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Rates of convergence in CLT and ASIP for sequences of expanding maps

Dmitry Dolgopyat, Yeor Hafouta

Abstract

We prove Berry-Esseen theorems and the almost sure invariance principle with rates for partial sums of the form $S_n=\sum_{j=0}^{n-1}f_j\circ T_{j-1}\circ\cdots\circ T_1\circ T_0$ where $f_j$ are functions with uniformly bounded ``variation" and $T_j$ is a sequence of expanding maps. Using symbolic representations similar result follow for maps $T_j$ in a small $C^1$ neighborhood of an Axiom A map and Hölder continuous functions $f_j$. All of our results are already new for a single map $T_j=T$ and a sequence of different functions $(f_j)$.

Rates of convergence in CLT and ASIP for sequences of expanding maps

Abstract

We prove Berry-Esseen theorems and the almost sure invariance principle with rates for partial sums of the form where are functions with uniformly bounded ``variation" and is a sequence of expanding maps. Using symbolic representations similar result follow for maps in a small neighborhood of an Axiom A map and Hölder continuous functions . All of our results are already new for a single map and a sequence of different functions .
Paper Structure (45 sections, 34 theorems, 213 equations)

This paper contains 45 sections, 34 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Non-autonomous dynamical systems
  3. Our results
  4. The layout of the paper
  5. Preliminaries
  6. Setup
  7. Changing the reference measures.
  8. Main results
  9. Examples
  10. Piecewise expanding maps on the interval
  11. High dimensional piecewise expanding maps.
  12. Covering maps and sequential SFT
  13. Smooth expanding maps
  14. Subshifts of finite type
  15. Two sided SFT
  16. ...and 30 more sections

Key Result

Lemma 2.3

Under (LY1) and (LY2) we have where $\textbf{1}$ denotes the function taking the constant value $1$, regardless of its domain.

Theorems & Definitions (74)

  • Example 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Theorem 3.1
  • ...and 64 more