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 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).

**[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**

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

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

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1993-1143014-5

Article copyright:
© Copyright 1993
American Mathematical Society