scholarly journals Semi-rigidity of horocycle flows over compact surfaces of variable negative curvature

1987 ◽  
Vol 7 (1) ◽  
pp. 49-72 ◽  
Author(s):  
J. Feldman ◽  
D. Ornstein
Keyword(s):  
Invariant Measure ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Compact Surface ◽  
Maximal Entropy ◽  
Unit Tangent Bundle ◽  
Compact Surfaces ◽  
Measure Of Maximal Entropy

AbstractLet g be the geodesic flow on the unit tangent bundle of a C3 compact surface of negative curvature. Let μ be the g-invariant measure of maximal entropy. Let h be a uniformly parametrized flow along the horocycle foliation, i.e., such a flow exists, leaves μ invariant, and is unique up to constant scaling of the parameter (Margulis). We show that any measure-theoretic conjugacy: (h, μ) → (h′, μ′) is a.e. of the form θ, where θ is a homeomorphic conjugacy: g → g′. Furthermore, any homeomorphic conjugacy g → g′; must be a C1 diffeomorphism.

scholarly journals Flexibility of measure-theoretic entropy of boundary maps associated to Fuchsian groups

2021 ◽  
pp. 1-13
Author(s):  
ADAM ABRAMS ◽  
SVETLANA KATOK ◽  
ILIE UGARCOVICI
Keyword(s):  
Invariant Measure ◽  
Explicit Formula ◽  
Negative Curvature ◽  
Compact Surface ◽  
Fuchsian Groups ◽  
Maximal Entropy ◽  
Constant Negative Curvature ◽  
Boundary Map ◽  
Fundamental Polygon ◽  
Measure Of Maximal Entropy

Abstract Given a closed, orientable, compact surface S of constant negative curvature and genus $g \geq 2$ , we study the measure-theoretic entropy of the Bowen–Series boundary map with respect to its smooth invariant measure. We obtain an explicit formula for the entropy that only depends on the perimeter of the $(8g-4)$ -sided fundamental polygon of the surface S and its genus. Using this, we analyze how the entropy changes in the Teichmüller space of S and prove the following flexibility result: the measure-theoretic entropy takes all values between 0 and a maximum that is achieved on the surface that admits a regular $(8g-4)$ -sided fundamental polygon. We also compare the measure-theoretic entropy to the topological entropy of these maps and show that the smooth invariant measure is not a measure of maximal entropy.

Lefschetz formulae for Anosov flows on 3-manifolds

1993 ◽  
Vol 13 (2) ◽  
pp. 335-347 ◽  
Author(s):  
Héctor Sánchez-Morgado
Keyword(s):  
Fundamental Group ◽  
Unitary Representation ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Constant Negative Curvature ◽  
Closed Orbits ◽  
Unit Tangent Bundle ◽  
Anosov Flows

AbstractFried has related closed orbits of the geodesic flow of a surface S of constant negative curvature to the R-torsion for a unitary representation of the fundamental group of the unit tangent bundle T1S. In this paper we extend those results to transitive Anosov flows and 2-dimensional attractors on 3-manifolds.

scholarly journals Geodesic flows on manifolds of negative curvature with smooth horospheric foliations

1991 ◽  
Vol 11 (4) ◽  
pp. 653-686 ◽  
Author(s):  
Renato Feres
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Geodesic Flows ◽  
Constant Negative Curvature ◽  
Unit Tangent Bundle ◽  
Tangent Bundles ◽  
Negative Sectional Curvature

AbstractWe improve and extend a result due to M. Kanai about rigidity of geodesic flows on closed Riemannian manifolds of negative curvature whose stable or unstable horospheric foliation is smooth. More precisely, the main results proved here are: (1) Let M be a closed C∞ Riemannian manifold of negative sectional curvature. Assume the stable or unstable foliation of the geodesic flow φt: V → V on the unit tangent bundle V of M is C∞. Assume, moreover, that either (a) the sectional curvature of M satisfies −4 < K ≤ −1 or (b) the dimension of M is odd. Then the geodesic flow of M is C∞-isomorphic (i.e., conjugate under a C∞ diffeomorphism between the unit tangent bundles) to the geodesic flow on a closed Riemannian manifold of constant negative curvature. (2) For M as above, assume instead of (a) or (b) that dim M ≡ 2(mod 4). Then either the above conclusion holds or φ1, is C∞-isomorphic to the flow , on the quotient Γ\, where Γ is a subgroup of a real Lie group ⊂ Diffeo () with Lie algebra is the geodesic flow on the unit tangent bundle of the complex hyperbolic space ℂHm, m = ½ dim M.

