What does a generic Markov operator look like?

Vershik, A.

doi:10.1090/S1061-0022-06-00928-9

What does a generic Markov operator look like?
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by A. M. Vershik
Translated by: N. Tsilevich

St. Petersburg Math. J. 17 (2006), 763-772

DOI: https://doi.org/10.1090/S1061-0022-06-00928-9

Published electronically: July 20, 2006

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Abstract:

Generic (i.e., forming an everywhere dense massive subset) classes of Markov operators in the space $L^2(X,\mu )$ with a finite continuous measure are considered. In a canonical way, each Markov operator is associated with a multivalued measure-preserving transformation (i.e., a polymorphism), and also with a stationary Markov chain; therefore, one can also talk of generic polymorphisms and generic Markov chains. Not only had the generic nature of the properties discussed in the paper been unclear before this research, but even the very existence of Markov operators that enjoy these properties in full or partly was known. The most important result is that the class of totally nondeterministic nonmixing operators is generic. A number of problems is posed; there is some hope that generic Markov operators will find applications in various fields, including statistical hydrodynamics.

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