2005/10/15 by A. M. Vershik, Vershik, A.
Computer Science · #28D05 #47A45 #60J27 #Computability, Logic, AI Algorithms #FOS: Mathematics #Functional Analysis (math.FA) #Fuzzy Logic and Control Systems #Neural Networks and Applications #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.math/0510320
openalex publication_date 2005/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider generic i.e., forming an everywhere dense massive subset classes of Markov operators in the space L2(X,μ) with a finite continuous measure. Since there is a canonical correspondence that associates 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. The most important and inexpected generic properties of Markov operators (or Markov chains or polymorphisms) is nonmixing and totally nondeterministicity. It was not known even existence of such Markov operators (the first example due to M.Rozenblatt). We suppose that this class coinsided with the class of special random perturbations of K-automorphisms. This theory is measure theoretic counterpart of the theory of nonselfadjoint contractions and its application.