Ursula Hamenstaedt
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.
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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 e…
Symbolic dynamics for the Teichmueller flow
本文通过构造有限型子移位及其Borel悬浮，建立到Teichmüller流的有限对一半共轭，从而证明分层分支上的Lebesgue测度是唯一最大熵测度。
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 …
Sinai-Ruelle-Bowen measure Entropy of geodesic flow on Convex Projective Surfaces
该研究聚焦凸实射影曲面上测地流的Sinai-Ruelle-Bowen测度熵，推导了熵与Hilbert面积、Blaschke度量下面积的极限关联特性。
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…
We show that the smooth measure is the unique measure of maximal entropy for the Teichmueller flow on the moduli space of abelian differentials.
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 study the Bowen topological entropy of generic and irregular points for certain dynamical systems. We define the topological entropy of noncompact sets for flows, analogous to Bowen's definition. We show that this entropy coincides with the Bowen topological entropy of the time-1 map on any set. We also show a Bowen's inequality for flows; namely, that the metric entropy with respect to every invariant measure for a continuous flow is an upper bound for the topological entropy of the set of generic points with respect to the same measure, and the equality is always true if the measure is er…
The Bernoulli property of Sinai-Ruelle-Bowen measures for flows
研究$C^{1+β}$流的弱混合超几何SRB测度的伯努利性质，证明其具有伯努利性质。
Let $(X, ϕ)$ be a compact metric flow without fixed points. We will be concerned with the entropy of flows which takes into consideration all possible reparametrizations of the flows. In this paper, by establishing the Brin-Katok's entropy formula for flows without fixed points in non-ergodic case, we prove the following result: for an ergodic $ϕ$-invariant measure $μ$, $$ h_{top}^{B}(ϕ, G_μ(ϕ))=h_μ(ϕ_{1}),$$ where $G_μ(ϕ)$ is the set of generic points for $μ$ and $h_{top}^{B}(ϕ, G_μ(ϕ))$ is the Bowen entropy on $G_μ(ϕ)$. This extends the classical result of Bowen in 1973 to fixed-point free f…
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 …
Let $X$ be a proper, geodesically complete CAT(0) space under a proper, non-elementary, isometric action by a group $Γ$ with a rank one element. We construct a generalized Bowen-Margulis measure on the space of unit-speed parametrized geodesics of $X$ modulo the $Γ$-action. Although the construction of Bowen-Margulis measures for rank one nonpositively curved manifolds and for CAT(-1) spaces is well-known, the construction for CAT(0) spaces hinges on establishing a new structural result of independent interest: Almost no geodesic, under the Bowen-Margulis measure, bounds a flat strip of any po…
We show that for many strata of Abelian differentials in low genus the sum of Lyapunov exponents for the Teichmueller geodesic flow is the same for all Teichmueller curves in that stratum, hence equal to the sum of Lyapunov exponents for the whole stratum. This behavior is due to the disjointness property of Teichmueller curves with various geometrically defined divisors on moduli spaces of curves.
Finiteness of Bowen-Margulis-Sullivan Measure for Gromov-Patterson-Sullivan Systems
本文提出强正递（SPR）性质，证明满足该性质的收敛群在关联流空间上具有有限的Bowen-Margulis-Sullivan测度，扩展了相对Anosov群的有限测度例子。
We study the geodesic flow of a class of 3-manifolds introduced by Benoist which have some hyperbolicity but are non-Riemannian, not CAT(0), and with non-C^1 geodesic flow. The geometries are nonstrictly convex Hilbert geometries in dimension three which admit compact quotient manifolds by discrete groups of projective transformations. We prove the Patterson-Sullivan density is canonical, with applications to counting, and construct explicitly the Bowen-Margulis measure of maximal entropy. The main result of this work is ergodicity of the Bowen-Margulis measure.
