  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.
