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.

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.

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.

Bounded geodesics and Hausdorff dimension

1994 ◽  
Vol 116 (3) ◽  
pp. 505-511 ◽  
Author(s):  
C. S. Aravinda
Keyword(s):  
Hausdorff Dimension ◽  
Tangent Bundle ◽  
Negative Curvature ◽  
Tangent Vector ◽  
Unit Tangent Vector ◽  
Constant Negative Curvature ◽  
Compact Closure ◽  
Riemannian Volume ◽  
Unit Tangent Bundle ◽  
Almost All

Let M be a Riemannian manifold of constant negative curvature and finite Riemannian volume. It is well-known that the geodesic flow on the unit tangent bundle SM of M is ergodic. In particular, it follows that for almost all (p, v)∈ SM, where p ∈M and v is a unit tangent vector at p, the geodesic through p in the direction of v is dense in M. A theorem of Dani [Dl] says that the set of all (p, v)∈SM for which the corresponding geodesic is bounded (namely those with compact closure in M) is ‘large’ in the sense that its Hausdorff dimension is equal to that of the unit tangent bundle itself. In fact, Dani generalized this result to a more general algebraic situation (cf. [D2]).

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 Factors of horocycle flows

1982 ◽  
Vol 2 (3-4) ◽  
pp. 465-489 ◽  
Author(s):  
Marina Ratner
Keyword(s):  
Finite Volume ◽  
Tangent Bundle ◽  
Negative Curvature ◽  
Constant Negative Curvature ◽  
Horocycle Flow ◽  
Unit Tangent Bundle

AbstractWe classify up to an isomorphism all factors of the classical horocycle flow on the unit tangent bundle of a surface of constant negative curvature with finite volume.

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.

scholarly journals Vortices over Riemann surfaces and dominated splittings

2021 ◽  
pp. 1-26
Author(s):  
THOMAS METTLER ◽  
GABRIEL P. PATERNAIN
Keyword(s):  
Euler Characteristic ◽  
Tangent Bundle ◽  
Riemann Surfaces ◽  
Dominated Splitting ◽  
Vortex Equations ◽  
Special Cases ◽  
Unit Tangent Bundle ◽  
Anosov Flows

Abstract We associate a flow $\phi $ with a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi $ always admits a dominated splitting and identify special cases in which $\phi $ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$ .

scholarly journals Natural Paracontact Magnetic Trajectories on Unit Tangent Bundles

Axioms ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 72
Author(s):  
Mohamed Tahar Kadaoui Abbassi ◽  
Noura Amri
Keyword(s):  
Gaussian Curvature ◽  
Tangent Bundle ◽  
Two Dimensional ◽  
Constant Gaussian Curvature ◽  
Metric Structures ◽  
Unit Tangent Bundle ◽  
Tangent Bundles ◽  
Full Classification ◽  
Kaluza Klein

In this paper, we study natural paracontact magnetic trajectories in the unit tangent bundle, i.e., those that are associated to g-natural paracontact metric structures. We characterize slant natural paracontact magnetic trajectories as those satisfying a certain conservation law. Restricting to two-dimensional base manifolds of constant Gaussian curvature and to Kaluza–Klein type metrics on their unit tangent bundles, we give a full classification of natural paracontact slant magnetic trajectories (and geodesics).

scholarly journals Legendre curves and the singularities of ruled surfaces obtained by using rotation minimizing frame

2021 ◽  
Vol 73 (5) ◽  
pp. 589-601
Author(s):  
M. Bekar ◽  
F. Hathout ◽  
Y. Yayli
Keyword(s):  
Tangent Bundle ◽  
Vector Fields ◽  
Ruled Surfaces ◽  
Unit Tangent Bundle ◽  
Legendre Curves ◽  
Rotation Minimizing Frame

UDC 514.7 In this paper, Legendre curves in unit tangent bundle are given using rotation minimizing vector fields. Ruled surfaces corresponding to these curves are represented. Singularities of these ruled surfaces are also analyzed and classified.

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