Ergodic properties that lift to compact group extensions
HTML articles powered by AMS MathViewer
- by E. Arthur Robinson PDF
- Proc. Amer. Math. Soc. 102 (1988), 61-67 Request permission
Abstract:
Let $T$ and $R$ be measure preserving, $T$ weakly mixing, $R$ ergodic, and let $S$ be conservative ergodic and nonsingular. Let $\tilde T$ be a weakly mixing compact abelian group extension of $T$. If $T \times S$ is ergodic then $\tilde T \times S$ is ergodic. A corollary is a new proof that if $T$ is mildly mixing then so is $\tilde T$. A similar statement holds for other ergodic multiplier properties. Now let $\tilde T$ be a weakly mixing type $\alpha$ compact affine $G$ extension of $T$ where $\alpha$ is an automorphism of $G$. If $T$ and $R$ are disjoint and $\alpha$ or $R$ has entropy zero, then $\tilde T$ and $R$ are disjoint. $\tilde T$ is uniquely ergodic if and only if $T$ is uniquely ergodic and $\alpha$ has entropy zero. If $T$ is mildly mixing and $\tilde T$ is weakly mixing then $\tilde T$ is mildly mixing. We also provide a new proof that if $\tilde T$ is weakly mixing then $\tilde T$ has the $K$-property if $T$ does.References
- Jon Aaronson, Category theorems for some ergodic multiplier properties, Israel J. Math. 51 (1985), no. 1-2, 151–162. MR 804481, DOI 10.1007/BF02772963
- Kenneth Berg, Quasi-disjointness in ergodic theory, Trans. Amer. Math. Soc. 162 (1971), 71–87. MR 284563, DOI 10.1090/S0002-9947-1971-0284563-1
- Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49. MR 213508, DOI 10.1007/BF01692494
- H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR 603625
- 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
- Paul R. Halmos, Lectures on ergodic theory, Publications of the Mathematical Society of Japan, vol. 3, Mathematical Society of Japan, Tokyo, 1956. MR 0097489
- A. del Junco and D. J. Rudolph, On ergodic actions whose self-joinings are graphs, CWI Report PM, R8408, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1984. MR 774226
- D. A. Lind, The structure of skew products with ergodic group automorphisms, Israel J. Math. 28 (1977), no. 3, 205–248. MR 460593, DOI 10.1007/BF02759810
- D. A. Lind, Ergodic affine transformations are loosely Bernoulli, Israel J. Math. 30 (1978), no. 4, 335–338. MR 507561, DOI 10.1007/BF02761998
- William Parry, Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math. 91 (1969), 757–771. MR 260975, DOI 10.2307/2373350
- M. S. Pinsker, Dynamical systems with completely positive or zero entropy, Soviet Math. Dokl. 1 (1960), 937–938. MR 0152628
- Daniel J. Rudolph, $k$-fold mixing lifts to weakly mixing isometric extensions, Ergodic Theory Dynam. Systems 5 (1985), no. 3, 445–447. MR 805841, DOI 10.1017/S0143385700003060
- Daniel J. Rudolph, Classifying the isometric extensions of a Bernoulli shift, J. Analyse Math. 34 (1978), 36–60 (1979). MR 531270, DOI 10.1007/BF02790007
- R. K. Thomas, Metric properties of transformations of $G$-spaces, Trans. Amer. Math. Soc. 160 (1971), 103–117. MR 293063, DOI 10.1090/S0002-9947-1971-0293063-4
- Peter Walters, Some invariant $\sigma$-algebras for measure-preserving transformations, Trans. Amer. Math. Soc. 163 (1972), 357–368. MR 291413, DOI 10.1090/S0002-9947-1972-0291413-7
- Peter Walters, Some transformations having a unique measure with maximal entropy, Proc. London Math. Soc. (3) 28 (1974), 500–516. MR 367158, DOI 10.1112/plms/s3-28.3.500
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 102 (1988), 61-67
- MSC: Primary 28D05
- DOI: https://doi.org/10.1090/S0002-9939-1988-0915717-4
- MathSciNet review: 915717