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

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 $p>0$,

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