What does a generic Markov operator look like.pdf

a r X i v : m a t h / 0 5 1 0 3 2 0 v 1 [ m a t h .F A ] 1 5 O c t 2 0 0 5 What does a generic Markov operator look like? A. M. Vershik? To the memory of O. A. Ladyzhenskaya Abstract We consider generic (i.e., forming an everywhere dense massive sub- set) classes of Markov operators in the space L2(X,μ) with a finite continuous measure. Since there is a canonical correspondence that as- sociates with each Markov operator a multivalued measure-preserving transformation (i.e., a polymorphism), as well as a stationary Markov chain, we can also speak about generic polymorphisms and generic Markov chains. It was not known not only that the suggested proper- ties are generic, but even whether there exist Markov operators that have simultaneously all or a part of them. The most important re- sult is that the class of totally nondeterministic nonmixing operators is generic. We pose a number of problems and express the hope that generic Markov operators will find applications in various fields, in- cluding statistical hydrodynamics. I was lucky to be friends with O.A., especially in the 70s, and some time I will write about this. In the late 60s and 70s, 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 me to start a joint research on multivalued solutions of equations. Our work resulted in a series of papers, see [13, 3]. We also had grandiose projects for further research, for example, of working on metric hydrodynamics, but they never were realized. Approximately at the same time, I started to develop general (multivalued) dynamics ([12]), 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. ?St. Petersburg Department of Steklov Institute of Mathematics. Fontanka 27, 191023 St. Petersburg, Russia. E-mail: versh