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From decay of correlations to recurrence times in systems with contracting directions

José F. Alves, João S. Matias

TL;DR

This work extends the bidirectional relationship between decay of correlations and recurrence times via Young structures to invertible systems with contracting directions. By developing maximal large deviations on Young towers through discretisation, quotient towers, and lifting, the authors establish sharp polynomial and exponential tail estimates for return times that mirror the observed decay rates of correlations. The results provide a robust framework showing that decay rates imply the existence of Young structures with corresponding recurrence tails, and conversely, enabling a unified treatment of statistical properties for partially hyperbolic systems and their SRB measures. The approach clarifies how nonuniform hyperbolicity, holonomy regularity, and tower dynamics interact to yield precise quantitative recurrence and large-deviation information, with implications for a broad class of dynamical systems exhibiting contracting directions.

Abstract

Classic results by L.-S. Young show that the decay of correlations for systems that admit inducing schemes can be obtained through the recurrence rates of the inducing scheme. Reciprocal results were obtained for non-invertible systems (without contracting directions). Here, we obtain reciprocal results also for invertible systems (with contracting directions).

From decay of correlations to recurrence times in systems with contracting directions

TL;DR

This work extends the bidirectional relationship between decay of correlations and recurrence times via Young structures to invertible systems with contracting directions. By developing maximal large deviations on Young towers through discretisation, quotient towers, and lifting, the authors establish sharp polynomial and exponential tail estimates for return times that mirror the observed decay rates of correlations. The results provide a robust framework showing that decay rates imply the existence of Young structures with corresponding recurrence tails, and conversely, enabling a unified treatment of statistical properties for partially hyperbolic systems and their SRB measures. The approach clarifies how nonuniform hyperbolicity, holonomy regularity, and tower dynamics interact to yield precise quantitative recurrence and large-deviation information, with implications for a broad class of dynamical systems exhibiting contracting directions.

Abstract

Classic results by L.-S. Young show that the decay of correlations for systems that admit inducing schemes can be obtained through the recurrence rates of the inducing scheme. Reciprocal results were obtained for non-invertible systems (without contracting directions). Here, we obtain reciprocal results also for invertible systems (with contracting directions).
Paper Structure (15 sections, 9 theorems, 92 equations)

This paper contains 15 sections, 9 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Young structures
  3. Partially hyperbolic systems
  4. Young towers
  5. Maximal large deviations in the tower
  6. Quotient tower
  7. Discretisation
  8. Maximal large deviations for the discretisation
  9. Polynomical case
  10. Exponential and stretched exponential cases
  11. Maximal large deviations for the original observable
  12. Maximal large deviations in the original system
  13. Lifting to a tower
  14. Maximal large deviations for the lifting
  15. Young structures and recurrence times

Key Result

Theorem 1

Let $f\colon M\to M$ be a $C^{1+\eta}$ diffeomorphism and let $K\subset M$ be a partially hyperbolic set. Suppose that $f$ admits an ergodic $SRB$ measure $\mu$ supported on $K$ with all Lyapunov exponents along $E^{cu}$ positive.

Theorems & Definitions (16)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • ...and 6 more