Local limit theorems for expanding maps
Dmitry Dolgopyat, Yeor Hafouta

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 where are uniformly H\"older functions and are expanding maps. Using a symbolic representation a similar result follows for maps in a small neighborhood of an Axiom A map and H\"older continuous functions . All of our results are already new when all maps are the same but observables 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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Taxonomy
TopicsMathematical Dynamics and Fractals · Advanced Topology and Set Theory · Advanced Differential Equations and Dynamical Systems
