Entropy points and applications

Ye, Xiangdong; Zhang, Guohua

doi:10.1090/S0002-9947-07-04357-7

Entropy points and applications
HTML articles powered by AMS MathViewer

by Xiangdong Ye and Guohua Zhang PDF

Trans. Amer. Math. Soc. 359 (2007), 6167-6186 Request permission

Abstract:

First notions of entropy point and uniform entropy point are introduced using Bowen’s definition of topological entropy. Some basic properties of the notions are discussed. As an application it is shown that for any topological dynamical system there is a countable closed subset whose Bowen entropy is equal to the entropy of the original system. Then notions of C-entropy point are introduced along the line of entropy tuple both in topological and measure-theoretical settings. It is shown that each C-entropy point is an entropy point, and the set of C-entropy points is the union of sets of C-entropy points for all invariant measures.

References

F. Blanchard, Fully positive topological entropy and topological mixing, Symbolic dynamics and its applications (New Haven, CT, 1991) Contemp. Math., vol. 135, Amer. Math. Soc., Providence, RI, 1992, pp. 95–105. MR 1185082, DOI 10.1090/conm/135/1185082
François Blanchard, A disjointness theorem involving topological entropy, Bull. Soc. Math. France 121 (1993), no. 4, 465–478 (English, with English and French summaries). MR 1254749, DOI 10.24033/bsmf.2216
F. Blanchard and Y. Lacroix, Zero entropy factors of topological flows, Proc. Amer. Math. Soc. 119 (1993), no. 3, 985–992. MR 1155593, DOI 10.1090/S0002-9939-1993-1155593-2
F. Blanchard, E. Glasner, and B. Host, A variation on the variational principle and applications to entropy pairs, Ergodic Theory Dynam. Systems 17 (1997), no. 1, 29–43. MR 1440766, DOI 10.1017/S0143385797069794
F. Blanchard, B. Host, A. Maass, S. Martinez, and D. J. Rudolph, Entropy pairs for a measure, Ergodic Theory Dynam. Systems 15 (1995), no. 4, 621–632. MR 1346392, DOI 10.1017/S0143385700008579
F. Blanchard and W. Huang, Entropy sets, weak mixing sets and entropy capacity, to appear in Discrete and Continuous Dynamical Systems.
Mike Boyle and Tomasz Downarowicz, The entropy theory of symbolic extensions, Invent. Math. 156 (2004), no. 1, 119–161. MR 2047659, DOI 10.1007/s00222-003-0335-2
M. Brin and A. Katok, On local entropy, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 30–38. MR 730261, DOI 10.1007/BFb0061408
Dou Dou, Xiangdong Ye, and Guohua Zhang, Entropy sequences and maximal entropy sets, Nonlinearity 19 (2006), no. 1, 53–74. MR 2191619, DOI 10.1088/0951-7715/19/1/004
Doris Fiebig, Ulf-Rainer Fiebig, and Zbigniew H. Nitecki, Entropy and preimage sets, Ergodic Theory Dynam. Systems 23 (2003), no. 6, 1785–1806. MR 2032488, DOI 10.1017/S0143385703000221
Eli Glasner, A simple characterization of the set of $\mu$-entropy pairs and applications, Israel J. Math. 102 (1997), 13–27. MR 1489099, DOI 10.1007/BF02773793
Eli Glasner, Ergodic theory via joinings, Mathematical Surveys and Monographs, vol. 101, American Mathematical Society, Providence, RI, 2003. MR 1958753, DOI 10.1090/surv/101
Eli Glasner and Benjamin Weiss, Topological entropy of extensions, Ergodic theory and its connections with harmonic analysis (Alexandria, 1993) London Math. Soc. Lecture Note Ser., vol. 205, Cambridge Univ. Press, Cambridge, 1995, pp. 299–307. MR 1325706, DOI 10.1017/CBO9780511574818.011
E. Glasner and B. Weiss, On the interplay between measurable and topological dynamics, Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, pp. 597–648. MR 2186250, DOI 10.1016/S1874-575X(06)80035-4
Wen Huang and Xiangdong Ye, A local variational relation and applications, Israel J. Math. 151 (2006), 237–279. MR 2214126, DOI 10.1007/BF02777364
Wen Huang, Xiangdong Ye, and Guohua Zhang, A local variational principle for conditional entropy, Ergodic Theory Dynam. Systems 26 (2006), no. 1, 219–245. MR 2201946, DOI 10.1017/S014338570500043X
W. Huang, X. Ye and G. H. Zhang, Relative entropy tuples, relative u.p.e. and c.p.e. extensions, Israel Journal of Mathematics, 158(2007), 249-284.
W. Huang, X. Ye and G. H. Zhang, Lowering topological entropy over subsets, In preparation.
A. B. Katok, Lyapunov exponents, entropy and periodic points for diffeomorphisms, Publ. Math. I.H.E.S., 51(1980), 137-173.
Pierre-Paul Romagnoli, A local variational principle for the topological entropy, Ergodic Theory Dynam. Systems 23 (2003), no. 5, 1601–1610. MR 2018614, DOI 10.1017/S0143385703000105
Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108, DOI 10.1007/978-1-4612-5775-2
Guohua Zhang, Relative entropy, asymptotic pairs and chaos, J. London Math. Soc. (2) 73 (2006), no. 1, 157–172. MR 2197376, DOI 10.1112/S0024610705022520