Topological groups with rokhlin properties

Colloquium Mathematicum

Volumn 110, Issue 1, 2008, Pages 51-80

  Glasner, Eli ^a   Weiss, Benjamin ^b  

^a TEL AVIV UNIVERSITY   (Israel)

^b HEBREW UNIVERSITY OF JERUSALEM   (Israel)

Author keywords

Cantor group; Dense conjugacy class; Fra ss structure; Group of homeomorphisms; Measure preserving transformation; Rokhlin property; Weak mixing

Indexed keywords

EID: 85017290797     PISSN: 00101354     EISSN: 17306302     Source Type: Journal    
DOI: 10.4064/cm110-1-2     Document Type: Article

References (57)

1. O. Ageev, The homogeneous spectrum problem in ergodic theory, Invent. Math. 160 (2005), 417-446.
2. O. Ageev, On spectral invariants in modern ergodic theory, in: Proc. Internat. Congress Math., Madrid, 2005, 417-446.
3. O. Ageev, Spectral rigidity of group actions and Kazhdan’ groups, preprint.
4. E. Akin, The General Topology of Dynamical Systems, Amer. Math. Soc., Providence, 1993.
5. E. Akin, On chain continuity, Discrete Cont. Dynam. Syst. 2 (1996), 111-120.
6. E. Akin, Good measures on Cantor space, Trans. Amer. Math. Soc. 357 (2005), 2681-2722.
7. E. Akin, J. Auslander, and K. Berg, When is a transitive map chaotic, in: Convergence in Ergodic Theory and Probability, de Gruyter, 1996, 25-40.
8. E. Akin, E. Glasner and B. Weiss, Generically there is but one self homeomorphism of the Cantor set,Trans. Amer. Math. Soc., to appear.
9. E. Akin, M. Hurley and J. Kennedy, Dynamics of topologically generic homeomor-phisms, Mem. Amer. Math. Soc. 164 (2003), no. 783.
10. S. Alpern and V. S. Prasad, Typical Dynamics of Volume Preserving Homeomor-phisms, Cambridge Univ. Press, 2000.
11. S. Bezuglyi, A. H. Dooley and J. Kwiatkowski, Topologies on the group of homeo-morphisms of a Cantor set, Topol. Methods Nonlinear Anal. 27 (2006), 299-331.
12. S. Bezuglyi, A. H. Dooley and J. Kwiatkowski, Topologies on the group of Borel automorphisms of a standard Borel space, ibid. 27 (2006), 333-385.
13. S. Bezuglyi, A. H. Dooley and K. Medynets, The Rokhlin lemma for homeomor-phisms of a Cantor set, Proc. Amer. Math. Soc. 134 (2005), 2957-2964.
14. S. Bezuglyi, A. H. Dooley and K. Medynets, Approximation in ergodic theory, Borel, and Cantor dynamics, in: Algebraic and Topological Dynamics, Contemp. Math. 385, Amer. Math. Soc., 2005, 39-64.
15. J. R. Choksi and M. G. Nadkarni, Baire category in space of measures, unitary operators, and transformations, in: Invariant Subspaces and Allied Topics, Narosa, New Delhi, 1990, 147-163.
16. J. R. Choksi and V. S. Prasad, Approximation and Baire category theorems in ergodic theory, in: Measure Theory and its Applications, Lecture Notes in Math. 1033, Springer, 1983, 94-113.
17. C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Reg. Conf. Ser. Math. 38, Amer. Math. Soc., 1978.
18. M. Foreman, D. J. Rudolph and B. Weiss, On the conjugacy relation in ergodic theory, C. R. Math. Acad. Sci. Paris. 343 (2006), 653-656.
19. M. Foreman and B. Weiss, An anti-classification theorem for ergodic measure pre-serving transformations, J. Eur. Math. Soc. 6 (2004), 277-292.
20. N. A. Friedman, Introduction to Ergodic Theory, van Nostrand Reinhold, New York, 1970.
21. H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Dio-phantine approximation, Math. Systems Theory 1 (1967), 1-49.
22. E. Glasner, Ergodic Theory via Joinings, Math. Surveys Monogr. 101, Amer. Math. Soc., 2003.
23. E. Glasner and J. King, A zero-one law for dynamical properties, in: Topological Dynamics and Applications (Minneapolis, MN, 1995), Contemp. Math. 215, Amer. Math. Soc., 1998, 231-242.
24. E. Glasner, J.-P. Thouvenot and B. Weiss, Every countable group has the weak Rohlin property, Bull. London Math. Soc. 38, (2006), 932-936.
