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An Abramov formula for stationary spaces of discrete groups

Published online by Cambridge University Press:  15 January 2013

YAIR HARTMAN ,
YURI LIMA  and
OMER TAMUZ
YAIR HARTMAN
Affiliation:
Weizmann Institute of Science, Faculty of Mathematics and Computer Science, PO Box 26, 76100, Rehovot, Israel email yair.hartman@weizmann.ac.ilyuri.lima@weizmann.ac.ilomer.tamuz@weizmann.ac.il
YURI LIMA
Affiliation:
Weizmann Institute of Science, Faculty of Mathematics and Computer Science, PO Box 26, 76100, Rehovot, Israel email yair.hartman@weizmann.ac.ilyuri.lima@weizmann.ac.ilomer.tamuz@weizmann.ac.il
OMER TAMUZ
Affiliation:
Weizmann Institute of Science, Faculty of Mathematics and Computer Science, PO Box 26, 76100, Rehovot, Israel email yair.hartman@weizmann.ac.ilyuri.lima@weizmann.ac.ilomer.tamuz@weizmann.ac.il
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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 $G$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 3 , June 2014 , pp. 837 - 853
Copyright
©2013 Cambridge University Press 

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