Partitions with independent iterates in random dynamical systems

TL;DR

This paper extends Krengel's characterization of weakly mixing transformations to random dynamical systems by analyzing partitions with independent iterates, using advanced ergodic theory tools.

Contribution

It generalizes Krengel's results to weakly mixing random dynamical systems, providing a new perspective on partitions with independent iterates.

Findings

Weakly independent partitions are dense in weakly mixing systems.

Extension of Krengel's theorem to random dynamical systems.

Use of advanced ergodic theory tools for analysis.

Abstract

Consider an invertible measure-preserving transformation of a probability space. A finite partition of the space is called weakly independent if there are infinitely many images of this partition under powers of the transformation that are jointly independent. Krengel proved that a transformation is weakly mixing if and only if weakly independent partitions of the underlying space are dense among all finite partitions. Using the tools developed in the later papers of del Junco-Reinhold-Weiss and del Junco-Begun we obtain Krengel- type results for weakly mixing random dynamical systems (or equivalently, skew products that are relatively weakly mixing).