Unique ergodicity of the horocycle flow: Variable negative curvature case

1975 ◽  
Vol 21 (2-3) ◽  
pp. 133-144 ◽  
Author(s):  
Brian Marcus
Keyword(s):  
Negative Curvature ◽  
Unique Ergodicity ◽  
Flow Variable ◽  
Horocycle Flow

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 A solution to Flinn’s conjecture on weakly expansive flows

2020 ◽  
pp. 1-7
Author(s):  
HIEN MINH HUYNH
Keyword(s):  
Riemann Surfaces ◽  
Negative Curvature ◽  
Constant Negative Curvature ◽  
Horocycle Flow ◽  
Compact Riemann Surfaces ◽  
Phd Thesis ◽  
Expansive Flows

In L. W. Flinn’s PhD thesis published in 1972, the author conjectured that weakly expansive flows are also expansive flows. In this paper we use the horocycle flow on compact Riemann surfaces of constant negative curvature to show that Flinn’s conjecture is not true.

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.

Teichmüller dynamics and unique ergodicity via currents and Hodge theory

2020 ◽  
Vol 2020 (768) ◽  
pp. 39-54
Author(s):  
Curtis T. McMullen
Keyword(s):  
Hodge Theory ◽  
Unique Ergodicity ◽  
Polygonal Billiards ◽  
Teichmuller Dynamics

AbstractWe present a cohomological proof that recurrence of suitable Teichmüller geodesics implies unique ergodicity of their terminal foliations. This approach also yields concrete estimates for periodic foliations and new results for polygonal billiards.

scholarly journals Hyperbolic Center of Mass for a System of Particles in a Two-Dimensional Space with Constant Negative Curvature: An Application to the Curved 2-Body Problem

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 531
Author(s):  
Pedro Pablo Ortega Palencia ◽  
Ruben Dario Ortiz Ortiz ◽  
Ana Magnolia Marin Ramirez
Keyword(s):  
Negative Curvature ◽  
Dimensional Space ◽  
Center Of Mass ◽  
Simple Expression ◽  
Two Dimensional ◽  
Constant Negative Curvature ◽  
Euclidean Case ◽  
System Of Particles ◽  
Material Points ◽  
The One

In this article, a simple expression for the center of mass of a system of material points in a two-dimensional surface of Gaussian constant negative curvature is given. By using the basic techniques of geometry, we obtained an expression in intrinsic coordinates, and we showed how this extends the definition for the Euclidean case. The argument is constructive and serves to define the center of mass of a system of particles on the one-dimensional hyperbolic sphere LR1.

Manifolds with wells of negative curvature

1991 ◽  
Vol 103 (1) ◽  
pp. 471-495 ◽  
Author(s):  
K. D. Elworthy ◽  
Steven Rosenberg
Keyword(s):  
Negative Curvature

scholarly journals On a Unique Ergodicity of Some Markov Processes

2011 ◽  
Vol 36 (4) ◽  
pp. 589-606 ◽  
Author(s):  
Rafał Kapica ◽  
Tomasz Szarek ◽  
Maciej Ślȩczka
Keyword(s):  
Markov Processes ◽  
Unique Ergodicity
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