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arXiv · 2407.08690

Local limit theorems for expanding maps

Dmitry Dolgopyat·Yeor Hafouta
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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 Hölder 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 Hölder 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.

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Dmitry Dolgopyat, Yeor Hafouta. 2024-07-11. Local limit theorems for expanding maps. https://arxiv.org/abs/2407.08690

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