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Polynomial decay of correlations of pseudo-Anosov diffeomorphisms

Dominic Veconi

TL;DR

This work constructs a smooth $C^{2+\varepsilon}$ realization $g$ of a pseudo-Anosov diffeomorphism on any compact surface, carefully slowing trajectories near singularities and conjugating by a mass-pushing map to ensure area preservation. The resulting system is topologically conjugate to the original pseudo-Anosov and supports a unique SRB measure $\mu_1$ with nonzero Lyapunov exponents a.e., and it exhibits Bernoulli dynamics, polynomial decay of correlations, a central limit theorem, and polynomial large deviations. The core method is to model $g$ by a Young tower, with both lower and upper polynomial tails for the inducing time, yielding precise rates of mixing and statistical limit laws. By varying the slowdown parameters, the paper demonstrates a flexible framework to realize a wide range of ergodic properties for nonuniformly hyperbolic diffeomorphisms on surfaces, extending previous results on Katok-type constructions. The results have significant implications for understanding statistical stability and fluctuations in smooth nonuniformly hyperbolic dynamics on surfaces.

Abstract

We give a construction of a smooth realization of a pseudo-Anosov diffeomorphism of a Riemannian surface, and show that it admits a unique SRB measure with polynomial decay of correlations, large deviations, and the central limit theorem. The construction begins with a linear pseudo-Anosov diffeomorphism whose singularities are fixed points. Near the singularities, the trajectories are slowed down, and then the map is conjugated with a homeomorphism that pushes mass away from the origin. The resulting map is a $C^{2+ε}$ diffeomorphism topologically conjugate to the original pseudo-Anosov map. To prove that this map has polynomial decay of correlations, our main technique is to use the fact that this map has a Young tower, and study the decay of the tail of the first return time to the base of the tower.

Polynomial decay of correlations of pseudo-Anosov diffeomorphisms

TL;DR

This work constructs a smooth realization of a pseudo-Anosov diffeomorphism on any compact surface, carefully slowing trajectories near singularities and conjugating by a mass-pushing map to ensure area preservation. The resulting system is topologically conjugate to the original pseudo-Anosov and supports a unique SRB measure with nonzero Lyapunov exponents a.e., and it exhibits Bernoulli dynamics, polynomial decay of correlations, a central limit theorem, and polynomial large deviations. The core method is to model by a Young tower, with both lower and upper polynomial tails for the inducing time, yielding precise rates of mixing and statistical limit laws. By varying the slowdown parameters, the paper demonstrates a flexible framework to realize a wide range of ergodic properties for nonuniformly hyperbolic diffeomorphisms on surfaces, extending previous results on Katok-type constructions. The results have significant implications for understanding statistical stability and fluctuations in smooth nonuniformly hyperbolic dynamics on surfaces.

Abstract

We give a construction of a smooth realization of a pseudo-Anosov diffeomorphism of a Riemannian surface, and show that it admits a unique SRB measure with polynomial decay of correlations, large deviations, and the central limit theorem. The construction begins with a linear pseudo-Anosov diffeomorphism whose singularities are fixed points. Near the singularities, the trajectories are slowed down, and then the map is conjugated with a homeomorphism that pushes mass away from the origin. The resulting map is a diffeomorphism topologically conjugate to the original pseudo-Anosov map. To prove that this map has polynomial decay of correlations, our main technique is to use the fact that this map has a Young tower, and study the decay of the tail of the first return time to the base of the tower.

Paper Structure

This paper contains 19 sections, 27 theorems, 115 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pseudo-Anosov maps
  4. Ergodic properties
  5. Main results
  6. Construction of the smooth pseudo-Anosov model
  7. Construction of $g$
  8. Smoothness and area invariance of $g$
  9. Other topological properties
  10. Pseudo-Anosov Diffeomorphisms are Young Diffeomorphisms
  11. Young diffeomorphisms
  12. Realizing $g$ as a Young diffeomorphism
  13. Behavior near singularities
  14. A lower bound on the tail of the return time
  15. An upper bound on the tail of the return time
  16. ...and 4 more sections

Key Result

Theorem 1

Let $M$ be a compact orientable surface, and let $f : M \to M$ be a homeomorphism. Then $f$ is isotopic to a homeomorphism $F : M \to M$ satisfying exactly one of the following three conditions:

Figures (2)

  • Figure 1: A 3-pronged singularity of a measured foliation with singularities.
  • Figure 2: A singular neighborhood with a 3-pronged singularity. The solid lines and broken lines respectively represent the stable and unstable foliations $\mathcal{F}^s$ and $\mathcal{F}^u$, for example.

Theorems & Definitions (54)

  • Theorem
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5: FS79
  • Proposition 2.6: FS79
  • Definition 2.7
  • Definition 2.8
  • ...and 44 more