Oscillating and nonsummable Radon-Nikodym cocycles along the forward geodesic of measure-class-preserving transformations

TL;DR

This paper constructs specific measures on the Cantor space where Radon-Nikodym cocycles either oscillate or decay nonsummably along a particular transformation's geodesic, answering open questions in the field.

Contribution

It introduces new examples of Radon-Nikodym cocycles with oscillating or nonsummable behavior, expanding understanding of measure-class-preserving transformations.

Findings

Constructed product measures with oscillating cocycles.

Provided examples with nonsummable decay of cocycles.

Connected cocycle behavior to random walk theory.

Abstract

We consider the least-deletion map on the Cantor space, namely the map that changes the first 1 in a binary sequence to 0, and construct product measures on $2^\mathbb{N}$ so that the corresponding Radon-Nikodym cocycles oscillate or converge to zero nonsummably along the forward geodesic of the map. These examples answer two questions of Tserunyan and Tucker-Drob. We analyze the oscillating example in terms of random walks on $\mathbb{Z}$, using the Chung-Fuchs theorem.