Local stability of ergodic averages

TL;DR

This paper provides explicit bounds for local stability of ergodic averages, enabling practical search for stable points and offering a new perspective on the pointwise ergodic theorem through proof-theoretic methods.

Contribution

It introduces explicit bounds on the stability of ergodic averages, contrasting with traditional non-constructive approaches, and applies proof mining techniques to ergodic theory.

Findings

Explicit bounds on n for local stability of ergodic averages

Comparison showing bounds differ from upcrossing inequalities

Application of proof mining to derive constructive results in ergodic theory

Abstract

The mean ergodic theorem is equivalent to the assertion that for every function K and every epsilon, there is an n with the property that the ergodic averages A_m f are stable to within epsilon on the interval [n,K(n)]. We show that even though it is not generally possible to compute a bound on the rate of convergence of a sequence of ergodic averages, one can give explicit bounds on n in terms of K and || f || / epsilon. This tells us how far one has to search to find an n so that the ergodic averages are "locally stable" on a large interval. We use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem, and show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to Bishop and Ivanov. Finally, we explain how our positive results can be viewed as an application of a body of general proof-theoretic methods falling under the heading of "proof mining."