Transfer operators and atomic decomposition

TL;DR

This paper introduces a novel approach using atomic decomposition and Banach spaces to analyze transfer operators for piecewise maps, demonstrating their quasi-compactness and enabling statistical properties like decay of correlations.

Contribution

It develops a new framework to study transfer operators with low regularity, showing they are quasi-compact and facilitating statistical limit theorems for broad classes of observables.

Findings

Transfer operators are quasi-compact under low regularity conditions.

Exponential decay of correlations is achievable.

Statistical limit theorems hold for unbounded observables.

Abstract

We use the method of atomic decomposition and a new family of Banach spaces to study the action of transfer operators associated to piecewise-defined maps. It turns out that these transfer operators are quasi-compact even when the associated potential, the dynamics and the underlying phase space have very low regularity. In particular it is often possible to obtain exponential decay of correlations, the Central Limit Theorem and almost sure invariance principle for fairly general observables, including unbounded ones.