Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Metric properties of transformations of $ G$-spaces


Author: R. K. Thomas
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] S. A. Juzvinskiĭ, Metric properties of the endomorphisms of compact groups, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 1295–1328 (Russian). MR 0194588
  • [2] V. A. Rohlin, Metric properties of endomorphisms of compact commutative groups, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 867–874 (Russian). MR 0168697
  • [3] 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
  • [4] Paul R. Halmos, Lectures on ergodic theory, Publications of the Mathematical Society of Japan, no. 3, The Mathematical Society of Japan, 1956. MR 0097489
  • [5] Ja. Sinaĭ, On the concept of entropy for a dynamic system, Dokl. Akad. Nauk SSSR 124 (1959), 768–771 (Russian). MR 0103256
    Ja. Sinaĭ, Flows with finite entropy, Dokl. Akad. Nauk SSSR 125 (1959), 1200–1202 (Russian). MR 0103257
  • [6] 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
  • [7] William Parry, Ergodic properties of affine transformations and flows on nilmanifolds., Amer. J. Math. 91 (1969), 757–771. MR 0260975, https://doi.org/10.2307/2373350
  • [8] J. Tits, Liesche Gruppen und Algebren, Mathematischen Institut, Bonn, 1965.
  • [9] G. W. Mackey, The theory of group representations, Lecture Notes (Summer, 1955), Department of Mathematics, University of Chicago, Chicago, Ill., 1955. MR 19, 117.
  • [10] Irving Kaplansky, Groups with representations of bounded degree, Canadian J. Math. 1 (1949), 105–112. MR 0028317

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 28A65

Retrieve articles in all journals with MSC: 28A65


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0293063-4
Keywords: Completely positive entropy, $ G$-space, $ \sigma $-commuting
Article copyright: © Copyright 1971 American Mathematical Society