Entropy points and applications

Authors:
Xiangdong Ye and Guohua Zhang

Journal:
Trans. Amer. Math. Soc. **359** (2007), 6167-6186

MSC (2000):
Primary 37A05, 37A35

Published electronically:
July 23, 2007

MathSciNet review:
2336322

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**1.**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**, 10.1090/conm/135/1185082**2.**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****3.**F. Blanchard and Y. Lacroix,*Zero entropy factors of topological flows*, Proc. Amer. Math. Soc.**119**(1993), no. 3, 985–992. MR**1155593**, 10.1090/S0002-9939-1993-1155593-2**4.**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**, 10.1017/S0143385797069794**5.**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**, 10.1017/S0143385700008579**6.**F. Blanchard and W. Huang,*Entropy sets, weak mixing sets and entropy capacity*, to appear in Discrete and Continuous Dynamical Systems.**7.**Mike Boyle and Tomasz Downarowicz,*The entropy theory of symbolic extensions*, Invent. Math.**156**(2004), no. 1, 119–161. MR**2047659**, 10.1007/s00222-003-0335-2**8.**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**, 10.1007/BFb0061408**9.**Dou Dou, Xiangdong Ye, and Guohua Zhang,*Entropy sequences and maximal entropy sets*, Nonlinearity**19**(2006), no. 1, 53–74. MR**2191619**, 10.1088/0951-7715/19/1/004**10.**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**, 10.1017/S0143385703000221**11.**Eli Glasner,*A simple characterization of the set of 𝜇-entropy pairs and applications*, Israel J. Math.**102**(1997), 13–27. MR**1489099**, 10.1007/BF02773793**12.**Eli Glasner,*Ergodic theory via joinings*, Mathematical Surveys and Monographs, vol. 101, American Mathematical Society, Providence, RI, 2003. MR**1958753****13.**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**, 10.1017/CBO9780511574818.011**14.**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**, 10.1016/S1874-575X(06)80035-4**15.**Wen Huang and Xiangdong Ye,*A local variational relation and applications*, Israel J. Math.**151**(2006), 237–279. MR**2214126**, 10.1007/BF02777364**16.**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**, 10.1017/S014338570500043X**17.**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.**18.**W. Huang, X. Ye and G. H. Zhang,*Lowering topological entropy over subsets*, In preparation.**19.**A. B. Katok,*Lyapunov exponents, entropy and periodic points for diffeomorphisms*, Publ. Math. I.H.E.S.,**51**(1980), 137-173.**20.**Pierre-Paul Romagnoli,*A local variational principle for the topological entropy*, Ergodic Theory Dynam. Systems**23**(2003), no. 5, 1601–1610. MR**2018614**, 10.1017/S0143385703000105**21.**Peter Walters,*An introduction to ergodic theory*, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR**648108****22.**Guohua Zhang,*Relative entropy, asymptotic pairs and chaos*, J. London Math. Soc. (2)**73**(2006), no. 1, 157–172. MR**2197376**, 10.1112/S0024610705022520

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
37A05,
37A35

Retrieve articles in all journals with MSC (2000): 37A05, 37A35

Additional Information

**Xiangdong Ye**

Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China

Email:
yexd@ustc.edu.cn

**Guohua Zhang**

Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China

Email:
ghzhang@mail.ustc.edu.cn

DOI:
https://doi.org/10.1090/S0002-9947-07-04357-7

Keywords:
Topological entropy,
entropy for a measure,
entropy tuple,
entropy point

Received by editor(s):
September 15, 2005

Received by editor(s) in revised form:
May 15, 2006

Published electronically:
July 23, 2007

Additional Notes:
Both authors were partially supported by Ministry of Education (no. 0358053), and the first author was partially supported by NSFC (no. 10531010)

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.