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 study expansive properties for the geodesic and horocycle flows on compact Riemann surfaces of constant negative curvature. It is well-known that the geodesic flow is expansive in the sense of Bowen-Walters and the horocycle flow is positive and negative separating in the sense of Gura. In this paper, we give a new proof of the expansiveness in the sense of Bowen-Walters for the geodesic flow and show that the horocycle flow is positive and negative kinematic expensive in the sense of Artigue as well as expansive in the sense of Katok-Hasselblatt but not expensive in the sense of Bowen-Walt…
We give an algebraic way of distinguishing the components of the exceptional strata of quadratic differentials in genus three and four. The complete list of these strata is (9, -1), (6,3,-1), (3,3,3, -1) in genus three and (12), (9,3), (6,6), (6,3,3) and (3,3,3,3) in genus four. This result is part of a more general investigation of disjointness of Teichmueller curves with divisors of Brill-Noether type on the moduli space of curves. As a result we show that for many strata of quadratic differentials in low genus the sum of Lyapunov exponents for the Teichmueller geodesic flow is the same for …
On the Myrberg Limit Sets and Bowen-Margulis-Sullivan Measures for Visibility Manifolds without Conjugate Points
本文研究了无共轭点的一致可见流形上Myrberg型动力学与测地流遍历性质的关系，证明了Myrberg极限集的Patterson-Sullivan测度正性等价于测地流的保守性，并将双曲流形的经典结果推广至无共轭点流形。
Using an approach due to Bowen, Franco showed that continuous expansive flows with specification have unique equilibrium states for potentials with the Bowen property. We show that this conclusion remains true using weaker non-uniform versions of specification, expansivity, and the Bowen property. We also establish a corresponding result for homeomorphisms. In the homeomorphism case, we obtain the upper bound from the level-2 large deviations principle for the unique equilibrium state. The theory presented in this paper provides the basis for an ongoing program to develop the thermodynamic for…
Topological and Metric Pressure for Singular Flows
本文通过引入重标度度量压和拓扑压的概念消除奇异性影响，证明了不同Bowen球定义的等价性及压的Katok公式，并建立了部分变分原理。
Entropy on regular sets and periodic-orbit growth for singular flows
该研究针对度量空间上的流，证明紧度量空间上流的有限拓扑熵由正则集紧子集承载，建立Bowen–Walters不等式并将其应用于多类奇异流，扩展了相关文献的结论。
Is every irreducible shift of finite type flow equivalent to a renewal system? For the first time, this variation of a classic problem formulated by Adler is investigated, and several partial results are obtained in an attempt to find the range of the Bowen--Franks invariant over the set of renewal systems of finite type. In particular, it is shown that the Bowen--Franks group is cyclic for every member of a class of renewal systems known to attain all entropies realised by shifts of finite type, and several classes of renewal systems with non--trivial values of the invariant are constructed.
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.
We propose a general framework for studying pseudo-Anosov homeomorphisms on translation surfaces. This new approach, among other consequences, allows us to compute the systole of the Teichmueller geodesic flow restricted to the hyperelliptic connected components, settling a question of Farb. We stress that all proofs and computations are performed without the help of a computer. As a byproduct, our methods give a way to describe the bottom of the lengths spectrum of the hyperelliptic components.
Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization
本文针对相对Anosov子群，通过Groves-Manning空间测地流重参数化流空间，证明了Bowen-Margulis-Sullivan测度有限、唯一且强混合，推广了Kleinian群经典结果至高秩。
Topological flows for hyperbolic groups
本文证明非初等双曲群上强双曲度量的Bowen-Margulis流是边界平方上唯一群不变且具最大Hausdorff维数的Radon测度类，并用于刻画粗略相似度量。
Exponential mixing of the Teichmüller flow on affine invariant manifolds
本文针对有限型非例外可定向曲面，利用符号编码方法重新证明了 Avila 和 Gouezel 的结果：Teichmüller 流关于任意遍历 SL(2,R)-不变 Borel 概率测度具有指数混合性。
Let $M=X/Γ$ be a geometrically finite negatively curved manifold with fundamental group $Γ$ acting on $X$ by isometries. The purpose of this paper is to study the mixing property of the geodesic flow on $T^1M$, the asymptotic equivalent as $R\longrightarrow+\infty$ of the number of closed geodesics on $M$ of length less than $R$ and of the orbital counting function $\sharp\{γ\inΓ |\ d(o,γ.o)\le R\}$. These properties are well known when the Bowen-Margulis measure on $T^1M$ is finite. We consider here divergent Schottky groups whose Bowen-Margulis measure is infinite and ergodic, and we precise…
It is well-known that SRB and equilibrium measures for uniformly hyperbolic flows admit a product structure in terms of measures on stable and unstable leaves with scaling properties given by the potential function. We describe a construction of these leaf measures analogous to the definition of Hausdorff measure, relying on the Pesin-Pitskel' description of topological pressure as a dimensional characteristic using Bowen balls. These leaf measures were constructed for discrete-time systems by the author, Ya. Pesin, and A. Zelerowicz. In the continuous-time setting here, the description of the…
We prove effective equidistribution of horospherical flows in $\operatorname{SO}(n,1)^\circ / Γ$ when $Γ$ is geometrically finite and the frame flow is exponentially mixing for the Bowen-Margulis-Sullivan measure. We also discuss settings in which such an exponential mixing result is known to hold. As part of the proof, we show that the Patterson-Sullivan measure satisfies some friendly-like properties when $Γ$ is geometrically finite.
