## Topological ergodic theory and mean rotation

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- by Steve Alpern and V. S. Prasad
- Proc. Amer. Math. Soc.
**118**(1993), 279-284 - DOI: https://doi.org/10.1090/S0002-9939-1993-1143014-5
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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).## References

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## Bibliographic 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