A general condition for lifting theorems

Robinson, E. Arthur

doi:10.1090/S0002-9947-1992-1040044-2

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6850(online) ISSN 0002-9947(print)

A general condition for lifting theorems

Author: E. Arthur Robinson
Journal: Trans. Amer. Math. Soc. 330 (1992), 725-755
MSC: Primary 28D05; Secondary 28D20
MathSciNet review: 1040044
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We define a general condition, called stability on extensions of measure preserving transformations . Stability is defined in terms of relative unique ergodicity, and as a joining property. Ergodic compact group extensions are stable, and moreover stable extensions satisfy lifting theorems similar to those satisfied by group extensions. In general, stable extensions have relative entropy zero. In the class of continuous flow extensions over strictly ergodic homeomorphisms, stable extensions are generic.

[1] Kenneth Berg, Quasi-disjointness in ergodic theory, Trans. Amer. Math. Soc. 162 (1971), 71–87. MR 0284563 (44 #1788), http://dx.doi.org/10.1090/S0002-9947-1971-0284563-1
[2] Kenneth R. Berg, Quasi-disjointness, products and inverse limits, Math. Systems Theory 6 (1972), 123–128. MR 0306443 (46 #5569)
[3] J. R. Blum and D. L. Hanson, On the mean ergodic theorem for subsequences, Bull. Amer. Math. Soc. 66 (1960), 308–311. MR 0118803 (22 #9572), http://dx.doi.org/10.1090/S0002-9904-1960-10481-8
[4] A. Rothstein and R. Burton, Isomorphism theorems in ergodic theory, Lecture notes, Department of Mathematics, Oregon State University.
[5] Jean Coquet and Pierre Liardet, A metric study involving independent sequences, J. Analyse Math. 49 (1987), 15–53. MR 928506 (89e:11043), http://dx.doi.org/10.1007/BF02792891
[6] Nathaniel A. Friedman, Mixing on sequences, Canad. J. Math. 35 (1983), no. 2, 339–352. MR 695088 (85a:28011), http://dx.doi.org/10.4153/CJM-1983-019-2
[7] Nathaniel A. Friedman, Higher order partial mixing, Conference in modern analysis and probability (New Haven, Conn., 1982), Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 111–130. MR 737394 (85h:28014), http://dx.doi.org/10.1090/conm/026/737394
[8] N. A. Friedman and D. S. Ornstein, On partially mixing transformations, Indiana Univ. Math. J. 20 (1970/1971), 767–775. MR 0267074 (42 #1976)
[9] N. A. Friedman and D. S. Ornstein, On mixing and partial mixing, Illinois J. Math. 16 (1972), 61–68. MR 0293059 (45 #2138)
[10] Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49. MR 0213508 (35 #4369)
[11] H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math. 83 (1961), 573–601. MR 0133429 (24 #A3263)
[12] Harry Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 31 (1977), 204–256. MR 0498471 (58 #16583)
[13] H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR 603625 (82j:28010)
[14] Hillel Furstenberg and Benjamin Weiss, The finite multipliers of infinite ergodic transformations, The structure of attractors in dynamical systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), Lecture Notes in Math., vol. 668, Springer, Berlin, 1978, pp. 127–132. MR 518553 (80b:28023)
[15] Shmuel Glasner, Relatively invariant measures, Pacific J. Math. 58 (1975), no. no.2, 393–410. MR 0380756 (52 #1653)
[16] -, Quasi-factors in ergodic theory, Israel J. Math. 45 (1983), 198-208.
[17] S. Glasner and B. Weiss, On the construction of minimal skew products, Israel J. Math. 34 (1979), no. 4, 321–336 (1980). MR 570889 (82f:54068), http://dx.doi.org/10.1007/BF02760611
[18] S. Glasner and B. Weiss, Processes disjoint from weak mixing, Trans. Amer. Math. Soc. 316 (1989), no. 2, 689–703. MR 946217 (90c:28026), http://dx.doi.org/10.1090/S0002-9947-1989-0946217-9
[19] Lee Kenneth Jones, A mean ergodic theorem for weakly mixing operators, Advances in Math. 7 (1971), 211–216 (1971). MR 0285690 (44 #2908)
[20] Roger Jones and William Parry, Compact abelian group extensions of dynamical systems. II, Compositio Math. 25 (1972), 135–147. MR 0338318 (49 #3083)
