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Decay of Correlations via induced Weak Gibbs Markov maps for non-Hölder observables

Asad Ullah, Helder Vilarinho

TL;DR

This work extends decay-of-correlations, Central Limit Theorem, and Large Deviations results to dynamical systems admitting an induced weak Gibbs–Markov map for a broad class of non-Hölder observables. The authors build a mixing tower over the induced map and develop a coupling framework that transfers statistical properties from the tower to the original system via a semi-conjugacy. By relating observable regularities on the base to tower regularities and bounding convergence to equilibrium through return-time tails, they obtain polynomial or exponential decay rates depending on tail behavior, along with conditions for CLT (notably when $a>2$) and Large Deviations. The results unify and extend prior work on non-Hölder observables, providing explicit rates tied to return-time tails and offering practical tools for proving statistical properties in a wider class of systems. The approach leverages induced tower structures and robust coupling arguments to bridge non-Hölder regularity with sharp probabilistic limit theorems in dynamical systems.

Abstract

We obtain estimates on the decay of correlations, Central Limit Theorem and Large Deviations for dynamical systems admitting an induced weak Gibbs--Markov map, for larger classes of observables with weaker regularity than Hölder, characterized by suitable moduli of continuity.

Decay of Correlations via induced Weak Gibbs Markov maps for non-Hölder observables

TL;DR

This work extends decay-of-correlations, Central Limit Theorem, and Large Deviations results to dynamical systems admitting an induced weak Gibbs–Markov map for a broad class of non-Hölder observables. The authors build a mixing tower over the induced map and develop a coupling framework that transfers statistical properties from the tower to the original system via a semi-conjugacy. By relating observable regularities on the base to tower regularities and bounding convergence to equilibrium through return-time tails, they obtain polynomial or exponential decay rates depending on tail behavior, along with conditions for CLT (notably when ) and Large Deviations. The results unify and extend prior work on non-Hölder observables, providing explicit rates tied to return-time tails and offering practical tools for proving statistical properties in a wider class of systems. The approach leverages induced tower structures and robust coupling arguments to bridge non-Hölder regularity with sharp probabilistic limit theorems in dynamical systems.

Abstract

We obtain estimates on the decay of correlations, Central Limit Theorem and Large Deviations for dynamical systems admitting an induced weak Gibbs--Markov map, for larger classes of observables with weaker regularity than Hölder, characterized by suitable moduli of continuity.
Paper Structure (7 sections, 15 theorems, 77 equations)

This paper contains 7 sections, 15 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and statement of main results
  3. Decay of Correlations for Tower Maps for larger classes of Observables
  4. Tower Maps
  5. Coupling
  6. Polynomial Return time
  7. Exponential Return time

Key Result

Theorem A

Consider a measure space $(M,\mathcal{B},m)$, endowed with some metric, and a measurable map $f: M\rightarrow M$ satisfying $f_{*}m \ll m$. Let $f^{R}: \Delta_{0} \rightarrow \Delta_{0}$ be an aperiodic induced WGM expanding map with a coprime block, $R \in L^{1}(m)$ and let $\mu$ be the unique ergo

Theorems & Definitions (22)

  • Definition 2.1
  • Theorem A
  • Corollary B: Central Limit Theorem
  • Corollary C: Large Deviations
  • Example 2.2
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Remark 3.5
  • ...and 12 more