Some problems on measure-preserving transformations

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.