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Polynomial decay of correlations for nonpositively curved surfaces

Yuri Lima, Carlos Matheus, Ian Melbourne

Abstract

We prove polynomial decay of correlations for geodesic flows on a class of nonpositively curved surfaces where zero curvature only occurs along one closed geodesic. We also prove that various statistical limit laws, including the central limit theorem, are satisfied by this class of geodesic flows.

Polynomial decay of correlations for nonpositively curved surfaces

Abstract

We prove polynomial decay of correlations for geodesic flows on a class of nonpositively curved surfaces where zero curvature only occurs along one closed geodesic. We also prove that various statistical limit laws, including the central limit theorem, are satisfied by this class of geodesic flows.

Paper Structure

This paper contains 22 sections, 31 theorems, 129 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Surfaces with nonpositive curvature
  3. Geodesic flows
  4. Surfaces of revolution
  5. Surfaces with degenerate closed geodesic
  6. Chernov axioms for exponential mixing
  7. Chernov axioms
  8. Dynamics of transitions in the neck $\mathcal{N}$
  9. Transition section $\Omega$ and map $f_0$
  10. Formula for the transition map $f_0$
  11. Transition times and derivatives of $f_0$
  12. Poincaré section and first return map $f$
  13. Construction of $\widetilde{\Sigma}$
  14. Construction of $\widehat{\Sigma}$
  15. Construction of $\Sigma_0$
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $r\in [4,\infty)$, and let $S$ be a closed Riemannian surface of nonpositive curvature obtained by isometrically gluing two negatively curved surfaces with boundaries to the boundaries of the surface of revolution with profile $1+|s|^r$, $|s|\leq 1$. Let $M=T^1S$ and $a=\tfrac{r+2}{r-2}\in(1,3]$

Figures (9)

  • Figure 1: Surfaces with degenerate closed geodesic $\gamma$ considered in Theorem \ref{['thm:main']}.
  • Figure 2: Clairaut relation: $c=\xi(s(t))\cos\psi(t)=\xi(s(t))^2 \theta'(t)$ is constant along $\gamma$.
  • Figure 3: An example of a surface $S$ with degenerate closed geodesic.
  • Figure 4: (a) Asymptotic vector. (b) Bouncing vector. (c) Crossing vector.
  • Figure 5: The pre-iterates of a curve intersecting $\Omega_-^=$ equals two curves of infinite length accumulating at $\Omega_+^=$.
  • ...and 4 more figures

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 2.2: Pesin Pesin-geodesic-flows
  • Lemma 2.4
  • Proposition 2.5
  • Theorem 2.6: Gerber & Wilkinson Gerber-Wilkinson
  • Theorem 3.1: Chernov & Zhang Chernov-Zhang
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • proof
  • ...and 43 more