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 Hyperbolic rank rigidity for manifolds of -pinched negative curvature

2018 ◽  
Vol 40 (5) ◽  
pp. 1194-1216
Author(s):  
CHRIS CONNELL ◽  
THANG NGUYEN ◽  
RALF SPATZIER
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Negative Curvature ◽  
Jacobi Field ◽  
Rigidity Result ◽  
Locally Symmetric Space ◽  
Rank One ◽  
Sectional Curvatures

A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature $-1$ with the geodesic. If, in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac{1}{4}]$ and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial counterpart to Hamenstädt’s hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.

Geodesic Flow on Ideal Polyhedra

1997 ◽  
Vol 49 (4) ◽  
pp. 696-707 ◽  
Author(s):  
Charalambos Charitos ◽  
Georgios Tsapogas
Keyword(s):  
Geodesic Flow ◽  
Negative Curvature ◽  
Closed Orbits

AbstractIn this work we study the geodesic flow on n-dimensional ideal polyhedra and establish classical (for manifolds of negative curvature) results concerning the distribution of closed orbits of the flow.

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.

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 Existence of Dirichlet infinite harmonic measures on the unit disc

1995 ◽  
Vol 138 ◽  
pp. 141-167
Author(s):  
Mitsuru Nakai
Keyword(s):  
Euclidean Space ◽  
Harmonic Measure ◽  
Unit Disc ◽  
Affirmative Answer ◽  
Dimensional Euclidean Space ◽  
Dirichlet Integral ◽  
Harmonic Measures

The primary purpose of this paper is to give an affirmative answer to a problem posed by Ohtsuka [13] whether there exists a p-harmonic measure on the unit disc in the 2-dimensional Euclidean space R2 with an infinite p-Dirichlet integral for the exponent 1 < p < 2.

scholarly journals Propriétés ergodiques, en mesure infinie, de certains systèmes dynamiques fibrés

1989 ◽  
Vol 9 (3) ◽  
pp. 433-453 ◽  
Author(s):  
Y. Guivarc'h
Keyword(s):  
Dynamical Systems ◽  
Invariant Measure ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Compact Manifolds ◽  
Constant Negative Curvature ◽  
Skew Products ◽  
Ergodic Properties ◽  
Anosov Systems

AbstractWe study the ergodic properties of a class of dynamical systems with infinite invariant measure. This class contains skew-products of Anosov systems with ℝd. The results are applied to theKproperty of skew-products and also to the ergodicity of the geodesic flow on abelian coverings of compact manifolds with constant negative curvature.

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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