25. E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity 6 (1993), 1067-1075.
26. E. Glasner and B. Weiss, Kazhdan’s property T and the geometry of the collection of invariant mea-sures, Geom. Funct. Anal. 7 (1997), 917-935.
27. E. Glasner and B. Weiss, The topological Rohlin property and topological entropy, Amer. J. Math. 123 (2001), 1055-1070.
28. P. R. Halmos, In general a measure preserving transformation is mixing, Ann. of Math. 45 (1944), 786-792.
29. P. R. Halmos, Lecture on Ergodic Theory, Chelsea, New York, 1960.
30. G. Higman, B. H. Neumann and H. Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (1949), 247-254.
31. G. Hjorth, Classification and Orbit Equivalence Relations, Math. Surveys Monogr. 75, Amer. Math. Soc., 2000.
32. G. Hjorth, On invariants for measure preserving transformations, Fund. Math. 169 (2001), 51-84.
33. M. Hochman, Genericity in topological dynamics, to appear; arXiv:math.DS/ 0605386, 2006.
34. W. Hodges, A Shorter Model Theory, Cambridge Univ. Press, 1977.
35. W. Hodges, I. Hodkinson, D. Lascar and S. Shelah, The small index property for ω-stable ω-categorical structures and for the random graph, J. London Math. Soc. 48 (1993), 204-218.
36. A. A. Ivanov, Generic expansions of !-categorical structures and semantics of gen-eralized quantifiers, J. Symbolic Logic 64 (1999), 775-789.
37. A. del Junco, Disjointness of measure-preserving transformations, minimal self-join-ings and category, in: Ergodic Theory and Dynamical Systems, I (College Park, MD, 1979-80), Progr. Math. 10, Birkhäuser, 1981, 81-89.
38. A. del Junco and M. Lemańczyk, Generic spectral properties of measure-preserving maps and applications, Proc. Amer. Math. Soc. 115 (1992), 725-736.
39. A. del Junco and D. Rudolph, Residual behavior of induced maps, Israel J. Math. 93 (1996), 387-398.
40. A. S. Kechris, Classical Descriptive Set Theory, Springer, 1995.
41. A. S. Kechris, Global Aspects of Ergodic Group Actions and Equivalence Relations, http://www.math.caltech.edu/people/kechris.html.
42. A. S. Kechris and C. Rosendal, Turbulence, amalgamation and generic automor-phisms of homogeneous structures, arXiv:math.LO/0409567.
43. D. Kerr and M. Pichot, Asymptotic abelianness, weak mixing, and property T, a preprint.
44. K. Kuratowski, Topology, Vol. 1, Academic Press, 1966.
45. D. Kuske and J. K. Truss, Generic automorphisms of the universal partial order, Proc. Amer. Math. Soc. 129 (2000), 1939-1948.
46. D. S. Ornstein, Bernoulli shifts with the same entropy are isomorphic, Adv. Math. 4 (1970), 337-352.
47. D. S. Ornstein and P. Shields, An uncountable family of K-automorphisms, ibid. 10 (1973), 63-88.
48. D. S. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math. 48 (1987), 1-141.
49. D.V. Osin, Small cancellations over hyperbolic groups and embedding theorems, arXiv:math.GR/0411039, 2004.
50. J. Oxtoby, Measure and Category, 2nd ed., Springer, 1980.
51. J. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical tran-sitivity, Ann. of Math. 42 (1941), 874-920.
52. C. Rosendal and S. Solecki, Automatic continuity of homomorphisms and fixed points on metric compacta, arXiv:math.LO/0604575, 2006.
53. J. J. Rotman, The Theory of Groups, 3rd ed., W. C. Brown, Dubuque, IA, 1980.
54. D. Rudolph, Residuality and orbit equivalence, in: Topological Dynamics and Applications (Minneapolis, MN, 1995), Contemp. Math. 215, Amer. Math. Soc., 1998, 243-254.
55. S. Solecki, Extending partial isometries, Israel J. Math. 150 (2005), 315-332.
56. J. K. Truss, Generic automorphisms of homogeneous structures, Proc. London Math. Soc. 65 (1992), 121-141.
57. B. Weiss, Actions of amenable groups, in: Topics in Dynamics and Ergodic Theory, London Math. Soc. Lecture Note Ser. 310, Cambridge Univ. Press, 2003, 226-262.