Topological ergodic theory and mean rotation

Authors:
Steve Alpern and V. S. Prasad

Journal:
Proc. Amer. Math. Soc. **118** (1993), 279-284

MSC:
Primary 58F11; Secondary 28D05

MathSciNet review:
1143014

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Abstract: Let denote the set of all homeomorphisms of a compact manifold that preserve a locally positive nonatomic Borel probability measure and are isotopic to the identity. The notion of the mean rotation vector for a torus homeomorphism has been extended by Fathi to a continuous map on . We show that any abstract ergodic behavior typical for automorphisms of as a Lebesgue space is also typical not only in but also in each closed subset of constant . By typical we mean dense in the appropriate space. Weak mixing is an example of such a typical abstract ergodic behavior. This contrasts sharply with a deep result of the KAM theory that for some rotation vectors , there is an open neighborhood of rotation by , in the space of smooth volume preserving -torus diffeomorphisms with , where each diffeomorphism in the open set is conjugate to rotation by (and hence cannot be weak mixing).

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1993-1143014-5

Article copyright:
© Copyright 1993
American Mathematical Society