommand{\PP}{{\mathds P}} \newcommand{\MM}{{\mathfrak M}} \def\A{{\mathcal A}} \def\B{{\mathcal B}} \def\C{{\mathcal C}} \def\D{{\mathbb D}} \def\E{{\mathcal E}} \def\F{{\mathcal F}} \def\H{{\mathcal H}} \def\L{{\mathcal L}} \def\M{{\mathcal M}} \def\N{{\mathcal N}} \def\P{{\mathcal P}} \def\R{{\mathcal R}} \def\S{{\mathcal S}} \def\T{{\mathcal T}} \def\gap{\hbox{\rm gap}} \def\rep{\hbox{\rm rep}} \newcommand{\II}{{\mathcal I}} \newcommand{\YY}{{\mathcal Y}} \newcommand{\XX}{{\mathcal X}} \newcommand{\OO}{{\mathcal O}} \renewcommand{\geq}{\geqslant} \numberwithin{equation}{section} \newtheorem{theo}{Theorem } \newtheorem{prop}[theo]{Proposition} \newtheorem{coro}[theo]{Corollary} \newtheorem{conj}[theo]{Conjecture} \newtheorem{lem}[theo]{Lemma} \theoremstyle{definition} \newtheorem{defi}[theo]{Definition} \newtheorem{rema}[theo]{Remark} \newtheorem{question}[theo]{Question} \title{Generic vectors for the horocycle flow on abelian covers of compact hyperbolic surfaces} \date{Rennes, June, 10th 2008} \author{Omri Sarig, Barbara Schapira} %\institute{ Universit\'e d'Amiens} \begin{document} %%%%%%%%%%%%%%%%%% Transparent 0 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame} \maketitle \end{frame} %%%%%%%%%%%%%%%%%% Transparent 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} %%%%%%%%%%%%%%%%% \subsection{Generic vectors} \begin{frame} $(X,\mathcal{B},\mu,\phi)$ measured dynamical system\\ $\mu$ infinite $\sigma$-finite Radon measure\\ \begin{defi}$x\in X$ is said {\em generic} w.r.t. $\mu$ if for all $f,g\in C_c(X)$, with $\int g\,dm>0$, \\ \uncover<2-3>{ $$ \frac{\int_0^T f\circ \phi^s x\,ds}{\int_0^T g\circ \phi^s x\,ds}\to \frac{\int_X f\,d\mu}{\int_X g\,d\mu}\quad \mbox{when}\quad T\to\infty $$ } \end{defi} \uncover<3>{ \begin{rema} If $x$ is generic for $\mu$, it is generic for $c \mu$ for any constant $c>0$ \end{rema} } \end{frame} %%%%%%%%%%%%%%%%%%%%%% Transparent 2 %%%%%%%%%%%%%%%%%%%%% \subsection{Geodesic and horocycle flows} \begin{frame} On $T^1\DD$, $(g^t)_{t\in\RR}$ geodesic flow, $(h^s)_{s\in\RR}$ (stable) horocycle flow\\ % dessins $$ g^t\circ h^s=h^{se^{-t}}\circ g^t $$ $(h^sv)_{s\in\RR}=W^{ss}(v)$ strong stable manifold of $v$.% for the action of $(g^t)$. \\ \uncover<2-4>{ \begin{prop} $S_0$ {\em compact } hyperbolic surface. } \uncover<3-4>{ $(g^t)_{t\in\RR}$ is {\em hyperbolic} } %(it has infinitely many periodic orbits, ergodic measures...) } \uncover<4>{$(h^s)_{s\in\RR}$ is {\em minimal}, {\em ergodic}, {\em uniquely ergodic}. The invariant measure is the Liouville measure. (Hedlund 1936, Furstenberg 1973)} % Lebesgue \end{prop} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%% Transparent 3 %%%%%%%%%%%%%%%%%%%%%%% \subsection{Abelian covers of compact hyperbolic surfaces} \begin{frame} Let $\Gamma \triangleleft \Gamma_0$ be a normal subgroup of $\Gamma_0$. Assume that $\Gamma_0/\Gamma\simeq \ZZ^d$. %make a picture \begin{tabular}{cc} \uncover<2-4>{ \begin{diagram} T^1\DD & & \\ & \rdTo^{\Gamma} & \\ \dTo^{\Gamma_0} & & T^1S\\%=T^1\DD/\Gamma \\ & \ldTo^{\Gamma_0/\Gamma} & \\ T^1 S_0=T^1\D/\Gamma_0 & & \\ \end{diagram}} $\quad$ & \begin{minipage}{6cm} \uncover<3-4>{At large scale, the dynamic of $(g^t)_{t\in\RR}$ looks like a random walk on $\ZZ^d$.