Local limit theorems for expanding maps

TL;DR

This paper establishes local limit theorems for sums involving sequences of expanding maps and H"older functions, extending classical results to non-stationary dynamical systems with variable observables.

Contribution

It introduces new local limit theorems for non-stationary expanding maps with varying observables, including a symbolic approach and reduction theory for sequential systems.

Findings

Proved local limit theorems for sums of non-stationary expanding maps.

Extended results to maps near Axiom A maps in the $C^1$ topology.

Provided a reduction theory for sequential dynamical systems.

Abstract

We prove local central limit theorems for partial sums of the form \newline $\,S_n=\sum_{j=0}^{n-1}f_j\circ T_{j-1}\circ\cdots\circ T_1\circ T_0$ where $f_j$ are uniformly H\"older functions and $T_j$ are expanding maps. Using a symbolic representation a similar result follows for maps $T_j$ in a small $C^1$ neighborhood of an Axiom A map and H\"older continuous functions $f_j$. All of our results are already new when all maps are the same $T_j=T$ but observables $(f_j)$ are different. The current paper compliments [43] where Berry--Esseen theorems are obtained. An important step in the proof is developing an appropriate reduction theory in the sequential case.