Metric properties of transformations of $G$-spaces
HTML articles powered by AMS MathViewer
- by R. K. Thomas PDF
- Trans. Amer. Math. Soc. 160 (1971), 103-117 Request permission
Abstract:
The measure-preserving transformation $T$ acts on a Lebesgue space $(M,\mathcal {B},\mu )$ which is also a $G$-space for a compact separable group $G$. It is proved that if the factor-transformation on the space of $G$-orbits has completely positive entropy and a certain condition regarding the relations between the actions of $G$ and $T$ is satisfied, then $T$ weakly mixing implies $T$ has completely positive entropy.References
- S. A. Juzvinskiĭ, Metric properties of the endomorphisms of compact groups, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 1295–1328 (Russian). MR 0194588
- V. A. Rohlin, Metric properties of endomorphisms of compact commutative groups, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 867–874 (Russian). MR 0168697
- V. A. Rohlin, Lectures on the entropy theory of transformations with invariant measure, Uspehi Mat. Nauk 22 (1967), no. 5 (137), 3–56 (Russian). MR 0217258
- Paul R. Halmos, Lectures on ergodic theory, Publications of the Mathematical Society of Japan, vol. 3, Mathematical Society of Japan, Tokyo, 1956. MR 0097489
- Ja. Sinaĭ, On the concept of entropy for a dynamic system, Dokl. Akad. Nauk SSSR 124 (1959), 768–771 (Russian). MR 0103256
- D. Z. Arov, Calculation of entropy for a class of group endomorphisms, Zap. Meh.-Mat. Fak. Har′kov. Gos. Univ. i Har′kov. Mat. Obšč. (4) 30 (1964), 48–69 (Russian). MR 0213507
- William Parry, Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math. 91 (1969), 757–771. MR 260975, DOI 10.2307/2373350 J. Tits, Liesche Gruppen und Algebren, Mathematischen Institut, Bonn, 1965. G. W. Mackey, The theory of group representations, Lecture Notes (Summer, 1955), Department of Mathematics, University of Chicago, Chicago, Ill., 1955. MR 19, 117.
- Irving Kaplansky, Groups with representations of bounded degree, Canad. J. Math. 1 (1949), 105–112. MR 28317, DOI 10.4153/cjm-1949-011-9
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 160 (1971), 103-117
- MSC: Primary 28A65
- DOI: https://doi.org/10.1090/S0002-9947-1971-0293063-4
- MathSciNet review: 0293063