An Abramov formula for stationary spaces of discrete groups
Yair Hartman, Yuri Lima, Omer Tamuz · Ergodic Theory and Dynamical Systems · 2013
Abstract Let $(G, \mu )$ be a discrete group equipped with a generating probability measure, and let $\Gamma $ be a finite index subgroup of $G$ . A $\mu $ -random walk on $G$ , starting from the identity, returns to $\Gamma $ with probability one. Let $\theta $ be the hitting measure, or the distribution of the position in which the random walk first hits $\Gamma $ . We prove that the Furstenberg entropy of a $(G, \mu )$ -stationary space, with respect to the action of $(\Gamma , \theta )$ , is equal to the Furstenberg entropy with respect to the action of $(G, \mu )$ , times the index of $\Gamma $ in $G$ . The index is shown to be equal to the expected return time to $\Gamma $ . As a corollary, when applied to the Furstenberg–Poisson boundary of $(G, \mu )$ , we prove that the random walk entropy of $(\Gamma , \theta )$ is equal to the random walk entropy of $(G, \mu )$ , times the index of $\Gamma $ in $G$ .