Ergodicity of Poisson products and applications

Details Ergodicity of Poisson products and applications Ergodicity of Poisson products and applications

2011-07-04

arxiv.org/abs/1107.0520v1

Mathematics

Dynamical Systems

10 pages

Scientific paper

For every $\sigma$-finite measure-preserving transformation $T$ acting on a space $X$ there is an associated probability preserving transformation $T_*$ which acts on discrete countable subsets of $X$. This is the Poisson suspension of $T$. We prove ergodicity of the \emph{Poisson-product} $T \times T_*$ under the assumption that $T$ is ergodic and conservative. From this we deduce some probabilistic results: The ergodicity of the "first return of left-most transformation" associated with a measure preserving transformation on $\mathbb{R}_+$, and the non-existence of a $T$-invariant Poisson-thinning. We discuss ergodicity for the Poisson-product of measure preserving group actions, and related spectral properties.

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