|
Entropy points and applications
Author(s):
Xiangdong
Ye;
Guohua
Zhang
Journal:
Trans. Amer. Math. Soc.
359
(2007),
6167-6186.
MSC (2000):
Primary 37A05, 37A35
Posted:
July 23, 2007
Retrieve article in:
PDF DVI PostScript
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.
References:
-
- 1.
- F. Blanchard, Fully positive topological entropy and topological mixing, Symbolic Dynamics and its Applications, AMS Contemporary Mathematics, 135(1992), 95-105. MR 1185082 (93k:58134)
- 2.
- F. Blanchard, A disjointness theorem involving topological entropy, Bull. de la Soc. Math. de France, 121(1993), 465-478. MR 1254749 (95e:54050)
- 3.
- F. Blanchard and Y. Lacroix, Zero-entropy factors of topological flows, Proc. Amer. Math. Soc., 119(1993), 985-992. MR 1155593 (93m:54066)
- 4.
- F. Blanchard, E. Glasner and B. Host, A variation on the variational principle and applications to entropy pairs, Ergodic Th. and Dynam. Sys., 17(1997), 29-43. MR 1440766 (98k:54073)
- 5.
- F. Blanchard, B. Host, A. Maass, S. Martínez and D. Rudolph, Entropy pairs for a measure, Ergodic Th. and Dynam. Sys., 15(1995), 621-632. MR 1346392 (96m:28024)
- 6.
- F. Blanchard and W. Huang, Entropy sets, weak mixing sets and entropy capacity, to appear in Discrete and Continuous Dynamical Systems.
- 7.
- M. Boyle and T. Downarowicz, The entropy theory of symbolic extensions, Invent. Math., 156 (2004), 119-161. MR 2047659 (2005d:37015)
- 8.
- M. Brin and A. B. Katok, On local entropy, Geometric dynamics (Rio de Janeiro, 1981), 30-38, Lecture Notes in Math. 1007, New York, Springer-Verlag (1983). MR 730261 (85c:58063)
- 9.
- D. Dou, X. Ye and G. H. Zhang, Entropy sequences and maximal entropy sets, Nonlinearity, 19(2006), 53-74. MR 2191619 (2006i:37037)
- 10.
- D. Fiebig, U. Fiebig and Z. Nitecki, Entropy and preimage sets, Ergodic Th. and Dynam. Sys., 23(2003), 1785-1806. MR 2032488 (2004k:37025)
- 11.
- E. Glasner, A simple characterization of the set of
 -entropy pairs and applications, Israel Journal of Mathematics, 192(1997), 13-27. MR 1489099 (98k:54076) - 12.
- E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs 101, American Mathematical Society, 2003. MR 1958753 (2004c:37011)
- 13.
- E. Glasner and B. Weiss, Topological entropy of extensions, Ergodic Theory and its connections with Harmonic Analysis, Cambridge University Press, 1995, 299-307. MR 1325706 (96b:54064)
- 14.
- E. Glasner and B. Weiss, On the interplay between measurable and topological dynamics, Handbook of Dynamical systems, Vol. 1B, Hasselblatt and Katok, eds., Elsevier, Amsterdam, 2006, 597-648. MR 2186250 (2006i:37005)
- 15.
- W. Huang and X. Ye, A local variational relation and applications, Israel Journal of Mathematics, 151(2006), 237-280. MR 2214126 (2006k:37033)
- 16.
- W. Huang, X. Ye and G. H. Zhang, A local variational principle for conditional entropy, Ergodic Th. and Dynam. Sys., 26(2006), 219-245. MR 2201946 (2006j:37015)
- 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.
- P. Romagnoli, A local variational principle for the topological entropy, Ergodic Th. and Dynam. Sys., 23(2003), 1601-1610. MR 2018614 (2004i:37030)
- 21.
- P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics 79, Springer-Verlag, New York-Berlin, 1982. MR 648108 (84e:28017)
- 22.
- G. H. Zhang, Relative entropy, asymptotic pairs and chaos, Journal of London Mathematical Society (2), 73(2006), 157-172. MR 2197376 (2006k:37035)
Similar Articles:
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:
10.1090/S0002-9947-07-04357-7
PII:
S 0002-9947(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
Posted:
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)
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
Forward Citation(s): Information for authors on submitting citations The following works have cited this article Dou Dou, Xiangdong Ye and Guohua Zhang, Entropy sequences and maximal entropy sets, Nonlinearity 19 (2006), 53-74. (English) MR 2191619 (2006i:37037)
HUANG Wen and YE Xiang-Dong, Complexity of Dynamical System and Tuples, Journal of the Graduate School of the Chinese Academy of Sciences 23 (2006), 701-707. (English)
WEN HUANG, XIANGDONG YE and GUOHUA ZHANG, A local variational principle for conditional entropy, Ergod. Th. & Dynam. Sys. 26 (2006), 219-245. (English) MR 2201946 (2006j:37015)
|
Comments: webmaster@ams.org