Let S be a nonexceptional oriented surface of finite type. We construct an uncountable family of probability measures on the space of area on holomorphic quadratic differentials over the moduli space for S containing the usual Lebesgue measure. These measures are invariant under the Teichmueller geodesic flow, and they are mixing, absolutely continuous with respect to the stable and unstable foliation adn exponentially recurrent to a compact set. Finally we show that the critical exponent of the mapping class group equals the dimension of the Teichmueller space for S. Moreover, this critical exponent coincides with the the logarithmic asymptotic of the number of closed Teichmueller geodesics in moduli space which meet a sufficiently large compact set.
Let Q be a connected component of a stratum in the space of quadratic differentials for a non-exceptional Riemann surface of finite type. We show that the probability measure on Q in the Lebesgue measure class which is invariant under the Teichmueller flow is obtained by Bowen's construction.
Let Q(S) be the moduli space of area one holomorphic quadratic differentials for an oriented surface S of genus g with m punctures and 3g-3+m>1. We show that the supremum over all compact subsets K of Q(S) of the asymptotic growth rate of the number of periodic orbits of the Teichmueller flow which are contained in K equals h=6g-6+2m. Moreover, h is also the supremum of the topological entropies of the restriction of the Teichmueller flow to compact invariant subsets of Q(S).
In this paper we give a geometric interpretation of the renormalization algorithm and of the continued fraction map that we introduced in arxiv:0905.0871 to give a characterization of symbolic sequences for linear flows in the regular octagon. We interpret this algorithm as renormalization on the Teichmueller disk of the octagon and explain the relation with Teichmueller geodesic flow. This connection is analogous to the classical relation between Sturmian sequences, continued fractions and geodesic flow on the modular surface. We use this connection to construct the natural extension and the …
For a non-exceptional oriented surface S let Q(S) be the moduli space of area one quadratic differentials. We show that there is a Borel subset E of Q(S) which is invariant under the Teichmueller flow F^t and of full measure for every invariant Borel probability measure, and there is a measurable conjugacy of the restriction of F^t to E into the Weil-Petersson flow. This conjugacy induces a continuous injection H of the space of invariant Borel probability measures for F^t into the space of invariant Borel probability measures for the Weil-Petersson flow. The map H is not surjective, but its i…
The logarithmic asymptotics for the growth of the number of periodic orbits, such that the norm of the corresponding renormalization matrix does not exceed a given constant, is computed for the Teichmueller flow on Veech's moduli space of zippered rectangles. The rate is equal to the entropy of the flow with respect to the absolutely continuous invariant measure.
We prove that on the typical translation surface the flow in almost every pair of directions are not isomorphic to each other and are in fact disjoint. It was not known if there were any translation surfaces other than torus covers with this property. We provide an application to the convergence of `circle averages' for the flow (away from a sequence of radii of density 0) for such surfaces. Even the density of a sequence of 'circles' was only known in a few special examples.
A stretched-exponential bound is obtained for the decay of correlations of the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations. A Corollary is the Central Limit Theorem for the Teichmueller flow on the moduli space of abelian differentials. The proof involves an approximation of the induction map by a Markov chain satisfying the Doeblin condition, the method of Sinai and Bunimovich--Sinai. The main estimate is Lemma 4. As a corollary of such "loss of memory" estimates, integrability of a small power of the Kontsevich-Zorich cocycle is obtained, thus yielding …
Consider a component Q of a stratum in the moduli space of area one abelian differentials on a surface of genus g. Call a property P for periodic orbits of the Teichmueller flow typical if the growth rate of orbits with this property is maximal. Typical are: The logarithms of the eigenvalues of the symplectic matrix defined by the orbit are arbitrarily close to the Lyapunov exponents of Q, and its trace field is a totally real splitting field of degree g over Q. If g>2 then periodic orbits whose SL(2,R)-orbit closure equals Q are typical. We also show that Q contains only finitely many algebra…
We extend Teichmueller dynamics to a flow on the total space of a flat bundle of deformation spaces of representations of the fundamental group of a fixed surface S in a Lie group G. The resulting dynamical system is a continuous version of the action of the mapping class group of S on the deformation space. We observe how ergodic properties of this action relate to this flow. When G is compact, this flow is strongly mixing over each component of the derormation space and of each stratum of the Teichmueller unit sphere bundle over the Riemann moduli space. We prove ergodicity for the analogous…
We introduce a twisted cohomology cocycle over the Teichmueller flow and prove a "spectral gap" for its Lyapunov spectrum with respect to the Masur-Veech measures. We then derive Hoelder estimates on spectral measures and bounds on the speed of weak mixing for almost all translation flows in every stratum of Abelian differentials on Riemann surfaces, as well as bounds on the deviation of ergodic averages for product translation flows on the product of a translation surface with a circle.
