A quantitative Mean Ergodic Theorem for uniformly convex Banach spaces

TL;DR

This paper establishes an explicit uniform bound on the local stability of ergodic averages in uniformly convex Banach spaces, extending prior results from Hilbert spaces to a broader class of spaces.

Contribution

It provides a quantitative, finitary version of the Mean Ergodic Theorem for uniformly convex Banach spaces, generalizing earlier results from Hilbert spaces.

Findings

Derived explicit uniform bounds for ergodic averages

Extended results from Hilbert spaces to uniformly convex Banach spaces

Generalized the finitary version of the Mean Ergodic Theorem

Abstract

We provide an explicit uniform bound on the local stability of ergodic averages in uniformly convex Banach spaces. Our result can also be viewed as a finitary version in the sense of T. Tao of the Mean Ergodic Theorem for such spaces and so generalizes similar results obtained for Hilbert spaces by Avigad, Gerhardy and Towsner (arXiv:0706.1512v2 [math.DS]) and T. Tao (arXiv:0707.1117v1 [math.DS]).