What Does a Generic Markov Operator Look Like?

Generic (i.e., forming an everywhere dense massive subset) classes of Markov operators in the space L2(X,μ) with a finite continuous measure are considered. In a canonical way, each Markov operator is associated with a multivalued measure-preserving transformation (i.e., a polymorphism), and also with a stationary Markov chain; therefore, one can also talk of generic polymorphisms and generic Markov chains. Not only had the generic nature of the properties discussed in the paper been unclear before this research, but even the very existence of Markov operators that enjoy these properties in full or partly was known. The most important result is that the class of totally nondeterministic nonmixing operators is generic. A number of problems is posed; there is some hope that generic Markov operators will find applications in various fields, including statistical hydrodynamics. I was fortunate to be friends with O.A., especially in the 1970s, and some time I will write about this. In the late 1960s and 1970s, she was very interested in dynamical systems, and this was an additional motive for our contacts. Working on the Hopf equation, she arrived at the necessity of considering multivalued (Markov) mappings and suggested that we start a joint research on multivalued solutions of equations. Our work resulted in a series of papers; see [4, 5]. We also had grandiose projects for further research, for example, working on metric hydrodynamics, but these were never realized. Approximately at the same time, I started to develop general (multivalued) dynamics ([1]), and recently, after a long interval, I have returned to this subject. In this paper, dedicated to the unforgettable O.A., I continue this topic. §1. Markov operators