A ratio ergodic theorem via tiling and uniformly syndetic markers

TL;DR

This paper presents a new Borel-based ratio ergodic theorem using tiling and syndetic markers, extending previous results to more complex non-invertible transformations and identifying precise local ergodic limits.

Contribution

It introduces a measureless version of Dowker's ratio ergodic theorem and generalizes the vanishing markers lemma for continuum-to-one Borel transformations.

Findings

Established a Borel ratio ergodic theorem with explicit limit identification.

Extended tiling methods to non-invertible Borel transformations.

Generalized the vanishing markers lemma for continuum-to-one transformations.

Abstract

We prove a purely Borel/measureless version of Dowker's ratio ergodic theorem, from which we derive a strengthening of Dowker's original theorem with a precise identification of the limit of local ergodic ratios. This is done by implementing the pointwise tiling idea of [Tse18] in the more complex setting of continuum-to-one Borel transformations. Along the way, we establish a vanishing markers lemma for these transformations, which generalizes its well-known counterpart for invertible transformations.