Expansiveness for flows on noncompact spaces
本文引入拓扑扩张流概念，证明其共轭不变性及与紧致情形的联系，刻画其结构，并推广Bowen-Walters不等式至非紧空间。
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…
Let C be a locally convex subset of a negatively curved Riemannian manifold M. We define the skinning measure on the outer unit normal bundle to C in M by pulling back the Patterson-Sullivan measures at infinity, and give a finiteness result of skinning measures, generalising the work of Oh and Shah, with different methods. We prove that the skinning measures, when finite, of the equidistant hypersurfaces to C equidistribute to the Bowen-Margulis measure on the unit tangent bundle of M, assuming only that the Bowen-Margulis measure is finite and mixing for the geodesic flow. Under additional a…
Let $G$ be a connected simple linear Lie group of rank one, and let $Γ<G$ be a discrete Zariski dense subgroup admitting a finite Bowen-Margulis-Sullivan measure $m^{\operatorname{BMS}}$. We show that the right translation action of the one dimensional diagonalizable subgroup is mixing on $(Γ\backslash G, m^{\operatorname{BMS}})$. Together with the work of Roblin, this proves ergodicity of the Burger-Roblin measure under the horospherical group $N$, establishes a classification theorem for $N$ invariant Radon measures on $Γ\backslash G$, and provides precise asymptotics for the Haar measure ma…
Thermodynamic formalism for Quasimorphisms: Bounded Cohomology and Statistics
本文发展了用于研究紧负曲率空间基本群准同态空间的热力学框架，证明该空间与关联的Gromov测地流的Bowen函数空间在Banach空间上同构，并建立准同态的中心极限定理和不变原理。
Ergodic dichotomy for subspace flows in higher rank
本文研究高秩子空间流的遍历性，提出类似 Hopf-Tsuji-Sullivan 二分性的判据，并推广到任意维数连通对角子群，为一般 Anosov 子群建立余维数二分性，推广了 Burger-Landesberg-Lee-Oh 的工作。
Let $Γ$ be a geometrically finite discrete subgroup in $\operatorname{SO}(d+1,1)^{\circ}$ with parabolic elements. We establish exponential mixing of the geodesic flow on the unit tangent bundle $\operatorname{T}^1(Γ\backslash \mathbb{H}^{d+1})$ with respect to the Bowen-Margulis-Sullivan measure, which is the unique probability measure on $\operatorname{T}^1(Γ\backslash \mathbb{H}^{d+1})$ with maximal entropy. As an application, we obtain a resonance free region for the resolvent of the Laplacian on $Γ\backslash \mathbb{H}^{d+1}$. Our approach is to construct a coding for the geodesic flow an…
Equidistribution of common perpendiculars in negative curvature
本文证明了在夹负曲率黎曼流形中，当Bowen-Margulis测度有限且混合时，公共垂线上的Lebesgue测度等分布到该测度，并给出了局部对称空间下的误差项及加权势函数的等分布结果。
Let S be an ergodic measure-preserving automorphism on a non-atomic probability space, and let T be the time-one map of a topologically weak mixing suspension flow over an irreducible subshift of finite type under a Holder ceiling function. We show that if the measure-theoretic entropy of S is strictly less than the topological entropy of T, then there exists an embedding from the measure-preserving automorphism into the suspension flow. As a corollary of this result and the symbolic dynamics for geodesic flows on compact surfaces of negative curvature developed by Bowen and Ratner, we also ob…
SRB measures for Anosov actions
本文研究闭流形上一般Anosov $\u211d^κ$作用的一类由Ruelle-Taylor共振理论定义的不变测度，证明其具备SRB测度的核心性质，并给出正Weyl腔作用传递时测度唯一且满支撑的结论。
In 2004, Taubes introduced the space of minimal hyperbolic germs with elements consisting of the first and second fundamental form of an equivariant immersed minimal disk in hyperbolic 3-space. Herein, we initiate a further study of this space by studying the behavior of a dynamically defined function which records the entropy of the geodesic flow on the associated Riemannian surface. We provide a useful estimate on this function which, in particular, yields a new proof of Bowen's theorem on the rigidity of the Hausdorff dimension of the limit set of quasi-Fuchsian groups. These follow from ne…
Let $X$ be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank one axis. We prove the Bowen-Margulis measure on the space of geodesics is the unique measure of maximal entropy for the geodesic flow, which has topological entropy equal to the critical exponent of the Poincaré series.