Let S be a non-exceptional oriented surface of finite type. We discuss the action of subgroups of the mapping class group of S on the CAT(0)-boundary of the completion of Teichmueller space with respect to the Weil-Petersson metric. We show that the set of invariant Borel probability measures for the Weil-Petersson flow on moduli space which are supported on closed orbits is dense in the space of invariant Borel probability measures.
If $(M,g)$ is a smooth compact rank $1$ Riemannian manifold without focal points, it is shown that the measure $μ_{\max}$ of maximal entropy for the geodesic flow is unique. In this article, we study the statistic properties and prove that this unique measure $μ_{\max}$ is mixing. Stronger conclusion that the geodesic flow on the unit tangent bundle $SM$ with respect to $μ_{\max}$ is Bernoulli is acquired provided $M$ is a compact surface with genus greater than one and no focal points.
A consequence of Ornstein theory is that the infinite entropy flows associated with Poisson processes and continuous-time irreducible Markov chains on a finite number of states are isomorphic as measure-preserving systems. We give an elementary construction of such an isomorphism that has an additional finitariness property, subject to the additional conditions that the Markov chain has a uniform holding rate and a mixing skeleton.
We study the Bernoulli property for a class of partially hyperbolic systems arising from skew products. More precisely, we consider a hyperbolic map $(T,M,μ)$, where $μ$ is a Gibbs measure, an aperiodic Hölder continuous cocycle $ϕ:M\to \mathbb R$ with zero mean and a zero-entropy flow $(K_t,N,ν)$. We then study the skew product $$ T_ϕ(x,y)=(Tx,K_{ϕ(x)}y), $$ acting on $(M\times N,μ\times ν)$. We show that if $(K_t)$ is of slow growth and has good equidistribution properties, then $T_ϕ$ remains Bernoulli. In particular, our main result applies to $(K_t)$ being a typical translation flow on a s…
We study skew products where the base is a hyperbolic automorphism of $\mathbb{T}^2$, the fiber is a smooth area preserving flow on $\mathbb{T}^2$ with one fixed point (of high degeneracy) and the skewing function is a smooth non coboundary with non-zero integral. The fiber dynamics can be represented as a special flow over an irrational rotation and a roof function with one power singularity. We show that for a full measure set of rotations the corresponding skew product is $K$ and not Bernoulli. As a consequence we get a natural class of volume-preserving diffeomorphisms of $\mathbb{T}^4$ wh…
Generalizing a result of Furstenberg, we show that for every infinite discrete group $G$, the Bernoulli flow $2^G$ is disjoint from every minimal $G$-flow. From this, we deduce that the algebra generated by the minimal functions $\mathfrak{A}(G)$ is a proper subalgebra of $\ell^\infty(G)$ and that the enveloping semigroup of the universal minimal flow $M(G)$ is a proper quotient of the universal enveloping semigroup $βG$. When $G$ is countable, we also prove that for any metrizable, minimal $G$-flow, there exists a free, minimal flow disjoint from it and that there exist continuum many mutuall…
We prove that the geodesic flow for the Weil-Petersson metric on the moduli space of Riemann surfaces is ergodic (in fact Bernoulli) and has finite, positive metric entropy.
Let $G$ be a semisimple Lie group with Haar measure $μ$ and let $Γ$ be an irreducible lattice in $G$. For $g\in G$, we consider left translation $L_g$ acting on $(G\backslashΓ,μ)$. We show that if $L_g$ is $K$ (which is equivalent to positive entropy of $L_g$) then $L_g$ is a Bernoulli automorphism. As a corollary, we also obtain analogous results for homogeneous flows.
Interval exchange maps are related to geodesic flows on translation surfaces; they correspond to the first return maps of the vertical flow on a transverse segment. The Rauzy-Veech induction on the space of interval exchange maps provides a powerful tool to analyze the Teichmueller geodesic flow on the moduli space of Abelian differentials. Several major results have been proved using this renormalization. Danthony and Nogueira introduced in 1988 a natural generalization of interval exchange transformations, namely the linear involutions. These maps are related to general measured foliations o…
We describe the scaling scenery associated to Bernoulli measures supported on separated self-affine sets under the condition that certain projections of the measure are absolutely continuous.