\\} \uncover<4>{Locally, the dynamic is hyperbolic.} \end{minipage} \end{tabular} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%% Transparent 4 %%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Asymptotic cycles and invariant measures} \begin{frame}%asymptotic cycles Let $F_0$ be a fundamental domain for the action of $\ZZ^d$ on $T^1 S$. If $w\in T^1 S$, there is a unique $\xi(g^tw)\in\ZZ^d$ s.t. $g^tw\in \xi(g^tw)\cdot F_0$. \uncover<2-4>{ \begin{defi} The {\em asymptotic cycle} of $w\in T^1 S$ is $\displaystyle\xi_\infty(w)=\lim_{t\to\infty}\frac{\xi(g^tw)}{t}\in\RR^d$. \end{defi}} %\uncover<3-5>{It is defined $m$-a.e., for $m$ any ergodic $(g^t)$-invariant measure on $T^1 S_0$.} \uncover<3-4>{ It is invariant by $(g^t)$, $(h^s)$, and $\ZZ^d$.\\} \uncover<4>{Let $C=\{$ asymptotic cycles $\}$. Closed convex set.\\ $Int(C)$ is diffeomorphic to $\RR^d$ (Babillot-Ledrappier 1998). } \end{frame} %%%%%%%%%%%%%%%%%%%%% Transparent 5 %%%%%%%%%%%%%%%%%%%%% \begin{frame}% Invariant measures $\M:=\{ (h^s)-\mbox{ergodic invariant Radon measures on } T^1 S$. \uncover<2-5>{ \begin{theo}[Babillot-Ledrappier, Sarig (02)] There is a surjective 1-1 correspondance $\Xi\in Int(C)\to m_\Xi\in\M$. Moreover, for $m_\Xi$-a.e. $w$, $\xi_\infty(w)=\Xi$. \end{theo}} \uncover<3-5>{ \begin{theo}[Sarig, Sch] $v\in T^1 S$ is generic for $m_\Xi$ iff $\xi_{\infty}(w)=\Xi\in Int(C)$. \end{theo}} %\uncover<4-5>{ %\begin{rema} $(h^s v)_{0\le s \le e^{T_n}}$ generic wrt $m_\Xi$ iff $\frac{\xi(g^{t_n}w)}{t_n}\to \Xi\in Int(C)$. %\end{rema} %} \uncover<4-5>{ If $\xi_{\infty}(v)\in\partial C$, $(h^s w)$ can be dense but not equidistributed. } \uncover<5>{ \begin{rema}[Eberlein] $(h^s v)_{s\in\RR}$ is {\em not} dense iff $d(g^tv,v)\ge t-constant$ when $t\to\infty$. It implies $\xi_{\infty}(v)\in\partial C$. \end{rema} } \end{frame} %%%%%%%%%%%%%%%%% \section{Around the result} %%%%%%%%%%%%%%%%%%%%%%%% Transparent 6 %%%%%%%%%%%%%%%%% \subsection{Thermodynamical formalism} %%% J'en suis la!! \begin{frame} \begin{defi}Let $m$ be a $(g^t)$-invariant measure on $T^1 S_0$. Its {\em average asymptotic cycle} is $\Xi(m)=\int_{T^1S_0}\xi_{\infty}(w)\,dm(w)$. \end{defi} \uncover<2-3>{ \begin{defi} Define the map $\displaystyle u\in\RR^d\mapsto P(u)=\sup\{h_m +\,\}$, where $m$: $(g^t)$-invariant measure on $T^1 S_0$. \end{defi}} \uncover<3>{ \begin{prop}[Babillot-Ledrappier 96] $P$ is analytic on $\RR^d$, and $\nabla P:\RR^d\to Int(C)$ is a diffeomorphism. \end{prop}} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%ù transparent 7 \begin{frame} Notation: $u=u_\varphi\in\RR^d$ $\leftrightarrow$ character $\varphi=\varphi_u:v\mapsto $. \uncover<2->{ Let $\mu_\varphi$ be the $(g^t)$ invariant ergodic measure s.t. $$ P(u_\varphi)=h_{\mu_\varphi}+\varphi(\Xi(\mu_\varphi))=h_{\mu_\varphi}+$$} \uncover<3->{It satisfies $\nabla P(u_\varphi)=\Xi(\mu_\varphi)=\Xi_\varphi$.