Vol. 31, No. 2, 1969

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Approximation of transformations with continuous spectrum

Rafael Van Severen Chacon

Vol. 31 (1969), No. 2, 293–302

In several recent papers a new approach has been developed in the theory of approximation of automorphisms. Using this approach, Katok and Stepin have developed a new method which is very powerful and which has enabled them to solve several problems which had remained open for some time. Among the results they obtained is a characterization of automorphisms which are not strongly mixing in terms of the speed with which they can be approximated. Counter-examples may be given to show that the speed of approximation cannot be used to characterize those automorphisms which have continuous spectrum. In this paper certain related concepts are developed which do make it possible to deal in general with automorphisms which have continuous spectrum, and to distinguish those which are not strongly mixing from those which are strongly mixing among them. Since it is well-known that automorphisms have continuous spectrum if and only if they are weakly mixing, the result serves to distinguish between strong and weak mixing.

Mathematical Subject Classification
Primary: 28.70
Received: 17 January 1969
Published: 1 November 1969
Rafael Van Severen Chacon