Generic Behavior of a Measure Preserving Transformation

Abstract

Del Junco--Lemańczyk showed that a generic measure preserving transformation satisfies a certain orthogonality conditions. More precisely, there is a dense $G_{\delta }$ subset of measure preserving transformations such that for every $T\in G$ and $k(1),k(2),\dots ,k(l)\in {\mathbb{Z}}^{+}$, $k^{\prime }(1),k^{\prime }(2),\dots ,k^{\prime }(l^{\prime })\in {\mathbb{Z}}^{+}$, the convolutions \[ σ_{T^{k(1)}} \ast\cdots\ast σ_{T^{k(l)}} \ \text{and} \ σ_{T^{k'(1)}} \ast\cdots \astσ_{T^{k'(l')}} \] are mutually singular, provided that $(k(1),k(2),\dots ,k(l))$ is not a rearrangement of $(k^{\prime }(1),k^{\prime }(2),\dots ,k^{\prime }(l^{\prime }))$. We will introduce an analogous orthogonality conditions for continuous unitary representations of $L^0(\mu ,\mathbb{T})$ which we denote by DL--condition. We connect the DL--condition with a result of Solecki which states that every continuous unitary representations of $L^0(\mu ,\mathbb{T})$ is a direct sum of action by pointwise multiplication on measure spaces $(X^{|\kappa |},{\lambda }_{\kappa })$ where $\kappa$ is an increasing finite sequence of non-zero integers. In particular, we show that the "probabilistic" DL-condition translates to "deterministic" orthogonality conditions on the measures ${\lambda }_{\kappa }$.

19 pages

References in corpus (2)

• Generic representations of abelian groups and extreme amenability
• Closed subgroups generated by generic measure automorphisms