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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Topological ergodic theory and mean rotation
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by Steve Alpern and V. S. Prasad PDF
Proc. Amer. Math. Soc. 118 (1993), 279-284 Request permission

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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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 118 (1993), 279-284
  • MSC: Primary 58F11; Secondary 28D05
  • DOI: https://doi.org/10.1090/S0002-9939-1993-1143014-5
  • MathSciNet review: 1143014