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Exponential Mixing for Hyperbolic Flows on Non-Compact Spaces

Nicola Bertozzi, Paulo Varandas, Claudio Bonanno

Abstract

We introduce a family of hyperbolic flows on non-compact phase spaces that includes the geodesic flow on the modular surface. For these systems we prove exponential decay of correlations for sufficiently regular observables with respect to its SRB measure. Our approach follows the dynamical method of Dolgopyat and subsequent developments for suspension flows with uniformly hyperbolic Poincaré maps satisfying a uniform non-integrability condition. To fit this framework, we construct a suspension model via a triple inducing scheme that yields a uniformly hyperbolic Poincaré map with a countable Markov partition. We show that the resulting roof function is cohomologous to one that is constant along stable leaves and satisfies the required non-integrability and tail conditions. As an application, we recover a dynamical proof on Ratner's exponential mixing for the geodesic flow on the modular surface.

Exponential Mixing for Hyperbolic Flows on Non-Compact Spaces

Abstract

We introduce a family of hyperbolic flows on non-compact phase spaces that includes the geodesic flow on the modular surface. For these systems we prove exponential decay of correlations for sufficiently regular observables with respect to its SRB measure. Our approach follows the dynamical method of Dolgopyat and subsequent developments for suspension flows with uniformly hyperbolic Poincaré maps satisfying a uniform non-integrability condition. To fit this framework, we construct a suspension model via a triple inducing scheme that yields a uniformly hyperbolic Poincaré map with a countable Markov partition. We show that the resulting roof function is cohomologous to one that is constant along stable leaves and satisfies the required non-integrability and tail conditions. As an application, we recover a dynamical proof on Ratner's exponential mixing for the geodesic flow on the modular surface.
Paper Structure (18 sections, 29 theorems, 190 equations, 5 figures)

This paper contains 18 sections, 29 theorems, 190 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. One-dimensional maps
  4. Poincaré maps
  5. Roof functions and suspension flows
  6. Statements
  7. A criterion for exponential decay of correlations
  8. An inducing scheme for the base transformation
  9. Inducing scheme for $f$
  10. Inducing scheme for $\mathcal{P}$
  11. $C^1_b$ regular disintegration
  12. The roof function
  13. Exponential Decay of Correlations
  14. Exponential decay for the induced flow
  15. Exponential decay for the original flow
  16. ...and 3 more sections

Key Result

Theorem 2.2

Let $\varphi^t$ be a suspension flow over the base $(( \mathbb R^+\setminus \lbrace 1 \rbrace ) \times \mathbb R^+ , {\mathcal{P}} , m)$ with roof function $\rho$, satisfying assumptions (A1) - (A6), (B), (C), and (D). Then, there exist constants $C, \delta\in \mathbb R^+$ such that, for all $u,v \i

Figures (5)

  • Figure 2.1: An illustration of the maps $f_0$ and its inverse $g_0$.
  • Figure 4.1: Construction of the induced map $F$ and intervals $I_s$.
  • Figure 4.2: An illustration of the second induced map ${\widehat{F}}$ and intervals $J_s^q$.
  • Figure 5.1: A close-up of intervals $J_2^1$ and $J_3^1$ and the graph of ${\widehat{F}}$ over them. Here, $x^{(2)} = [\phi_2^1]^{2n-l}(x)$ and $x^{(3)} = [\phi_3^1]^{2n-l}(x)$.
  • Figure 5.2: A representation of the interval $J_{s_0}^{q_0}$ and its partition into intervals of the form $J_{s_0s_1}^{q_0q_1}$. Note that this structure is a replica inside of $J_{s_0}^{q_0}$ of the partition of $\left( g_0(1) , 1 \right)$ by the sets $J_s^q$.

Theorems & Definitions (61)

  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 3.1
  • Remark 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 51 more