} \uncover<4->{Moreover, $$\mu_\varphi-a.s. \quad \xi_\infty(v)=\Xi_\varphi.$$}% ergodicity of $\mu_\varphi$. %\uncover<5-6>{Define the map %$\displaystyle %\xi\in C\mapsto H(\xi):=\sup\{h_m(g^t),\,\Xi(m)=\xi\}\,$} % where $m$ varies among all $(g^t)$ invariant measures on $T^1 S_0$.} %\uncover<6>{We have $H(\Xi_\varphi)=h_{\mu_\varphi)$ and $u_\varphi=-\nabla H(u_\varphi)$. %It is a concave, smooth map. Moreover, $\displaystyle H(\Xi_\varphi)=h_{\mu_\varphi}(g)$, $u_\varphi=-\nabla H(\Xi_\varphi)$, $\lim_{\xi\to\partial C}H(\xi)=0$. } \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%% Transparent 8 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Construction of $(h^s)$-invariant measures} \begin{frame} % Construction of invariant measures Lift $\mu_\varphi$ to $\tilde{\mu}_\varphi$. We get a $\Gamma_0$- and $(g^t)$-invariant measure on $T^1 \D$. \uncover<2-6>{$T^1 \D\simeq S^1\times S^1\times \RR$, and $\displaystyle \tilde{\mu_\varphi}\sim \nu_\varphi\times\nu_\varphi\times dt$ \\ } \uncover<3-6>{ Define $\displaystyle \tilde{m}_\varphi\sim ds\times\nu_\varphi\times dt$ : $\Gamma$- and $(h^s)$-invariant measure on $T^1\D$. } \uncover<4-6>{On $T^1 S=T^1\D/\Gamma$, we get $m_\varphi$: infinite Radon measure, $(h^s)$-invariant, full support.} \uncover<5-6>{ It satisfies $$g^t_*m_\varphi=e^{-P(u_\varphi)t}m_\varphi, \quad\mbox{and}\quad D_*m_\varphi=e^{\varphi(D)}m_\varphi\quad\mbox{ for} \quad D\in \ZZ^d.$$ } \uncover<6>{ By construction, $$m_\varphi-a.s., \quad\xi_\infty(w)=\Xi_\varphi.$$} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%% Transparent 9 %%%%%%%%%%%%%%%%%%ù \subsection{The Bowen-Series coding of the geodesic flow} \begin{frame} %\begin{rema} Our result 'should' be true on abelian covers of surfaces with variable negative curvature. %But the proof uses the Bowen-Series coding of $(g^t)$, valid on hyperbolic surfaces. %\end{rema} \begin{lem}[Bowen-Series] The geodesic flow on $T^1 S_0$ is conjugate to a top. mixing special flow over a subshift of finite type. The conjugacy is $1-1$ everywhere except on {\bf countably} many geodesics. \end{lem} \uncover<2-4>{ \begin{lem}[Pollicott, Pollicott-Sharp, Babillot-Ledrappier] $(g^t)$ on $T^1S$ is conjugate to the action of $\RR$ by translation on $(\Sigma\times \ZZ^d\times\RR)/\sim$, where $(x,\xi,t)\sim (\sigma x, \xi+f(x), t-r(x))$, $f$ and $r$ H\"older. } \uncover<3-4>{ This conjugacy $\pi:(\Sigma\times \ZZ^d\times\RR)/\sim \to T^1 S$ is surjective and $1-1$ except on a {\bf countable} number of geodesics. } \uncover<4>{The symbolic strong stable manifolds are well understood. } %Moreover, for $w=\pi(x,\xi,t)$ and $w'=\pi(x',\xi',t')$, %if there exist $p,q\ge 0$ such that $(x)_p^\infty=(x')_q^\infty$, $t-t'=r^*_p(x)-r^*_q(x')$, $\xi-\xi'=f_q(x')-f_p(x)$, %then $w$ and $w'$ are on the same horocycle. } \end{lem} \end{frame} \section{Asymptotic cycle of a generic vector} %%%%%%%%%%%%%%%%%%%%%%%%%%%%% Transparent 10 %%%%%%%%%%%%%%%%%%ù \subsection{A short wrong proof} \begin{frame} Let $v$ be a generic vector w.r.t. $m_\Xi$. \uncover<2-9>{Show that $\xi_{\infty}(v)=\Xi$. } \uncover<3-9>{$*$ $F_0\subset T^1 S$ : fundamental domain for the action of $\ZZ^d$,\\ $*$ $\lambda_v^T$ : normalized orbital probability measure on $(h^sv)_{0\le s\le T}\cap F_0$.