Random Entropy and Recurrence

1. Motivation and introduction. In [1], the concept of random entropy associated with a Zd random group action was introduced and studied. Every such Zd random group action is generated via a cocycle. (For the readers with a probabilistic background, a cocycle is a generalization of an ordinary random walk, the main difference being the fact that cocycles are generally indexed by Zd rather than by Z. A one-dimensional cocycle is nothing but an ordinary random walk; we give precise definitions in Section 2.) In the one-dimensional case, it is easy to see that having positive random entropy is equivalent to the transience of the associated random walk. It therefore seems reasonable to try to connect the concept of random entropy as developed in [1] and the transience of the generating cocycle. In this paper, we show that the one-dimensional connection holds in general. The paper is completely self-contained. Section 2 contains the setup, including all necessary definitions and the main results, while Sections 3 and 4 contain the proofs.