Equidistribution of divergent geodesics in negative curvature
本文研究非紧有限体积负曲率流形及几何有限树商空间中的发散测地线，证明其随复杂度增大分别趋向最大熵测度与 Bowen–Margulis 测度等分布。
Limits of geodesic push-forwards of horocycle invariant measures
本文证明SL(2,R)作用中horocycle与geodesic子群遍历平均的条件收敛，改进测地推进弱收敛，并给出Birkhoff一般性与Oseledets正则性较弱版本的简短证明。
Periodic orbits in the thin part of strata
该研究针对满足条件的闭定向曲面上的二次或阿贝尔微分模空间层的连通分支，证明了模空间紧子集补集内泰希米勒流周期轨道数的渐近增长率下界为层复维数减一。
On a compact manifold of any dimension $d\geq 3$, we show that joint non-integrability of the stable and unstable foliation of a hyperbolic attractor with one-dimensional expanding direction, for a vector field of class $C^2$, implies exponential mixing with respect to its physical measure. Consequently, the set of Axiom A vector fields which mix exponentially with respect to the physical measure of its non-trivial attractors contains a $C^1$-open and $C^2$-dense subset of the set of all Axiom A vector fields. Moreover, for codimension one $C^2$ Anosov flows in any dimension $d\geq 3$, if the …
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 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…
We prove that a singular-hyperbolic attractor of a 3-dimensional flow is chaotic, in two strong different senses. Firstly, the flow is expansive: if two points remain close for all times, possibly with time reparametrization, then their orbits coincide. Secondly, there exists a physical (or Sinai-Ruelle-Bowen) measure supported on the attractor whose ergodic basin covers a full Lebesgue (volume) measure subset of the topological basin of attraction. Moreover this measure has absolutely continuous conditional measures along the center-unstable direction, is a $u$-Gibbs state and an equilibrium …
Local mixing of one-parameter diagonal flows on Anosov homogeneous spaces
本文证明了连通半单实代数群上Zariski稠密Anosov子群商空间中，单参数对角流关于Bowen--Margulis--Sullivan测度的局部混合，给出了带横截偏差的相关函数指数衰减一致界，并支撑了矩阵系数衰减的渐近公式应用。
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 …
Exponential mixing of frame flows for convex cocompact locally symmetric spaces
本研究针对秩1连通无中心实单代数群的扎里斯基稠密无挠凸紧子群，证明其对应的标架流关于Bowen-Margulis-Sullivan测度具有指数混合性，推广并强化了已有相关结果。
Strong symbolic dynamics for geodesic flow on CAT(-1) spaces and other metric Anosov flows
该研究证明紧致或凸余紧局部CAT(-1)空间上的测地流可通过带Hölder屋顶函数的不可约有限型移位悬挂流编码，且其技术可推广到射影Anosov表示的测地流，还得到了关于平衡测度的强动力学结果。
Gibbs measures for geodesic flow on CAT(-1) spaces
本文针对 CAT(-1) 空间上的测地流，利用加权拟共形 Patterson 密度构造具有局部乘积结构的 Gibbs 测度，并在有限情形下证明其唯一平衡态性质，且无需势函数在共享线段上取值相同的条件。
Properly discontinuous actions, growth indicators, and conformal measures for transverse subgroups
本文研究连通半单实代数群的横向子群，证明其在 Weyl chamber 流空间上的作用是真正间断的，定义了 Bowen-Margulis-Sullivan 测度，建立了遍历二分法，并引入增长指标扩展了 Quint 的工作。
We investigate the distribution of orbits of a non-elementary discrete hyperbolic group acting on the n-dimensional hyperbolic space and its geometric boundary. In particular, we show that if the group $Γ$ admits a finite Bowen-Margulis-Sullivan measure (for instance, if it is geometrically finite), then every $Γ$-orbit in the hyperbolic space is equidistributed with respect to the Patterson-Sullivan measure supported on the limit set of $Γ$.The appendix by Maucourant is the extension of a part of his thesis where he obtains the same result as a simple application of Roblin's theorem. Our appr…
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