Topological ergodic theory and mean rotation
Authors:
Steve Alpern and V. S. Prasad
Journal:
Proc. Amer. Math. Soc. 118 (1993), 279284
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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 A. Fathi, Structure of the group of measure preserving homeomorphisms preserving a good measure on a compact manifold, Ann. Sci. École Norm. Sup. (4) 13 (1980), 4593. MR 584082 (81k:58042)
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 J. Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory and Dynamical Systems 8 (1988), 99107. MR 967632 (90d:58124)
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 P. Halmos, Lectures on ergodic theory, Publ. Math. Soc. Japan, Tokyo, 1956; reprint, Chelsea, New York, 1960. MR 0097489 (20:3958)
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 M. R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math. 49 (1979), 1233. MR 538680 (81h:58039)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199311430145
PII:
S 00029939(1993)11430145
Article copyright:
© Copyright 1993
American Mathematical Society
