Polynomial decay of correlations in the generalized baker's transformation

TL;DR

This paper introduces a family of area-preserving generalized baker's transformations with polynomial decay of correlations, providing a geometric framework and a unified approach to analyze mixing rates in non-uniformly hyperbolic systems.

Contribution

It constructs a new class of baker's maps with sharp polynomial mixing rates using a geometric approach based on a single-variable cut function.

Findings

All polynomial decay rates can be achieved within the class.

The analysis relies on constructing Young towers for the expanding factor.

Examples extend known 1-D non-uniformly expanding maps.

Abstract

We introduce a family of area preserving generalized baker's transformations acting on the unit square and having sharp polynomial rates of mixing for Holder data. The construction is geometric, relying on the graph of a single variable "cut function". Each baker's map B is non-uniformly hyperbolic and while the exact mixing rate depends on B, all polynomial rates can be attained. The analysis of mixing rates depends on building a suitable Young tower for an expanding factor. The mechanisms leading to a slow rate of correlation decay are especially transparent in our examples due to the simple geometry in the construction. For this reason we propose this class of maps as an excellent testing ground for new techniques for the analysis of decay of correlations in non-uniformly hyperbolic systems. Finally, some of our examples can be seen to be extensions of certain 1-D non-uniformly expanding maps that have appeared in the literature over the last twenty years thereby providing a unified treatment of these interesting and well-studied examples.