A ratio ergodic theorem via tiling and uniformly syndetic markers

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.