Entropy for group endomorphisms and homogeneous spaces

Bowen, Rufus

doi:10.1090/S0002-9947-1971-0274707-X

Entropy for group endomorphisms and homogeneous spaces
HTML articles powered by AMS MathViewer

by Rufus Bowen PDF

Trans. Amer. Math. Soc. 153 (1971), 401-414 Request permission

Erratum: Trans. Amer. Math. Soc. 181 (1973), 509-510.

Abstract:

Topological entropy ${h_d}(T)$ is defined for a uniformly continuous map on a metric space. General statements are proved about this entropy, and it is calculated for affine maps of Lie groups and certain homogeneous spaces. We compare ${h_d}(T)$ with measure theoretic entropy $h(T)$; in particular $h(T) = {h_d}(T)$ for Haar measure and affine maps $T$ on compact metrizable groups. A particular case of this yields the well-known formula for $h(T)$ when $T$ is a toral automorphism.

References

R. L. Adler, A. G. Konheim, and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309–319. MR 175106, DOI 10.1090/S0002-9947-1965-0175106-9
Rufus Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc. 154 (1971), 377–397. MR 282372, DOI 10.1090/S0002-9947-1971-0282372-0
L. Wayne Goodwyn, Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc. 23 (1969), 679–688. MR 247030, DOI 10.1090/S0002-9939-1969-0247030-3

The product theorem for topological entropy

James R. Munkres, Elementary differential topology, Annals of Mathematics Studies, No. 54, Princeton University Press, Princeton, N.J., 1963. Lectures given at Massachusetts Institute of Technology, Fall, 1961. MR 0163320, DOI 10.1515/9781400882656
William Parry, Entropy and generators in ergodic theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0262464
Ja. Sinaĭ, On the concept of entropy for a dynamic system, Dokl. Akad. Nauk SSSR 124 (1959), 768–771 (Russian). MR 0103256
V. A. Rohlin, Exact endomorphisms of a Lebesgue space, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 499–530 (Russian). MR 0143873
A. L. Genis, Metric properties of the endomorphisms of an $n$-dimensional torus, Dokl. Akad. Nauk SSSR 138 (1961), 991–993 (Russian). MR 0202907
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
K. Krzyżewski and W. Szlenk, On invariant measures for expanding differentiable mappings, Studia Math. 33 (1969), 83–92. MR 245761, DOI 10.4064/sm-33-1-83-92

Convolution of invariant measures

Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR 0073104
Harvey B. Keynes, Lifting of topological entropy, Proc. Amer. Math. Soc. 24 (1970), 440–445. MR 274710, DOI 10.1090/S0002-9939-1970-0274710-4
Claude Chevalley, Theory of Lie groups. I, Princeton University Press, Princeton, N. J., 1946 1957. MR 0082628, DOI 10.1515/9781400883851
D. V. Anosov and Ja. G. Sinaĭ, Certain smooth ergodic systems, Uspehi Mat. Nauk 22 (1967), no. 5 (137), 107–172 (Russian). MR 0224771
Lynn H. Loomis, An introduction to abstract harmonic analysis, D. Van Nostrand Co., Inc., Toronto-New York-London, 1953. MR 0054173
William Parry, Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math. 91 (1969), 757–771. MR 260975, DOI 10.2307/2373350