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Residuality of dynamical morphisms. (English) Zbl 0963.37007

A unified approach to three well-known theorems of ergodic theory is given: the theorem of Krieger that every measure-preserving ergodic transformation \(T\) with finite entropy, \(h(T)<\infty\), has a finite generator; the theorem of Sinai that if \(T_1\) is a Bernoulli shift with \(h(T_1)\leq h(T)\) then \(T\) is homeomorphic to \(T_1\); and the theorem of Ornstein that two Bernoulli shifts with the same entropy are isomorphic.
The maps \(\Phi\) producing isomorphisms resp. homeomorphisms are identified with the corresponding probability measures on the product space which are invariant and ergodic under the transformation \(T\times T_1\) and concentrated on the graph of \(\Phi\). If we consider the usual metric space \(M\) of all probability measures on the product space, then it is proved that in all three cases the set of isomorphisms resp. homeomorphisms is the countable intersection of dense open subsets of a suitable subspace of \(M\) which possess the Baire property that countable intersections of open dense subsets are dense. Hence these sets are nonempty.

MSC:

37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
37A05 Dynamical aspects of measure-preserving transformations
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