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A ratio ergodic theorem for commuting, conservative, invertible transformations with quasi-invariant measure summed over symmetric hypercubes

Published online by Cambridge University Press:  22 June 2007

JACOB FELDMAN
JACOB FELDMAN
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA (e-mail: feldman@math.berkeley.edu)
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Abstract

Let $T(1),\dots,T(d)$ be conservative, invertible, non-singular, commuting transformations on the Polish measure space $(X,m)$. Then for $f$ and $p$ in $L^1(m)$ with $p>0$,

\[ \frac{{\hat T}(1)_{-N}^N \dotsb {\hat T}(d)_{-N}^Nf}{{\hat T}(1)_{-N}^N \dotsb {\hat T}(d)_{-N}^Np}\to E[f | {\mathcal I}]/E[p| {\mathcal I}]\quad \text{as }N\to\infty. \]

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 4 , August 2007 , pp. 1135 - 1142
Copyright
2007 Cambridge University Press

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