Time-preserving conjugacies of geodesic flows

AbstractIn this note we study Borel-probability measures on the unit tangent bundle ofa compact negatively curved manifold M that are invariant under the geodesic flow. We interpret the entropy of such a measure as a Hausdorff dimension with respect to a natural family of distances on the ideal boundary of the universal covering of M. This in term yields necessary and sufficient conditions for the existence of time preserving conjugacies of geodesic flows.

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Cocycles, Hausdorff measures and cross ratios

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385797086379 ◽

1997 ◽

Vol 17 (5) ◽

pp. 1061-1081 ◽

Cited By ~ 4

Author(s):

URSULA HAMENSTÄDT

Keyword(s):

Riemannian Manifold ◽

Universal Covering ◽

Ideal Boundary ◽

Hausdorff Measures ◽

Hölder Continuous ◽

Negatively Curved ◽

Unit Tangent Bundle ◽

Unstable Manifolds ◽

Invariant Cocycles ◽

The Ideal

Let $f$ be a flip-invariant Hölder continuous function on the unit tangent bundle $T^1 M$ of a closed negatively curved Riemannian manifold $M$. We show that conditionals on strong unstable manifolds of the Gibbs equilibrium state defined by $f$ can be realized as Hausdorff measures. Moreover, cohomology classes of flip invariant cocycles are in one-to-one correspondence to cross ratios on the space of four pairwise distinct points of the ideal boundary of the universal covering $\tilde M$ of $M$.

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Sufficient conditions for expansive group action

Stochastics and Dynamics ◽

10.1142/s0219493720500227 ◽

2019 ◽

Vol 20 (03) ◽

pp. 2050022 ◽

Cited By ~ 1

Author(s):

Ali Barzanouni

Keyword(s):

Solvable Group ◽

Group Action ◽

Uniform Space ◽

Sufficient Conditions ◽

Open Cover ◽

Probability Measures ◽

Continuous Action ◽

Paracompact Space ◽

Finite Set ◽

Borel Probability

Existence of expansivity for group action [Formula: see text] depends on algebraic properties of [Formula: see text] and the topology of [Formula: see text]. We give an expansive action of a solvable group on [Formula: see text] while there is no expansive action of a solvable group on a dendrite [Formula: see text]. We prove that a continuous action [Formula: see text] on a compact metric space [Formula: see text] is expansive if and only if there exists an open cover [Formula: see text] such that for any other open cover [Formula: see text], [Formula: see text] for some finite set [Formula: see text]. In this paper, we introduce the notion of topological expansivity of a group action [Formula: see text] on a [Formula: see text]-paracompact space [Formula: see text]. If a [Formula: see text]-paracompact space [Formula: see text] admits topologically expansive action, then [Formula: see text] is Hausdorff space. We also show that a continuous action [Formula: see text] of a finitely generated group [Formula: see text] on a compact Hausdorff uniform space [Formula: see text] is expansive with an expansive neighborhood [Formula: see text] if and only if for every [Formula: see text] there is an entourage [Formula: see text] such that for every two [Formula: see text]-pseudo orbit [Formula: see text] if [Formula: see text] for all [Formula: see text], then [Formula: see text] for all [Formula: see text]. Finally, we introduce measure [Formula: see text]-expansive actions on a uniform space. The set of all [Formula: see text]-expansive measures with common expansive neighborhood for a group action [Formula: see text] is a convex, closed and [Formula: see text]-invariant subset of the set of all Borel probability measures on [Formula: see text]. Also, we show that a group action [Formula: see text] is expansive if all Borel probability measures are [Formula: see text]-expansive or all Dirac measures [Formula: see text], [Formula: see text], have a common expansive neighborhood.

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On characterizing the set of martingale measures in discrete time

Stochastics and Dynamics ◽

10.1142/s0219493715500173 ◽

2015 ◽

Vol 15 (03) ◽

pp. 1550017 ◽

Cited By ~ 2

Author(s):

Abdelkarem Berkaoui

Keyword(s):

Discrete Time ◽

Sufficient Conditions ◽

Sample Space ◽

Probability Measures ◽

Finite Sample ◽

Martingale Measures ◽

Space Case ◽

Necessary And Sufficient ◽

Adapted Process ◽

Vector Valued

We state necessary and sufficient conditions on a set of probability measures to be the set of martingale measures for a vector valued, bounded and adapted process. In the absence of the maximality condition, we prove the existence of the smallest set of martingale measures. We apply such results to the finite sample space case.

