Polymorphisms, Markov processes, quasi-similarity

Abstract

In this paper we develop the theory of 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. We formulate the important notion of quasi-similarity and consider quasi-similarity between polymorphisms and automorphisms. The question is as follows: is it possible to have a quasi-similarity between a measure-preserving automorphism $T$ and a polymorphism $\Pi$ (that is not an automorphism)? In less definite terms: what kind of equivalence can exist between deterministic and random (Markov) dynamical systems? We give the answer: every nonmixing prime polymorphism is quasi-similar to an automorphism with positive entropy, and every $K$-automorphism $T$ is quasi-similar to a polymorphism $\Pi$ that is a special random perturbation of the automorphism $T$.