\\ } \uncover<4-9>{$*$ $m_\Xi$ is restricted to $F_0$ and normalized, to be considered as a probability measure on $F_0$. } \uncover<5-9>{ {\em $v$ generic} means that $\lambda_v^T\to m_\Xi$ when $T\to\infty$. } \begin{diagram}%{ccc} \uncover<6-9>{\int_{F_0}\frac{\xi(g^tw)}{t}d\lambda_v^T(w)} & \uncover<7-9>{ \rTo^{T\to +\infty} & \int_{F_0}\frac{\xi(g^tw)}{t}dm_{\Xi}(w)} \\ \uncover<8-9>{\dTo^{t\to\infty}} & & \uncover<9>{\dTo^{t\to\infty}}\\ \uncover<8-9>{ \int_{F_0}\xi_\infty(w)d\lambda_v^T=\xi_\infty(v) } & & \uncover<9>{ \int_{F_0}\xi_{\infty}dm_{\Xi}=\Xi}\\ \end{diagram} \end{frame} \subsection{Strategy of the proof } %%%%%%%%%%%%%%%%%%%%%%%%%%%%% Transparent 11 %%%%%%%%%%%%%%%%%%ù \begin{frame} $\bullet$ Fix $\varepsilon>0$. \uncover<2->{$\bullet$ $\xi_\infty=\Xi$ $m_\Xi$-a.s. so that $\exists N\in\NN$ s.t. $m_{\Xi}\{|\frac{\xi(g^N\omega)}{N}-\Xi|>\varepsilon\}\le \varepsilon$.} \uncover<3->{$\bullet$ Fix $N$. $\frac{\xi_N}{N}=\frac{\xi\circ g^N}{N}$ is piecewise continuous and $\lambda_T^v\to m_\Xi$, } \uncover<4->{ so that for $T$ large, $\lambda_T^v\{|\frac{\xi_N(\omega)}{N}-\Xi|>\varepsilon\}\le 2\varepsilon$. Fix $T$} \uncover<5->{ Observation : for $t>>T$, $\displaystyle \frac{\xi_t(w)}{t}\simeq \frac{\sum_{k=0}^{[t/N]}\xi_N(g^{kN}w)}{\sum_0^{[t/N]}N}$.} \uncover<6->{\begin{eqnarray*} |\frac{\xi_t(v)}{t}-\Xi| & \simeq & |\lambda_T^v(\frac{\xi_t(w)}{t}-\Xi)|\simeq |\lambda_T^v(\frac{\sum_{k=0}^{[t/N]}\xi_N(g^{kN}w)}{\sum_0^{[t/N]}N}-\Xi)|\\} \uncover<7->{ & \le & \sum_{k=0}^{[t/N]}\frac{N}{t}\lambda_T^v(|\frac{\xi_N(g^{kN}(v))}{N}-\Xi|)}\\ \uncover<8>{ &\le &\sup_{0\le k\le [t/N]}\lambda_T^v(|\frac{\xi_N(g^{kN}(v))}{N}-\Xi|)} \end{eqnarray*} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%% Transparent 12 %%%%%%%%%%%%%%%%%%ù \subsection{Holonomies on the horocycles} \begin{frame} \begin{lem}[Key lemma]There exists $C>0$ s.t. uniformly in $N\in\NN$, $k\in\NN$, $T\ge e^{kN}$, $\displaystyle \lambda_T^v\{|\frac{\xi_N(g^{kN}w)}{N}-\Xi|>\varepsilon\}\le C\varepsilon\,.$ \end{lem} \uncover<2->{We know that $\lambda_T^v\{|\frac{\xi_N(w)}{N}-\Xi|>\varepsilon\}\le 2\varepsilon$.} \uncover<3>{Construct holonomies of the horospheric foliation \begin{diagram} a &=&[a_0\dots a_{N-1}] &\dots \dots & [a_{kN} \dots a_{(k+1)N-1}] \dots \dots \\ \dTo^{\kappa_{N,k}} & & \\ \kappa_{N,K}(a)&=& [a_{kN} \dots a_{(k+1)N-1}] &\dots \dots & [a_0\dots a_{N-1}] \dots \dots \\ \end{diagram}} \end{frame} \begin{frame} This map preserves strong stable manifolds. \uncover<2->{It can be defined on $T^1 S\simeq (\Sigma\times \ZZ^d\times\RR)/\sim$ and it preserves horocycles.