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
Fulltext 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
(80d:28033), http://dx.doi.org/10.1112/jlms/s218.2.305
 [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
(80m:28019)
 [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
(93a:28012), http://dx.doi.org/10.1112/blms/23.5.477
 [At]
Giles
Atkinson, Recurrence of cocycles and random walks, J. London
Math. Soc. (2) 13 (1976), no. 3, 486–488. MR 0419727
(54 #7745)
 [Br]
Morton
Brown, A mapping theorem for untriangulated manifolds,
Topology of 3manifolds and related topics (Proc. The Univ. of Georgia
Institute, 1961), PrenticeHall, Englewood Cliffs, N.J., 1962,
pp. 92–94. MR 0158374
(28 #1599)
 [CV]
Charles
O. Christenson and William
L. Voxman, Aspects of topology, Marcel Dekker Inc., New York,
1977. Pure and applied Mathematics, Vol. 39. MR 0487938
(58 #7521)
 [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
(81k:58042)
 [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
(90d:58124), http://dx.doi.org/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 (20 #3958)
 [He]
MichaelRobert
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 (81h:58039)
 [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
(41 #5594)
 [Mo]
Jürgen
Moser, A rapidly convergent iteration method and nonlinear
differential equations. II, Ann. Scuola Norm. Sup. Pisa (3)
20 (1966), 499–535. MR 0206461
(34 #6280)
 [OU]
J.
C. Oxtoby and S.
M. Ulam, Measurepreserving homeomorphisms and metrical
transitivity, Ann. of Math. (2) 42 (1941),
874–920. MR 0005803
(3,211b)
 [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
(92e:28006)
 [Sc]
Klaus
Schmidt, Cocycles on ergodic transformation groups, Macmillan
Company of India, Ltd., Delhi, 1977. Macmillan Lectures in Mathematics,
Vol. 1. MR
0578731 (58 #28262)
 [Al1]
 S. Alpern, Approximation to and by measure preserving homeomorphisms, J. London Math. Soc. (2) 18 (1978), 305315. MR 509946 (80d:28033)
 [Al2]
 , Generic properties of measure preserving homeomorphisms, Ergodic Theory, Lecture Notes in Math., vol. 729, Springer, Berlin and New York, 1979, pp. 1627. MR 550406 (80m:28019)
 [AP]
 S. Alpern and V. S. Prasad, Typical recurrence for lifts of mean rotation zero annulus homeomorphisms, Bull. London Math. Soc. 23 (1991), 477481. MR 1141019 (93a:28012)
 [At]
 G. Atkinson, Recurrence of cocycles and random walks, J. London Math. Soc. (2) 13 (1976), 486488. MR 0419727 (54:7745)
 [Br]
 M. Brown, A mapping theorem for untriangulated manifolds, Topology of Manifolds and Related Results (M. K. Fort, ed.), PrenticeHall, Englewood Cliffs, NJ, 1963, pp. 9294. MR 0158374 (28:1599)
 [CV]
 C. Christenson and W. Voxman, Aspects of topology, Marcel Dekker, New York, 1977. MR 0487938 (58:7521)
 [Fa]
 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)
 [Fr]
 J. Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory and Dynamical Systems 8 (1988), 99107. MR 967632 (90d:58124)
 [Ha]
 P. Halmos, Lectures on ergodic theory, Publ. Math. Soc. Japan, Tokyo, 1956; reprint, Chelsea, New York, 1960. MR 0097489 (20:3958)
 [He]
 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)
 [KS]
 A. Katok and A. Stepin, Metric properties of measure preserving homeomorphisms, Uspekhi Mat. Nauk. 25 (1970), 193220; English transl., Russian Math. Surveys 25, 191220. MR 0260974 (41:5594)
 [Mo]
 J. Moser, A rapidly convergent iteration method, Part II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 20 (1966), 499535. MR 0206461 (34:6280)
 [OU]
 J. C. Oxtoby and S. Ulam, Measure preserving homeomorphisms and metrical transitivity, Ann. of Math. (2) 42 (1941), 874920. MR 0005803 (3:211b)
 [Ru]
 D. J. Rudolph, Fundamentals of measurable dynamics. Ergodic theory on Lebesgue spaces, Oxford Univ. Press, London, 1990. MR 1086631 (92e:28006)
 [Sc]
 K. Schmidt, Cocycles of ergodic transformation groups, Macmillan Lectures in Math., Macmillan, New Delhi, 1977. MR 0578731 (58:28262)
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
DOI:
http://dx.doi.org/10.1090/S00029939199311430145
PII:
S 00029939(1993)11430145
Article copyright:
© Copyright 1993 American Mathematical Society
