Growth and mixing

Krzysztof Frączek,Leonid Polterovich

doi:10.3934/jmd.2008.2.315

Abstract

Given a bi-Lipschitz measure-preserving homeomorphism of a ﬁnite dimensional compact metric measure space, consider the sequence of 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

Highlights

Introduction and main resultsLet (M, ρ, μ) be a compact metric space endowed with a probability Borel measure μ with supp(μ) = M
In the present note we discuss a link between dynamics of φ ∈ G and geometry of the cyclic subgroup of G generated by φ (the growth rate of Γ(φn) as n → ∞.) On the geometric side, we focus on the quantity
Let us return to the study of the interplay between growth and mixing: we explore the inuence of the rate of mixing on the growth of Γn(φ)

Summary

Introduction and main results

The cyclic subgroup {φn} has the identity map as its limit point with respect to C0-topology This theorem has the following application to bi-Lipschitz ergodic theory 1/d ν nd Organization of the paper: In Section 2 we prove the universal lower growth bound given in Theorem 1.2 for a bi-Lipschitz homeomorphism which mixes a Lipschitz function (the case of homeomorphism which mixes an L2function is considered). In Appendix we prove Kolmogorov-Tihomirov type estimate (2)

Recurrence via Arzela-Ascoli compactness

Almost orthonormal systems of Lipschitz functions

From Lipschitz to Hölder observables

Existence of an adjoint sequence

Slowly growing diffeomorphisms

Growth of the Rudin-Shapiro shift