Polymorphisms, Markov processes, random perturbations of K-automorphisms

Abstract

In this paper we develop the theory of {\it polymorphisms} of measure spaces, which is a generalization of the theory of measure-preserving transformations; we describe the main notions and discuss relations to the theory of Markov processes, operator theory, ergodic theory, etc. Using these tools, we solve the problem, which appeared in dynamics in the 70s: what kind of equivalence can exist between deterministic and random dynamical systems; our refining of this question is as follows: is it possible to have the quasi-similarity of a measure-preserving automorphism and a polymorphism. We prove that it is possible and the automorphism must be a $K$-automorphism and the polymorphism must be a special random perturbation of the automorphism; more exactly, the polymorphism is a random walk over stable leaves of the $K$-automorphism. The main part of our analysis concerns quasi-deterministic Markov processes and their tail (residual) $σ$-fields; such processes correspond to the most interesting class of polymorphisms: prime nonmixing polymorphisms.