The unique ergodigity of the horocycle flow

1973 ◽  
pp. 95-115 ◽  
Author(s):  
Harry Furstenberg
Keyword(s):  
Horocycle Flow

Invariant measures for the horocycle flow on periodic hyperbolic surfaces

2007 ◽  
Vol 160 (1) ◽  
pp. 281-315 ◽  
Author(s):  
François Ledrappier ◽  
Omri Sarig
Keyword(s):  
Invariant Measures ◽  
Hyperbolic Surfaces ◽  
Horocycle Flow

Extreme instability of the horocycle flow

1991 ◽  
Vol 57 (1) ◽  
pp. 37-63
Author(s):  
Serge E. Troubetzkoy
Keyword(s):  
Horocycle Flow

On unipotent flows in ℋ(1,1)

2009 ◽  
Vol 30 (2) ◽  
pp. 379-398 ◽  
Author(s):  
KARIANE CALTA ◽  
KEVIN WORTMAN
Keyword(s):  
Moduli Space ◽  
Probability Measures ◽  
Specific Class ◽  
Horocycle Flow ◽  
Abelian Differentials ◽  
Unipotent Flows ◽  
Genus Two

AbstractWe study the action of the horocycle flow on the moduli space of abelian differentials in genus two. In particular, we exhibit a classification of a specific class of probability measures that are invariant and ergodic under the horocycle flow on the stratum ℋ(1,1).

scholarly journals Statistical behaviour of the leaves of Riccati foliations

2009 ◽  
Vol 30 (1) ◽  
pp. 67-96 ◽  
Author(s):  
CH. BONATTI ◽  
X. GÓMEZ-MONT ◽  
R. VILA-FREYER
Keyword(s):  
Lebesgue Measure ◽  
Geodesic Flow ◽  
Conformal Structure ◽  
Holomorphic Foliation ◽  
Horocycle Flow ◽  
Linear Ordinary Differential Equations ◽  
Positive Time ◽  
Measure Class ◽  
Dimension One

AbstractWe introduce the geodesic flow on the leaves of a holomorphic foliation with leaves of dimension one and hyperbolic, corresponding to the unique complete metric of curvature −1 compatible with its conformal structure. We do these for the foliations associated to Riccati equations, which are the projectivization of the solutions of linear ordinary differential equations over a finite Riemann surface of hyperbolic type S, and may be described by a representation ρ:π1(S)→GL(n,ℂ). We give conditions under which the foliated geodesic flow has a generic repeller–attractor statistical dynamics. That is, there are measures μ− and μ+ such that for almost any initial condition with respect to the Lebesgue measure class the statistical average of the foliated geodesic flow converges for negative time to μ− and for positive time to μ+ (i.e. μ+ is the unique Sinaï, Ruelle and Bowen (SRB)-measure and its basin has total Lebesgue measure). These measures are ergodic with respect to the foliated geodesic flow. These measures are also invariant under a foliated horocycle flow and they project to a harmonic measure for the Riccati foliation, which plays the role of an attractor for the statistical behaviour of the leaves of the foliation.

Horocycle flow on a surface of negative curvature is separating

1984 ◽  
Vol 36 (2) ◽  
pp. 632-635
Author(s):  
A. A. Gura
Keyword(s):  
Negative Curvature ◽  
Horocycle Flow

scholarly journals Unique ergodicity of the horocycle flow on Riemannnian foliations

2018 ◽  
Vol 40 (6) ◽  
pp. 1459-1479
Author(s):  
F. ALCALDE CUESTA ◽  
F. DAL’BO ◽  
M. MARTÍNEZ ◽  
A. VERJOVSKY
Keyword(s):  
Main Tool ◽  
Hyperbolic Surface ◽  
Unique Ergodicity ◽  
Alternative Proof ◽  
Compact Metric Space ◽  
Horocycle Flow ◽  
Full Support ◽  
Classic Result ◽  
Minimal Foliations ◽  
Special Case

A classic result due to Furstenberg is the strict ergodicity of the horocycle flow for a compact hyperbolic surface. Strict ergodicity is unique ergodicity with respect to a measure of full support, and therefore it implies minimality. The horocycle flow has been previously studied on minimal foliations by hyperbolic surfaces on closed manifolds, where it is known not to be minimal in general. In this paper, we prove that for the special case of Riemannian foliations, strict ergodicity of the horocycle flow still holds. This, in particular, proves that this flow is minimal, which establishes a conjecture proposed by Matsumoto. The main tool is a theorem due to Coudène, which he presented as an alternative proof for the surface case. It applies to two continuous flows defining a measure-preserving action of the affine group of the line on a compact metric space, precisely matching the foliated setting. In addition, we briefly discuss the application of Coudène’s theorem to other kinds of foliations.

scholarly journals Remarks on the dynamics of the horocycle flow for homogeneous foliations by hyperbolic surfaces

2015 ◽  
Vol 33 (4) ◽  
pp. 431-451 ◽  
Author(s):  
Fernando Alcalde Cuesta ◽  
Françoise Dal’Bo
Keyword(s):  
Hyperbolic Surfaces ◽  
Horocycle Flow
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