Discrete conformal groups and measurable dynamics (original) (raw)
The Weil-Petersson geodesic flow is ergodic
Annals of Mathematics, 2012
We prove that the geodesic flow for the Weil-Petersson metric on the moduli space of Riemann surfaces is ergodic (and in fact Bernoulli) and has finite, positive metric entropy. 1. Background on Teichmüller theory, Quasifuchsian space, and Weil-Petersson geometry Much of the discussion in this section is based on McMullen's paper [24]. Useful background can be found in [27] and the course notes [23].
Dynamics of geodesic flows on hyperbolic compact surfaces with some elliptic points
Dynamical Systems, 2014
We extend the results in [R. Adler, L. Flatto, Geodesic flows, interval maps and symbolic dynamics, Bull Am Math Soc. 1975;25(42): 229–334.] to compact surfaces of genus greater than one and with some elliptic points. Similar to hyperbolic case, a map TR conjugate to the cross-section map and defined on R, a union of finite rectangles in the plane, exists. This gives rise to a subshift of finite type and a sofic shift and their projections to axis known as Bowen–Series maps. Additionally, we will give a simple geometrical presentation of synchronizing and Markov words for , and will show that when considered as a sofic map, it is not of almost finite type. Moreover, if h is the entropy of any of those shift spaces, then ln (8g + 2r − 7) ≤ h ≤ ln (8g + 2r − 5), where g is the genus and r is the number of elliptic points.
Geodesic flows in manifolds of nonpositive curvature
Smooth Ergodic Theory and Its Applications, 2001
Introduction-a quick historical survey of geodesic flows on negatively curved spaces. II. Preliminaries on Riemannian manifolds A. Riemannian metric and Riemannian volume element B. Levi Civita connection and covariant differentiation along curves C. Parallel translation of vectors along curves D. Curvature E. Geodesics and geodesic flow F. Riemannian exponential map and Jacobi vector fields G. Isometries and local isometries H. Geometry of the tangent bundle with the Sasaki metric III. Manifolds of nonpositive sectional curvature A. Definition of nonpositive curvature by triangle comparisons B. Growth of Jacobi vector fields C. The Riemannian exponential map is a covering map. Theorem of Cartan-Hadamard. D. Examples : Riemannian symmetric spaces E. Convexity properties and the Cartan Fixed Point Theorem F. Fundamental group of a nonpositively curved manifold. G. Rank of a nonpositively curved manifold IV. Sphere at infinity of a simply connected manifold of nonpositive sectional curvature A. Asymptotic geodesics and cone topology for M (∞) B. Busemann functions and horospheres _______________________________________ Supported in part by NSF Grant DMS-9625452 2 C. Extension of isometries to homeomorphisms of the sphere at infinity. D. Relating the action of the geodesic flow of M on T 1 M to the action of π 1 (M) on M (∞) V. Measures on the sphere at infinity A. Harmonic measures {ν p : p ∈ M } B. Patterson-Sullivan measures {µ p : p ∈ M } C. Lebesgue measures {λ p : p ∈ M } D. Barycenter map for probability measures. VI. Anosov foliations in the unit tangent bundle T 1 M A. Stable and unstable Jacobi vector fields. B. The stable and unstable foliations E s and E u in T(T 1 M) C. The strong stable and strong unstable foliations E ss and E uu in T(T 1 M). D. Conditions for the foliations E ss and E uu to be Anosov. VII. Some outstanding problems of geometry and dynamics A. The Katok entropy conjecture B. Smoothness of Anosov foliations and Riemannian symmetric spaces C. The geodesic conjugacy problem D. Harmonic and asymptotically harmonic spaces E. Early partial solutions. VIII. The work of Besson-Courtois-Gallot A. Statement of the main result. B. Corollaries of the main result. C. Sketch of the proof of the main result. IX. References I.
In 1954, F. Mautner gave a simple representation theoretic argument that for compact surfaces of constant negative curvature, invariance of a function along the geodesic flow implies invariance along the horocycle flows (these are facts which imply ergodicity of the geodesic flow itself), [M]. Many generalizations of this Mautner phenomenon exist in representation theory, [St1]. Here, we establish a new generalization, Theorem 2.1, whose novelty is mostly its method of proof, namely the Anosov-Hopf ergodicity argument from dynamical systems. Using some structural properties of Lie groups, we also show that stable ergodicity is equivalent to the unique ergodicity of the strong stable manifold foliations in the context of affine diffeomorphisms.
New criteria for ergodicity and non-uniform hyperbolicity
2009
In this work we obtain a new criterion to establish ergodicity and non-uniform hyperbolicity of smooth measures of diffeomorphisms. This method allows us to give a more accurate description of certain ergodic components. The use of this criterion in combination with topological devices such as blenders lets us obtain global ergodicity and abundance of non-zero Lyapunov exponents in some contexts. In the partial hyperbolicity context, we obtain that stably ergodic diffeomorphisms are C^1-dense among volume preserving partially hyperbolic diffeomorphisms with two-dimensional center bundle. This is motivated by a well known conjecture of C. Pugh and M. Shub.
Topological dynamics of the Weil–Petersson geodesic flow
Advances in Mathematics, 2010
We prove topological transitivity for the Weil-Petersson geodesic flow for real two-dimensional moduli spaces of hyperbolic structures. Our proof follows a new approach that combines the density of singular unit tangent vectors, the geometry of cusps and convexity properties of negative curvature. We also show that the Weil-Petersson geodesic flow has: horseshoes, invariant sets with positive topological entropy, and that there are infinitely many hyperbolic closed geodesics, whose number grows exponentially in length. Furthermore, we note that the volume entropy is infinite. * 2000 Mathematics Subject Classification Primary: 37D40, 32G15; Secondary: 53D25.
On hyperbolic measures and periodic orbits
Discrete and Continuous Dynamical Systems, 2006
We prove that if a diffeomorphism on a compact manifold preserves a nonatomic ergodic hyperbolic Borel probability measure, then there exists a hyperbolic periodic point such that the closure of its unstable manifold has positive measure. Moreover, the support of the measure is contained in the closure of all such hyperbolic periodic points. We also show that if an ergodic hyperbolic probability measure does not locally maximize entropy in the space of invariant ergodic hyperbolic measures, then there exist hyperbolic periodic points that satisfy a multiplicative asymptotic growth and are uniformly distributed with respect to this measure.