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

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.