[21] Jonathan L. King, Joining-rank and the structure of finite rank mixing transformations, J. Analyse Math. 51 (1988), 182–227. MR 963154 (89k:28009), http://dx.doi.org/10.1007/BF02791123
[22] Wolfgang Krieger, On unique ergodicity, and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 327–346. MR 0393402 (52 #14212)
[23] George W. Mackey, Borel structure in groups and their duals, Trans. Amer. Math. Soc. 85 (1957), 134–165. MR 0089999 (19,752b), http://dx.doi.org/10.1090/S0002-9947-1957-0089999-2
[24] Edwin E. Moise, Geometric topology in dimensions 2 and 3, Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, Vol. 47. MR 0488059 (58 #7631)
[25] Mahesh G. Nerurkar, Ergodic continuous skew product actions of amenable groups, Pacific J. Math. 119 (1985), no. 2, 343–363. MR 803124 (86m:28011)
[26] William Parry, Ergodic properties of affine transformations and flows on nilmanifolds., Amer. J. Math. 91 (1969), 757–771. MR 0260975 (41 #5595)
[27] M. S. Pinsker, Dynamical systems with completely positive or zero entropy, Soviet Math. Dokl. 1 (1960), 937–938. MR 0152628 (27 #2603)
[28] E. Arthur Robinson Jr., The maximal abelian subextension determines weak mixing for group extensions, Proc. Amer. Math. Soc. 114 (1992), no. 2, 443–450. MR 1062835 (92e:28010), http://dx.doi.org/10.1090/S0002-9939-1992-1062835-X
[29] E. Arthur Robinson Jr., Ergodic properties that lift to compact group extensions, Proc. Amer. Math. Soc. 102 (1988), no. 1, 61–67. MR 915717 (88i:28035), http://dx.doi.org/10.1090/S0002-9939-1988-0915717-4
[30] Daniel J. Rudolph, An example of a measure preserving map with minimal self-joinings, and applications, J. Analyse Math. 35 (1979), 97–122. MR 555301 (81e:28011), http://dx.doi.org/10.1007/BF02791063
[31] Daniel J. Rudolph, Classifying the isometric extensions of a Bernoulli shift, J. Analyse Math. 34 (1978), 36–60 (1979). MR 531270 (80g:28020), http://dx.doi.org/10.1007/BF02790007
[32] Daniel J. Rudolph, 𝑘-fold mixing lifts to weakly mixing isometric extensions, Ergodic Theory Dynam. Systems 5 (1985), no. 3, 445–447. MR 805841 (87a:28023), http://dx.doi.org/10.1017/S0143385700003060
[33] Daniel J. Rudolph, 𝑍ⁿ and 𝑅ⁿ cocycle extensions and complementary algebras, Ergodic Theory Dynam. Systems 6 (1986), no. 4, 583–599. MR 873434 (88b:28032), http://dx.doi.org/10.1017/S0143385700003710
[34] -, Asymptotically Brownian skew products give nonloosely Bernoulli -automorphisms, preprint.
[35] Jean-Paul Thouvenot, Quelques propriétés des systèmes dynamiques qui se décomposent en un produit de deux systèmes dont l’un est un schéma de Bernoulli, Israel J. Math. 21 (1975), no. 2-3, 177–207 (French, with English summary). Conference on Ergodic Theory and Topological Dynamics (Kibbutz, Lavi, 1974). MR 0399419 (53 #3263)
[36] Haruo Totoki, Ergodic theory, Lecture Notes Series, No. 14, Matematisk Institut, Aarhus Universitet, Aarhus, 1969. MR 0254216 (40 #7425)
[37] V. S. Varadarajan, Groups of automorphisms of Borel spaces, Trans. Amer. Math. Soc. 109 (1963), 191–220. MR 0159923 (28 #3139), http://dx.doi.org/10.1090/S0002-9947-1963-0159923-5
[38] Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York, 1982. MR 648108 (84e:28017)
[39] Peter Walters, Some invariant 𝜎-algebras for measure-preserving transformations, Trans. Amer. Math. Soc. 163 (1972), 357–368. MR 0291413 (45 #506), http://dx.doi.org/10.1090/S0002-9947-1972-0291413-7
[40] Peter Walters, Some transformations having a unique measure with maximal entropy, Proc. London Math. Soc. (3) 28 (1974), 500–516. MR 0367158 (51 #3400)
[41] Benjamin Weiss, Strictly ergodic models for dynamical systems, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 2, 143–146. MR 799798 (87c:28026), http://dx.doi.org/10.1090/S0273-0979-1985-15399-6
[42] Robert J. Zimmer, Extensions of ergodic group actions, Illinois J. Math. 20 (1976), no. 3, 373–409. MR 0409770 (53 #13522)
[43] Robert J. Zimmer, Ergodic actions with generalized discrete spectrum, Illinois J. Math. 20 (1976), no. 4, 555–588. MR 0414832 (54 #2924)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|