The ergodic decomposition defined by actions of amenable groups
Volume 165 / 2021
Abstract
Let $G$ be a locally compact separable metric group, and assume that $G$ is amenable. Let $m_{_{\mathrm {L}}}$ be a left Haar measure on $G$, and let $\alpha =(F_n)_{n\in \mathbb {N}}$ be a tempered Følner sequence. Let $(X,d)$ be a locally compact separable metric space, and let $w:G\times X\to X$ be a left action which is jointly measurable and continuous in the first variable. Our goal in this work is to obtain an ergodic decomposition of $X$ defined by $w$.
An essential tool in obtaining the decomposition is an ergodic theorem of Elon Lindenstrauss [Invent. Math. 146 (2001)].
The decomposition is obtained by studying the convergence properties of the sequences $\big (\frac {1}{m_{_{\mathrm {L}}}(F_n)}\int _{F_n}h(gx)\,\mathrm {d}m_{_{\mathrm {L}}}(g)\big )_{n\in \mathbb {N}}$, $x\in X$, $h\in C_0(X)=$ the Banach space of all real-valued continuous functions that vanish at infinity (along compact sets), where $gx$ stands for $w(g,x)$, $g\in G$, $x\in X$.
