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 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.

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.

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.

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 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.

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.

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.

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