Generic Behavior of a Measure Preserving Transformation

Del Junco–Lema'nczyk 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’(1), k’(2), \dots, k’(l’)\in $\mathbb{Z}$^+$, the convolutions $${\sigma }_{T^{k(1)}}\ast \cdots \ast {\sigma }_{T^{k(l)}}\text{ and }{\sigma }_{T^{k^{\mathrm{\prime }}(1)}}\ast \cdots \ast {\sigma }_{T^{k^{\mathrm{\prime }}(l^{\mathrm{\prime }})}}$$ are mutually singular, provided that $(k(1), k(2), \dots, k(l))$isnotarearrangementof$(k’(1), k’(2), \dots, k’(l’))$. 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 $(X^{\mathrm{\mid }\kappa \mathrm{\mid }},{\lambda }_{\kappa })$ 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 ${\lambda }_{\kappa }$.