Generic Behavior of a Measure Preserving Transformation

TL;DR

This paper explores the generic orthogonality properties of measure preserving transformations and extends these concepts to continuous unitary representations of $L^0(,)$, establishing a connection between probabilistic and deterministic orthogonality conditions.

Contribution

It introduces DL--conditions for unitary representations of $L^0(,)$ and links these to existing results, providing a new perspective on orthogonality in dynamical systems.

Findings

Dense $G_\u03delta$ subset of transformations with orthogonality properties

DL--conditions for continuous unitary representations established

Connection between probabilistic and deterministic orthogonality conditions

Abstract

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 \[ \sigma_{T^{k(1)}} \ast\cdots\ast \sigma_{T^{k(l)}} \ \text{and} \ \sigma_{T^{k'(1)}} \ast\cdots \ast\sigma_{T^{k'(l')}} \] 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 $L^0(\mu,\mathbb{T})$ 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 $\lambda_\kappa$.