} \uncover<3->{$ \kappa_{N,k}$ switches $ \{|\frac{\xi_N(w)}{N}-\Xi|>\varepsilon\}$ and $\{|\frac{\xi_N(g^{kN}w)}{N}-\Xi|>\varepsilon\}$. } \uncover<4->{ Key point: it is absolutely continuous with uniformly bounded Radon-Nikodym derivatives. $$ \lambda_T^v\{|\frac{\xi_N(g^{kN}w)}{N}-\Xi|>\varepsilon\}=C^{\pm 1}\lambda_T^v\{|\frac{\xi_N(w)}{N}-\Xi|>\varepsilon\}\le C\varepsilon $$ } \end{frame} \section{Genericity of a vector with prescribed asymptotic cycle} \begin{frame} Let $v\in T^1 S$ s.t. $\xi_\infty(v)=\Xi\in Int(C)$. Show that $v$ is generic w.r.t. $m_\Xi$. \uncover<2->{The (symbolic) proof uses works of Lalley (87,89), Babillot-Ledrappier (96,98), Ledrappier-Sarig (05). } \uncover<3->{ \begin{prop} $\forall \varepsilon>0$, $\exists K_\varphi(\varepsilon)$ compact neighbourhood of $\Xi_\varphi$ s.t. for $f,g:T^1S\to \R_+$ continuous, and $T$ large, $$ \frac{\xi_{\ln T}(\omega)}{\ln T}\in K_\varphi(\varepsilon)\implies \frac{\int_0^{T}f\circ h^s\omega\,ds} {\int_0^{T}g\circ h^s\omega\,ds}=e^{\pm\varepsilon} \frac{\int_{T^1S}f\,dm_{\varphi}} {\int_{T^1S}g\,dm_{\varphi}}$$ \end{prop}} \uncover<4->{Method: give a precise asymptotic equivalent of $\int_O^T {\bf 1}_E\circ h^s\omega ds$ , for $E$ 'simple' sets. } \uncover<5->{Remark : the result allows to consider vectors s.t. $\frac{\xi_{\ln T}(w)}{\ln T}$ has several limit points.} \end{frame} \begin{frame} {\em Step 1:} $\displaystyle \int_0^T 1_E\circ h^s w\,ds\asymp \sum_{i=0}^N l(E\cap g^{-T^*}W^{ss}_{loc}(w_i^*))$, where \\ $W^{ss}_{loc}(x,\xi,t)=\{(x',\xi',t')\in W^{ss}(x,\xi,t), \,\, s.t.\,\, (x')_0^\infty=(x)_0^\infty\}$, $T^*=\ln(T/T_0)$, $N=N(T_0)$, $w_i^*=w_i^*(T)$.\\ \uncover<2->{{\em Step 2:} (Ledrappier-Sarig) From $\Sigma$ to $\Sigma^+$. If $w_i^*=\pi(x_i^*,\xi_i^*,t_i^*)$, $E=[a]\times \{\xi\}\times [\alpha,\beta]$, transform $l(E\cap g^{-T^*}W^{ss}_{loc}(w_i^*))$ to a symbolic sum over orbits of $\Sigma^+$ studied in Babillot-Ledrappier. } \uncover<3->{{\em Step 3} (Babillot-Ledrappier) : $\forall \varepsilon>0$, $\exists K_\varphi(\varepsilon)$, s.t for $T$ large, $$ \frac{\xi_i^*}{T^*}\in K_\varphi(\varepsilon)\implies l(E\cap g^{-T^*}W^{ss}_{loc}(w_i^*))\sim C e^{\pm C_\varphi\varepsilon}m_\varphi(E)\frac{e^{T^*H(\frac{\xi_i^*}{T^*})}}{(T^*)^{d/2}}$$ } \uncover<4->{{\em Step 4:} Divide $\int_0^T 1_E\circ h^s w\,ds$ by $\int_0^T 1_F\circ h^s w\,ds$, count the epsilon's, pass from $1_E$, $1_F$ to continuous maps, ...} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Methods and generalization} \begin{frame} \begin{question} Our proof uses the Bowen-Series coding. The statement {\em 'should'} be true on abelian covers of surfaces with variable negative curvature. More geometrical proof ? \end{question} \uncover<2-3>{ \begin{question} On regular (non abelian) covers of compact hyperbolic surfaces, the $(h^s)$-invariant ergodic measures are classified (Ledrappier-Sarig). Is it possible to characterize the generic points? \end{question}} \uncover<3>{ \begin{question} On other geometrically infinite surfaces ? Other dynamical systems ? \end{question}} % coding, constant curvature, BL result, \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\begin{figure}[!h] %\centerline{\includegraphics[height=8cm,width=7cm]{mafigure11.eps}} %\end{figure} \end{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yyyyyy