Some problems on measure-preserving transformations

TL;DR

This paper explores various complex problems in ergodic theory, including transformations, mixing properties, and open questions, providing insights and experimental observations relevant to measure-preserving systems.

Contribution

It introduces new results on multiple mixing, discusses open problems, and connects theoretical concepts with experimental observations in ergodic theory.

Findings

Proven multiple mixing property for systems with Gordin ergodic group

Experimental observation of dominant infinite threads in harmonic Z2-sequences

Discussion of the absence of Lehrer and Weiss property for Z^2-lattice actions

Abstract

This text is addressed to students. It is a short story about some problems in ergodic theory, both related and independent. We discuss the factorization of transformations into the product of three involutions; Furstenberg's theorem on weak multiple mixing in the mean along progressions; ergodic Furstenberg-Roth's theorem. In connection with Ledrappier's $Z^2$-action, we consider harmonic $Z_2$-sequences on $Z^2$ and experimentally observe the dominant infinite threads arising in them. The property of multiple mixing for systems with the Gordin ergodic group is proven. We discuss the absence of the Lehrer and Weiss property for the action of $\mathbb{Z}^2$-lattices, and Bergelson's question on sequence mixing. Some new open problems, unusual spectrum of a measure-preserving construction with related Katok's paradoxical effect are mentioned.