TY - JOUR

T1 - An Abramov formula for stationary spaces of discrete groups

AU - Hartman, Yair

AU - Lima, Yuri

AU - Tamuz, Omer

N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 2014/6

Y1 - 2014/6

N2 - Let (G,μ) be a discrete group equipped with a generating probability measure, and let Γ be a finite index subgroup of G. A μ -random walk on G, starting from the identity, returns to Γ with probability one. Let θ be the hitting measure, or the distribution of the position in which the random walk first hits Γ. We prove that the Furstenberg entropy of a (G,μ)-stationary space, with respect to the action of (Γ, θ), is equal to the Furstenberg entropy with respect to the action of (G,μ), times the index of Γ in G. The index is shown to be equal to the expected return time to Γ. As a corollary, when applied to the Furstenberg-Poisson boundary of (G,μ), we prove that the random walk entropy of (Γ, θ) is equal to the random walk entropy of (G,μ), times the index of Γ in G.

AB - Let (G,μ) be a discrete group equipped with a generating probability measure, and let Γ be a finite index subgroup of G. A μ -random walk on G, starting from the identity, returns to Γ with probability one. Let θ be the hitting measure, or the distribution of the position in which the random walk first hits Γ. We prove that the Furstenberg entropy of a (G,μ)-stationary space, with respect to the action of (Γ, θ), is equal to the Furstenberg entropy with respect to the action of (G,μ), times the index of Γ in G. The index is shown to be equal to the expected return time to Γ. As a corollary, when applied to the Furstenberg-Poisson boundary of (G,μ), we prove that the random walk entropy of (Γ, θ) is equal to the random walk entropy of (G,μ), times the index of Γ in G.

UR - http://www.scopus.com/inward/record.url?scp=84899812602&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84899812602&partnerID=8YFLogxK

U2 - 10.1017/etds.2012.167

DO - 10.1017/etds.2012.167

M3 - Article

AN - SCOPUS:84899812602

VL - 34

SP - 837

EP - 853

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 3

ER -