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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [Al1] Steve Alpern, Approximation to and by measure preserving homeomorphisms, J. London Math. Soc. (2) 18 (1978), no. 2, 305–315. MR 509946, 10.1112/jlms/s2-18.2.305
  • [Al2] Steve Alpern, Generic properties of measure preserving homeomorphisms, Ergodic theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978), Lecture Notes in Math., vol. 729, Springer, Berlin, 1979, pp. 16–27. MR 550406
  • [AP] Steve Alpern and V. S. Prasad, Typical recurrence for lifts of mean rotation zero annulus homeomorphisms, Bull. London Math. Soc. 23 (1991), no. 5, 477–481. MR 1141019, 10.1112/blms/23.5.477
  • [At] Giles Atkinson, Recurrence of co-cycles and random walks, J. London Math. Soc. (2) 13 (1976), no. 3, 486–488. MR 0419727
  • [Br] Morton Brown, A mapping theorem for untriangulated manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 92–94. MR 0158374
  • [CV] Charles O. Christenson and William L. Voxman, Aspects of topology, Marcel Dekker, Inc., New York-Basel, 1977. Pure and applied Mathematics, Vol. 39. MR 0487938
  • [Fa] A. Fathi, Structure of the group of homeomorphisms preserving a good measure on a compact manifold, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 1, 45–93. MR 584082
  • [Fr] John Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dynam. Systems 8* (1988), no. Charles Conley Memorial Issue, 99–107. MR 967632, 10.1017/S0143385700009366
  • [Ha] Paul R. Halmos, Lectures on ergodic theory, Publications of the Mathematical Society of Japan, no. 3, The Mathematical Society of Japan, 1956. MR 0097489
  • [He] Michael-Robert Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math. 49 (1979), 5–233 (French). MR 538680
  • [KS] A. B. Katok and A. M. Stepin, Metric properties of homeomorphisms that preserve measure, Uspehi Mat. Nauk 25 (1970), no. 2 (152), 193–220 (Russian). MR 0260974
  • [Mo] Jürgen Moser, A rapidly convergent iteration method and non-linear differential equations. II, Ann. Scuola Norm. Sup. Pisa (3) 20 (1966), 499–535. MR 0206461
  • [OU] J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2) 42 (1941), 874–920. MR 0005803
  • [Ru] Daniel J. Rudolph, Fundamentals of measurable dynamics, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1990. Ergodic theory on Lebesgue spaces. MR 1086631
  • [Sc] Klaus Schmidt, Cocycles on ergodic transformation groups, Macmillan Company of India, Ltd., Delhi, 1977. Macmillan Lectures in Mathematics, Vol. 1. MR 0578731

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58F11, 28D05

Retrieve articles in all journals with MSC: 58F11, 28D05

Additional Information

Article copyright: © Copyright 1993 American Mathematical Society