scholarly journals Entropy estimates for geodesic flows

1982 ◽  
Vol 2 (3-4) ◽  
pp. 513-524 ◽  
Author(s):  
P. Sarnak
Keyword(s):  
Riemannian Manifold ◽  
Topological Entropy ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Compact Riemannian Manifold ◽  
Geodesic Flows ◽  
Unit Tangent Bundle ◽  
Entropy Estimates ◽  
Measure Entropy

AbstractLet M be a compact Riemannian manifold of (variable) negative curvature. Let h be the topological entropy and hμ the measure entropy for the geodesic flow on the unit tangent bundle to M. Estimates for h and hμ in terms of the ‘geometry’ of M are derived. Connections with and applications to other geometric questions are discussed.

A new description of the Bowen—Margulis measure

1989 ◽  
Vol 9 (3) ◽  
pp. 455-464 ◽  
Author(s):  
Ursula Hamenstädt
Keyword(s):  
Unstable Manifold ◽  
Tangent Bundle ◽  
Compact Manifold ◽  
Negative Curvature ◽  
Universal Covering ◽  
Unstable Foliation ◽  
Unit Tangent Bundle

AbstractThe Bowen-Margulis measure on the unit tangent bundle of the universal covering of a compact manifold of negative curvature is determined by its restriction to the leaves of the strong unstable foliation. We describe this restriction to any strong unstable manifold W as a spherical measure with respect to a natural distance on W.

When is an Anosov flow geodesic?

1992 ◽  
Vol 12 (2) ◽  
pp. 227-232
Author(s):  
Leon W. Green
Keyword(s):  
Flow Field ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Vector Fields ◽  
Three Dimensional ◽  
Dimensional Manifold ◽  
Anosov Flow ◽  
Unit Tangent Bundle

AbstractLet X, H+, H− be vector fields tangent, respectively, to an Anosov flow and its expanding and contracting foliations in a compact three-dimensional manifold, with γ, α+, α− one forms dual to them. If α+([H+, H−]) = α−([H+, H−]) and γ([H+, H−]) = α−([X, H−]) − α+([X, H+]), then the manifold has the structure of the unit tangent bundle of a Riemannian orbifold with geodesic flow field X.

scholarly journals Coding of geodesics and Lorenz-like templates for some geodesic flows

2016 ◽  
Vol 38 (3) ◽  
pp. 940-960
Author(s):  
PIERRE DEHORNOY ◽  
TALI PINSKY
Keyword(s):  
Periodic Orbits ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Hyperbolic Metric ◽  
Geodesic Flows ◽  
Unit Tangent Bundle

We construct a template with two ribbons that describes the topology of all periodic orbits of the geodesic flow on the unit tangent bundle to any sphere with three cone points with hyperbolic metric. The construction relies on the existence of a particular coding with two letters for the geodesics on these orbifolds.

scholarly journals Propriétés ergodiques, en mesure infinie, de certains systèmes dynamiques fibrés

1989 ◽  
Vol 9 (3) ◽  
pp. 433-453 ◽  
Author(s):  
Y. Guivarc'h
Keyword(s):  
Dynamical Systems ◽  
Invariant Measure ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Compact Manifolds ◽  
Constant Negative Curvature ◽  
Skew Products ◽  
Ergodic Properties ◽  
Anosov Systems

AbstractWe study the ergodic properties of a class of dynamical systems with infinite invariant measure. This class contains skew-products of Anosov systems with ℝd. The results are applied to theKproperty of skew-products and also to the ergodicity of the geodesic flow on abelian coverings of compact manifolds with constant negative curvature.

Regularity at infinity of compact negatively curved manifolds

1994 ◽  
Vol 14 (3) ◽  
pp. 493-514
Author(s):  
Ursula Hamenstädt
Keyword(s):  
Tangent Bundle ◽  
Compact Manifold ◽  
Negative Curvature ◽  
Periodic Points ◽  
Geodesic Flows ◽  
Stable Foliation ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Unit Tangent Bundle ◽  
Measure Class

AbstractIt is shown that three different notions of regularity for the stable foliation on the unit tangent bundle of a compact manifold of negative curvature are equivalent. Moreover if is a time-preserving conjugacy of geodesic flows of such manifolds M, N then the Lyapunov exponents at corresponding periodic points of the flows coincide. In particular Δ also preserves the Lebesgue measure class.

Sign in / Sign up

Export Citation Format

Share Document