scholarly journals Equidistribution results for geodesic flows

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.

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.

Time-preserving conjugacies of geodesic flows

1992 ◽  
Vol 12 (1) ◽  
pp. 67-74 ◽  
Author(s):  
Ursula Hamenstädt
Keyword(s):  
Sufficient Conditions ◽  
Universal Covering ◽  
Probability Measures ◽  
Geodesic Flows ◽  
Ideal Boundary ◽  
Negatively Curved ◽  
Unit Tangent Bundle ◽  
Borel Probability ◽  
Necessary And Sufficient ◽  
The Ideal

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.

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 Genus-one Birkhoff sections for geodesic flows

2014 ◽  
Vol 35 (6) ◽  
pp. 1795-1813 ◽  
Author(s):  
PIERRE DEHORNOY
Keyword(s):  
Tangent Bundle ◽  
Geodesic Flow ◽  
Singular Points ◽  
Geodesic Flows ◽  
Return Maps ◽  
Unit Tangent Bundle ◽  
First Return Maps ◽  
Genus One

We prove that the geodesic flow on the unit tangent bundle to every hyperbolic 2-orbifold that is a sphere with three or four singular points admits explicit genus-one Birkhoff sections, and we determine the associated first return maps.

The accessibility property of expansive geodesic flows without conjugate points

2008 ◽  
Vol 28 (1) ◽  
pp. 229-244
Author(s):  
RAFAEL OSWALDO RUGGIERO
Keyword(s):  
Riemannian Manifold ◽  
Geodesic Flow ◽  
Conjugate Points ◽  
Stable Set ◽  
Geodesic Flows ◽  
Continuous Path ◽  
Two Dimensional ◽  
Unstable Set ◽  
Unit Tangent Bundle ◽  
Continuous Curves

AbstractLet (M,g) be a compact, smooth Riemannian manifold without conjugate points whose geodesic flow is expansive. We show that the geodesic flow of (M,g) has the accessibility property, namely, given two pointsθ1,θ2in the unit tangent bundle, there exists a continuous path joiningθ1,θ2formed by the union of a finite number of continuous curves, each of which is contained either in a strong stable set or in a strong unstable set of the dynamics. Since expansive geodesic flows of compact surfaces have no conjugate points, the accessibility property holds for every two-dimensional expansive geodesic flow.

scholarly journals Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra II

2021 ◽  
pp. 1-10
Author(s):  
ALINE CERQUEIRA ◽  
CARLOS G. MOREIRA ◽  
SERGIO ROMAÑA
Keyword(s):  
Hausdorff Dimension ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Riemannian Metric ◽  
Hausdorff Dimensions ◽  
Hyperbolic Set ◽  
Negatively Curved ◽  
Smooth Perturbation ◽  
Unit Tangent Bundle ◽  
Markov Spectrum

Abstract Let $g_0$ be a smooth pinched negatively curved Riemannian metric on a complete surface N, and let $\Lambda _0$ be a basic hyperbolic set of the geodesic flow of $g_0$ with Hausdorff dimension strictly smaller than two. Given a small smooth perturbation g of $g_0$ and a smooth real-valued function f on the unit tangent bundle to N with respect to g, let $L_{g,\Lambda ,f}$ (respectively $M_{g,\Lambda ,f}$ ) be the Lagrange (respectively Markov) spectrum of asymptotic highest (respectively highest) values of f along the geodesics in the hyperbolic continuation $\Lambda $ of $\Lambda _0$ . We prove that for generic choices of g and f, the Hausdorff dimensions of the sets $L_{g,\Lambda , f}\cap (-\infty , t)$ vary continuously with $t\in \mathbb {R}$ and, moreover, $M_{g,\Lambda , f}\cap (-\infty , t)$ has the same Hausdorff dimension as $L_{g,\Lambda , f}\cap (-\infty , t)$ for all $t\in \mathbb {R}$ .

Expansive geodesic flows in manifolds with no conjugate points

1997 ◽  
Vol 17 (1) ◽  
pp. 211-225 ◽  
Author(s):  
RAFAEL O. RUGGIERO
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Compact Riemannian Manifold ◽  
Product Structure ◽  
Conjugate Points ◽  
Geodesic Flows ◽  
Closed Geodesics ◽  
Local Product ◽  
Unit Tangent Bundle

Let $M$ be a compact Riemannian manifold with no conjugate points such that its geodesic flow is expansive. We show that there exists a local product structure in the unit tangent bundle of the manifold which is invariant under the geodesic flow. In particular, we have that the set of closed geodesics is dense and that the flow is topologically transitive.

scholarly journals Hyperbolic behaviour of geodesic flows on manifolds with no focal points

1983 ◽  
Vol 3 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Keith Burns
Keyword(s):  
Tangent Bundle ◽  
Geodesic Flow ◽  
Flow Direction ◽  
Geodesic Flows ◽  
Focal Points ◽  
Characteristic Exponents ◽  
Liouville Measure ◽  
Unit Tangent Bundle

AbstractIt is shown that the unit tangent bundle of a compact uniform visibility manifold with no focal points contains a subset of positive Liouville measure on which all the characteristic exponents of the geodesic flow (except in the flow direction) are non-zero. This completes Pesin's proof that the geodesic flow of such a manifold is Bernoulli.

scholarly journals Checking ergodicity of some geodesic flows with infinite Gibbs measure

1981 ◽  
Vol 1 (1) ◽  
pp. 107-133 ◽  
Author(s):  
Mary Rees
Keyword(s):  
Gibbs Measure ◽  
Symbolic Dynamics ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Discrete Group ◽  
Geodesic Flows ◽  
Unit Tangent Bundle ◽  
Divergence Type ◽  
Natural Measures ◽  
Group Of Isometries

AbstractThis paper concerns a problem which arose from a paper of Sullivan. Let Γ be a discrete group of isometries of hyperbolic space Hd+1. We study the question of when the geodesic flow on the unit tangent bundle UT (Hd+1/Γ) of Hd+1/Γ is ergodic with respect to certain natural measures. As a consequence, we study the question of when Γ is of divergence type. Ergodicity when the non-wandering set of UT (Hd+1/Γ) is compact is already known from the theory of symbolic dynamics, due to Bowen, or from Sullivan's work. For such a Γ, we consider a subgroup Γ1 of Γ with Γ/Γ1 ≅ℤυ and prove the geodesic flow on UT (Hd+1/Γ1) is ergodic (with respect to one of these natural measures) if and only if υ ≤ 2.

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