A ratio ergodic theorem via tiling and uniformly syndetic markers
Dynamical Systems
2025-09-23 v3 Logic
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.
Keywords
Cite
@article{arxiv.2208.06463,
title = {A ratio ergodic theorem via tiling and uniformly syndetic markers},
author = {Benjamin D. Miller and Anush Tserunyan},
journal= {arXiv preprint arXiv:2208.06463},
year = {2025}
}
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Final version