# American Institute of Mathematical Sciences

2005, 13(5): 1305-1324. doi: 10.3934/dcds.2005.13.1305

## Polymorphisms, Markov processes, quasi-similarity

 1 St. Petersburg Department of Steklov Institute of Mathematics, 27 Fontanka, St. Petersburg, 191023, Russian Federation

Received  October 2004 Revised  March 2005 Published  September 2005

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 $T$.
Citation: A. M. Vershik. Polymorphisms, Markov processes, quasi-similarity. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1305-1324. doi: 10.3934/dcds.2005.13.1305
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