Let $\{T^t\}$ be a smooth flow with positive speed and positive topological entropy on a compact smooth three dimensional manifold, and let $μ$ be an ergodic measure of maximal entropy. We show that either $\{T^t\}$ is Bernoulli, or $\{T^t\}$ is isomorphic to the product of a Bernoulli flow and a rotational flow. Applications are given to Reeb flows.
Tichler proved that a manifold admitting a smooth closed one-form fibers over a circle. More generally a manifold admitting $k$ independent closed one-forms fibers over a torus $T^k$. In this article we explain a version of this construction for manifolds with boundary using the techniques of $b$-calculus. We explore new applications of this idea to Fluid Dynamics and more concretely in the study of stationary solutions of the Euler equations. In the study of Euler flows on manifolds, two dichotomic situations appear. For the first one, in which the Bernoulli function is not constant, we provi…
In two fundamental classical papers, Masur and Veech have independently proved that the Teichmueller geodesic flow acts ergodically on each connected component of each stratum of the moduli space of quadratic differentials. It is therefore interesting to have a classification of the ergodic components. Veech has proved that these strata are not necessarily connected. In a recent work, Kontsevich and Zorich have completely classified the components in the particular case where the quadratic differentials are given by the global square of Abelian differentials. Here we are interested in the comp…
Given a tree $T$ and a group $\Ga$ of automorphisms of $T$, we study the markovian properties of the geodesic flow on the quotient by $\Ga$ of the space of geodesics of $T$. For instance, when $T$ is the Bruhat-Tits tree of a semi-simple connected algebraic group $\underline{G}$ of rank one over a non archimedian local field $\wh K$, and $\Ga$ is a (possibly non uniform) lattice in $\underline{G}(\wh K)$, we prove that the type preserving geodesic flow is Bernoulli with finite entropy. Under some mild assumptions, we prove that if the quotient geodesic flow is mixing for a probability Patterso…
We use the relation between the volumes of the strata of meromorphic quadratic differentials with at most simple poles on the Riemann sphere and counting functions of the number of (bands of) closed geodesics in associated flat metrics with singularities to prove a very explicit formula for the volume of each such stratum conjectured by M. Kontsevich a decade ago. The proof is based on a formula for the Lyapunov exponents of the geodesic flow on the moduli space, which gives a recursive formula from which the volumes could be recovered. An appendix with an ergodic theorem proved by Jon Chaika …
The squarefree flow is a natural dynamical system whose topological and ergodic properties are closely linked to the behavior of squarefree numbers. We prove that the squarefree flow carries a unique measure of maximal entropy and express this measure explicitly in terms of a skew-product of a Kronecker and a Bernoulli system. Using this characterization and a number-theoretic argument, we then show that the unique maximum entropy measure fails to possess the Gibbs property.
In this paper, we study dynamics of geodesic flows over closed surfaces of genus greater than or equal to 2 without focal points. Especially, we prove that there is a large class of potentials having unique equilibrium states, including scalar multiples of the geometric potential, provided the scalar is less than 1. Moreover, we discuss ergodic properties of these unique equilibrium states. We show these unique equilibrium states are Bernoulli, and weighted regular periodic orbits are equidistributed relative to these unique equilibrium states.
Equilibrium states for geodesic flows over closed rank 1 manifolds were studied recently by Burns, Climenhaga, Fisher and Thompson. For sufficiently regular potentials, it was shown that if the singular set does not carry full pressure then the equilibrium state is unique. The main result of this paper is that these equilibrium states have the Kolmogorov property. In particular, these measures are mixing of all orders and have positive entropy. For the Bowen-Margulis measure, we go further and obtain the Bernoulli property from the K-property using classic arguments from Ornstein theory. Our a…
In this article, we consider a closed rank one Riemannian manifold $M$ without focal points. Let $P(t)$ be the set of free-homotopy classes containing a closed geodesic on $M$ with length at most $t$, and $\# P(t)$ its cardinality. We obtain the following Margulis-type asymptotic estimates: \[\lim_{t\to \infty}\#P(t)/\frac{e^{ht}}{ht}=1\] where $h$ is the topological entropy of the geodesic flow. In the appendix, we also show that the unique measure of maximal entropy of the geodesic flow has the Bernoulli property.
The paper gives first quantitative estimates on the modulus of continuity of the spectral measure for weak mixing suspension flows over substitution automorphisms, which yield information about the "fractal" structure of these measures. The main results are, first, a Hoelder estimate for the spectral measure of almost all suspension flows with a piecewise constant roof function; second, a log-Hoelder estimate for self-similar suspension flows; and, third, a Hoelder asymptotic expansion of the spectral measure at zero for such flows. Our second result implies log-Hoelder estimates for the spect…
