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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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 $ {\mathcal{H}_0}({M^n},\mu )$ denote the set of all homeomorphisms of a compact manifold $ {M^n}$ that preserve a locally positive nonatomic Borel probability measure $ \mu $ 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 $ \theta $ on $ {\mathcal{H}_0}({M^n},\mu )$. We show that any abstract ergodic behavior typical for automorphisms of $ ({M^n},\mu )$ as a Lebesgue space is also typical not only in $ {\mathcal{H}_0}({M^n},\mu $ but also in each closed subset of constant $ \theta $. By typical we mean dense $ {G_\delta }$ 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 $ \overrightarrow v $, there is an open neighborhood of rotation by $ \overrightarrow v $, in the space of smooth volume preserving $ n$-torus diffeomorphisms with $ \theta = \overrightarrow v $, where each diffeomorphism in the open set is conjugate to rotation by $ \overrightarrow v $ (and hence cannot be weak mixing).

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