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Information geometry of Busemann-barycenter for probability measures

International Journal of Mathematics ◽

10.1142/s0129167x15410074 ◽

2015 ◽

Vol 26 (06) ◽

pp. 1541007 ◽

Cited By ~ 1

Author(s):

Mitsuhiro Itoh ◽

Hiroyasu Satoh

Keyword(s):

Natural Extension ◽

Information Geometry ◽

Acta Math ◽

Space Structure ◽

Hadamard Manifold ◽

Probability Measures ◽

Positive Density ◽

Ideal Boundary ◽

Push Forward ◽

The Ideal

Using Busemann function of an Hadamard manifold X we define the barycenter map from the space 𝒫+(∂X, dθ) of probability measures having positive density on the ideal boundary ∂X to X. The space 𝒫+(∂X, dθ) admits geometrically a fiber space structure over X from Fisher information geometry. Following the arguments in [E. Douady and C. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math.157 (1986) 23–48; G. Besson, G. Courtois and S. Gallot, Entropies et rigidités des espaces localement symétriques de coubure strictement négative, Geom. Funct. Anal.5 (1995) 731–799; Minimal entropy and Mostow's rigidity theorems, Ergodic Theory Dynam. Systems16 (1996) 623–649], we exhibit that under certain geometrical hypotheses a homeomorphism Φ of the ideal boundary ∂X induces, by the aid of push-forward, an isometry of X whose extension is Φ.

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Equidistribution results for geodesic flows

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2012.153 ◽

2013 ◽

Vol 34 (3) ◽

pp. 742-764

Author(s):

ABDELHAMID AMROUN

Keyword(s):

Geodesic Flow ◽

Large Deviation ◽

Riemannian Metric ◽

Upper Bounds ◽

Equilibrium States ◽

Topological Pressure ◽

Probability Measures ◽

Geodesic Flows ◽

Lower And Upper Bounds ◽

Unit Tangent Bundle

AbstractUsing the works of Mañé [On the topological entropy of the geodesic flows.J. Differential Geom.45(1989), 74–93] and Paternain [Topological pressure for geodesic flows.Ann. Sci. Éc. Norm. Supér.(4)33(2000), 121–138] we study the distribution of geodesic arcs with respect to equilibrium states of the geodesic flow on a closed manifold, equipped with a$\mathcal {C}^{\infty }$Riemannian metric. We prove large-deviation lower and upper bounds and a contraction principle for the geodesic flow in the space of probability measures of the unit tangent bundle. We deduce a way of approximating equilibrium states for continuous potentials.

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Regularity at infinity of compact negatively curved manifolds

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700007999 ◽

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.

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Connecting ergodicity and dimension in dynamical systems

Ergodic Theory and Dynamical Systems ◽

10.1017/s014338570000568x ◽

1990 ◽

Vol 10 (3) ◽

pp. 451-462 ◽

Cited By ~ 23

Author(s):

C. D. Cutler

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Metric Spaces ◽

Sufficient Conditions ◽

Probability Measures ◽

Pointwise Dimension ◽

Continuous Maps ◽

Borel Probability ◽

The Relationship

AbstractIn this paper we make precise the relationship between local or pointwise dimension and the dimension structure of Borel probability measures on metric spaces. Sufficient conditions for exact-dimensionality of the stationary ergodic distributions associated with a dynamical system are obtained. A counterexample is provided to show that ergodicity alone is not sufficient to guarantee exactdimensionality even in the case of continuous maps or flows.

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ELLIPTIC OPERATORS AND SOLUTIONS OF COHOMOLOGICAL EQUATIONS FOR GEODESIC FLOWS WITH HYPERBOLIC BEHAVIOR

International Journal of Mathematics ◽

10.1142/s0129167x91000387 ◽

1991 ◽

Vol 02 (06) ◽

pp. 701-709

Author(s):

SVETLANA KATOK

Keyword(s):

Negative Curvature ◽

Infinitesimal Generator ◽

Geodesic Flows ◽

Constant Negative Curvature ◽

Smooth Functions ◽

Negatively Curved ◽

Finite Dimensional ◽

Unit Tangent Bundle ◽

Hyperbolic Behavior ◽

Cohomological Equations

In this paper we study the space of smooth functions on the unit tangent bundle SM to a compact negatively curved surface M that are eigenfunctions of the infinitesimal generator of the action of SO(2) on SM, and that have zero integrals over all periodic orbits of the geodesic flow on SM. It is proved that the space of such functions is finite dimensional. In the case of constant negative curvature a complete description of this space is obtained.

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