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Superstability of hyperbolic automorphisms and unitary dilatations of Markov operators. (English. Russian original) Zbl 0655.60010

Vestn. Leningr. Univ., Math. 20, No. 3, 22-29 (1987); translation from Vestn. Leningr. Univ., Ser. I 1987, No. 3, 28-33 (1987).
Summary: In the paper one constructs a random perturbation of hyperbolic systems, which generates a Markov process with the following property: on its boundary a shift in the space of trajectories induces a transformation, metrically isomorphic to the initial one and, at the same time, the process is completely nondeterministic, i.e., does not have deterministic quotient processes.
An example is connected with the theory of dilatations of positive contractions and with the Sz.-Nagy-Foias classes. One has a probabilistic interpretation, generalizing M. Rosenblatt’s example [Markov processes. Structure and asymptotic behaviour. (1971; Zbl 0236.60002)]. One poses the problem of the determination of all dynamical systems with a similar property (superstability).

MSC:

60B99 Probability theory on algebraic and topological structures
60J99 Markov processes

Citations:

Zbl 0236.60002
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