Limit Theorems for Generalized Baker's Transformations

TL;DR

This paper investigates decay of correlations and limit theorems for generalized baker's transformations, revealing sharp mixing rates and convergence to various stable distributions, including non-normal ones, using advanced mathematical techniques.

Contribution

It introduces the first application of anisotropic Banach space methods and operator renewal theory to generalized baker's transformations, providing new limit theorem results.

Findings

Sharp rates of mixing for Lipschitz functions

Limit theorems showing convergence to stable distributions

Stable distributions with any skewness parameter in [-1,1] obtained as limits

Abstract

In this paper we study decay of correlations and limit theorems for generalized baker's transformations. Our examples are piecewise non-uniformly hyperbolic maps on the unit square that posses two spatially separated lines of indifferent fixed points. We obtain sharp rates of mixing for Lipschitz functions on the unit square and limit theorems for H\"older observables on the unit square. Some of our limit theorems exhibit convergence to non-normal stable distributions for H\"older observables. We observe that stable distributions with any skewness parameter in the allowable range of $[-1,1]$ can be obtained as a limit and derive an explicit relationship between the skewness parameter and the values of the H\"older observable along the lines of indifferent fixed points. This paper is the first application of anisotropic Banach space methods and operator renewal theory to generalized baker's transformations. Our decay of correlations results recover the results of Bose and Murray. Our results on limit theorems are new for generalized baker's transformations.