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 $G_δ$ subset of measure preserving transformations such that for every $T\in G$ and $k(1), k(2), \dots, k(l)\in \mathbb{Z}^+$, $k'(1), k'(2), \dots, k'(l')\in \mathbb{Z}^+$, 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 $(k'(1), k'(2), \dots, k'(l'))$. We will introduce an analogous orthogonality conditions for continuous unitary representations of $L^0(μ,\mathbb{T})$ which we denote by DL--condition. We connect the DL--condition with a result of Solecki which states that every continuous unitary representations of $L^0(μ,\mathbb{T})$ is a direct sum of action by pointwise multiplication on measure spaces $(X^{|κ|},λ_κ)$ where $κ$ 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 $λ_κ$.