Growth and mixing

TL;DR

This paper investigates the growth rate of Lipschitz norms of iterated measure-preserving homeomorphisms on compact metric spaces, establishing universal lower bounds related to the space's dimension and mixing properties.

Contribution

It provides new lower bounds on the growth of Lipschitz norms for measure-preserving homeomorphisms, including sharp bounds for rapid mixing scenarios.

Findings

Universal lower bound depending on space dimension

Lower bound for rapid mixing cases

Example using Rudin-Shapiro sequence demonstrating sharpness

Abstract

Given a bi-Lipschitz measure-preserving homeomorphism of a compact metric measure space of finite dimension, consider the sequence formed by the Lipschitz norms of its iterations. We obtain lower bounds on the growth rate of this sequence assuming that our homeomorphism mixes a Lipschitz function. In particular, we get a universal lower bound which depends on the dimension of the space but not on the rate of mixing. Furthermore, we get a lower bound on the growth rate in the case of rapid mixing. The latter turns out to be sharp: the corresponding example is given by a symbolic dynamical system associated to the Rudin-Shapiro sequence.