scholarly journals Harmonic measures and the foliated geodesic flow for foliations with negatively curved leaves

2014 ◽  
Vol 36 (2) ◽  
pp. 355-374 ◽  
Author(s):  
SÉBASTIEN ALVAREZ
Keyword(s):  
Gibbs Measure ◽  
Geodesic Flow ◽  
Bijective Correspondence ◽  
Negatively Curved ◽  
Harmonic Measures

In this paper we define a notion of Gibbs measure for the geodesic flow tangent to a foliation with negatively curved leaves and associated to a particular potential $H$. We prove that there is a canonical bijective correspondence between these measures and Garnett’s harmonic measures.

scholarly journals Gibbs measures for foliated bundles with negatively curved leaves

2016 ◽  
Vol 38 (4) ◽  
pp. 1238-1288 ◽  
Author(s):  
SÉBASTIEN ALVAREZ
Keyword(s):  
Invariant Measure ◽  
Gibbs Measure ◽  
Geodesic Flow ◽  
Gibbs Measures ◽  
Holonomy Group ◽  
Bijective Correspondence ◽  
Weighted Averages ◽  
Negatively Curved ◽  
Harmonic Measures ◽  
Foliated Bundle

In this paper we develop a notion of Gibbs measure for the geodesic flow tangent to a foliated bundle over a compact negatively curved base. We also develop a notion of$F$-harmonic measure and prove that there exists a natural bijective correspondence between these two concepts. For projective foliated bundles with$\mathbb{C}\mathbb{P}^{1}$-fibers without transverse invariant measure, we show the uniqueness of these measures for any Hölder potential on the base. In that case we also prove that$F$-harmonic measures are realized as weighted limits of large balls tangent to the leaves and that their conditional measures on the fibers are limits of weighted averages on the orbits of the holonomy group.

Conditional measure and flip invariance of Bowen-Margulis and harmonic measures on manifolds of negative curvature

1995 ◽  
Vol 15 (4) ◽  
pp. 807-811 ◽  
Author(s):  
Chengbo Yue
Keyword(s):  
Harmonic Measure ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Conditional Measure ◽  
Rigidity Result ◽  
Harmonic Measures

AbstractKifer and Ledrappier have asked whether the harmonic measures {νx} on manifolds of negative curvature are equivalent to the conditional measures of the harmonic measure v of the geodesic flow associated with the fibration {SxM}x∈M. We settle this question with a rigidity result. We also clear up the same problem concerning the Patterson-Sullivan measure and the Bowen–Margulis measure.

scholarly journals Dynamics of the geodesic flow of a foliation

1988 ◽  
Vol 8 (4) ◽  
pp. 637-650 ◽  
Author(s):  
Paweł G. Walczak
Keyword(s):  
Riemannian Manifold ◽  
Lyapunov Exponents ◽  
Geodesic Flow ◽  
Smooth Measure ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Foliated Riemannian Manifold

AbstractThe geodesic flow of a foliated Riemannian manifold (M, F) is studied. The invariance of some smooth measure is established under some geometrical conditions on F. The Lyapunov exponents and the entropy of this flow are estimated. As an application, the non-existence of foliations with ‘short’ second fundamental tensors is obtained on compact negatively curved manifolds.

scholarly journals Geodesic flows of negatively curved manifolds with smooth stable and unstable foliations

1988 ◽  
Vol 8 (2) ◽  
pp. 215-239 ◽  
Author(s):  
Masahiko Kanai
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Geodesic Flows ◽  
Constant Negative Curvature ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Closed Riemannian Manifold

AbstractWe are concerned with closed C∞ riemannian manifolds of negative curvature whose geodesic flows have C∞ stable and unstable foliations. In particular, we show that the geodesic flow of such a manifold is isomorphic to that of a certain closed riemannian manifold of constant negative curvature if the dimension of the manifold is greater than two and if the sectional curvature lies between − and −1 strictly.

Stable ergodicity of the time-one map of a geodesic flow

1998 ◽  
Vol 18 (6) ◽  
pp. 1545-1587 ◽  
Author(s):  
AMIE WILKINSON
Keyword(s):  
Geodesic Flow ◽  
Curved Surface ◽  
Negatively Curved ◽  
Stable Ergodicity

We prove that the time-one map of the geodesic flow for a closed, negatively curved surface is stably ergodic.

scholarly journals Escape of mass and entropy for geodesic flows

2017 ◽  
Vol 39 (2) ◽  
pp. 446-473
Author(s):  
FELIPE RIQUELME ◽  
ANIBAL VELOZO
Keyword(s):  
Phase Transition ◽  
Geodesic Flow ◽  
Upper Semicontinuity ◽  
Geodesic Flows ◽  
Parabolic Subgroups ◽  
Hölder Continuous ◽  
Upper Semicontinuous ◽  
Negatively Curved ◽  
Geometrically Finite ◽  
Loss Of Mass

In this paper, we study the ergodic theory of the geodesic flow on negatively curved geometrically finite manifolds. We prove that the measure-theoretical entropy is upper semicontinuous when there is no loss of mass. In the case where mass is lost, the critical exponents of parabolic subgroups of the fundamental group have a significant meaning. More precisely, the failure of upper-semicontinuity of the entropy is determined by the maximal parabolic critical exponent. We also study the pressure of positive Hölder-continuous potentials going to zero through the cusps. We prove that the pressure map $t\mapsto P(tF)$ is differentiable until it undergoes a phase transition, after which it becomes constant. This description allows us, in particular, to compute the entropy of the geodesic flow at infinity.

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 $t\in \mathbb {R}$ .

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