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

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 analyse the oscillating example in terms of random walks on $\mathbb{Z}$, using the Chung–Fuchs theorem.