Advertisement

Israel Journal of Mathematics

, Volume 151, Issue 1, pp 237–279 | Cite as

A local variational relation and applications

  • Wen Huang
  • Xiangdong Ye
Article

Abstract

In [BGH] the authors show that for a given topological dynamical system (X,T) and an open coveru there is an invariant measure μ such that infhμ(T,ℙ)≥htop(T,U) where infimum is taken over all partitions finer thanu. We prove in this paper that if μ is an invariant measure andhμ(T,ℙ) > 0 for each ℙ finer thanu, then infhμ(T,ℙ > 0 andhtop(T,U) > 0. The results are applied to study the topological analogue of the Kolmogorov system in ergodic theory, namely uniform positive entropy (u.p.e.) of ordern (n≥2) or u.p.e. of all orders. We show that for eachn≥2 the set of all topological entropyn-tuples is the union of the set of entropyn-tuples for an invariant measure over all invariant measures. Characterizations of positive entropy, u.p.e. of ordern and u.p.e. of all orders are obtained.

We could answer several open questions concerning the nature of u.p.e. and c.p.e.. Particularly, we show that u.p.e. of ordern does not imply u.p.e. of ordern+1 for eachn≥2. Applying the methods and results obtained in the paper, we show that u.p.e. (of order 2) system is weakly disjoint from all transitive systems, and the product of u.p.e. of ordern (resp. of all orders) systems is again u.p.e. of ordern (resp. of all orders).

Keywords

Invariant Measure Topological Entropy Full Support Ergodic Measure Positive Entropy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B1] F. Blanchard,Fully positive topological entropy and topological mixing, inSymbolic Dynamics and its Applications, Contemporary Mathematics135 (1992), 95–105.CrossRefGoogle Scholar
  2. [B2] F. Blanchard,A disjointness theorem involving topological entropy, Bulletin de la Société Mathématique de France121 (1993), 465–478.MathSciNetCrossRefGoogle Scholar
  3. [BL] F. Blanchard and Y. Lacroix,Zero-entropy factors of topological flows, Proceedings of the American Mathematical Society119 (1993), 985–992.MathSciNetCrossRefGoogle Scholar
  4. [B-R] F. Blanchard, B. Host, A. Maass, S. Martinez and D. Rudolph,Entropy pairs for a measure, Ergodic Theory and Dynamical Systems15 (1995), 621–632.MathSciNetCrossRefGoogle Scholar
  5. [BGH] F. Blanchard, E. Glasner and B. Host,A variation on the variational principle and applications to entropy pairs, Ergodic Theory and Dynamical Systems17 (1997), 29–43.MathSciNetCrossRefGoogle Scholar
  6. [BHM] F. Blanchard, B. Host and A. Maass,Topological complexity, Ergodic Theory and Dynamical Systems20 (2000), 641–662.MathSciNetCrossRefGoogle Scholar
  7. [F] H. Furstenberg,Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, 1981.Google Scholar
  8. [G] E. Glasner,A simple characterization of the set of μ-entropy pairs and applications, Israel Journal of Mathematics102 (1997), 13–27.MathSciNetCrossRefGoogle Scholar
  9. [GW1] E. Glasner and B. Weiss,Strictly ergodic, uniform positive entropy models, Bulletin de la Société Mathématique de France122 (1994), 399–412.MathSciNetCrossRefGoogle Scholar
  10. [GW2] E. Glasner and B. Weiss,Topological entropy of extensions, inErgodic Theory and its Connections with Harmonic Analysis, Cambridge University Press, 1995, pp. 299–307.Google Scholar
  11. [GW3] E. Glasner and B. Weiss,On the interplay between measurable and topological dynamics, Preprint 2004.Google Scholar
  12. [GW4] E. Glasner and B. Weiss,Quasi-factors of zero entropy systems, Journal of the American Mathematical Society8 (1995), 665–686.MathSciNetzbMATHGoogle Scholar
  13. [HLSY] W. Huang, S. Li, S. Shao and X. Ye,Null systems and sequence entropy pairs, Ergodic Theory and Dynamical Systems23 (2003), 1505–1523.MathSciNetCrossRefGoogle Scholar
  14. [HMRY] W. Huang, A. Maass, P. P. Romagnoli and X. Ye,Entropy pairs and a local abramov formula for measure-theoretic entropy for a cover, Ergodic Theory and Dynamical Systems24 (2004), 1127–1153.MathSciNetCrossRefGoogle Scholar
  15. [HMY] W. Huang, A. Maass and X. Ye,Sequence entropy pairs and complexity pairs for a measure, Annales de l'Institut Fourier54 (2004), 1005–1028.MathSciNetCrossRefGoogle Scholar
  16. [HY] W. Huang and X. Ye,Topological complexity, return times and weak disjointness, Ergodic Theory and Dynamical Systems24 (2004), 825–846.MathSciNetCrossRefGoogle Scholar
  17. [HSY] W. Huang, S. Shao and X. Ye,Mixing via sequence entropy, Contemporary mathematics, to appear.Google Scholar
  18. [LS] M. Lemanczyk and A. Siemaszko,A note on the existence of a largest topological factor with zero entropy, Proceedings of the American Mathematical Society129 (2001), 475–482.MathSciNetCrossRefGoogle Scholar
  19. [P] W. Parry,Topics in Ergodic Theory, Cambridge Tracks in Mathematics, Cambridge University Press, Cambridge-New York, 1981.zbMATHGoogle Scholar
  20. [PS] K. K. Park and A. Siemaszko,Relative topological Pinsker factors and entropy pairs, Monatshefte für Mathematik134 (2001), 67–79.MathSciNetCrossRefGoogle Scholar
  21. [R] V. A. Rohlin,On the fundament ideas of measure theory, Matematicheskii Sbornik (N.S.)25(67), 1 (1949) 107–150; Engl. transl., American Mathematical Society Translations, Series 110 (1962), 1–54.Google Scholar
  22. [S] S. Shelah,A combinatorial problem; stability and order for models and theories in infinitary languages, Pacific Journal of Mathematics41 (1972), 247–261.MathSciNetCrossRefGoogle Scholar
  23. [W1] B. Weiss,Single Orbit Dynamics, AMS Regional Conference Series in Mathematics, No. 95, 2000.Google Scholar
  24. [W2] B. Weiss,Topological transitivity and ergodic measures, Mathematical Systems Theory5 (1971), 71–75.MathSciNetCrossRefGoogle Scholar
  25. [X] J. Xiong,A simple proof of a theorem of Misiurewicz, Journal of China University of Science and Technology19 (1989), 21–24.MathSciNetzbMATHGoogle Scholar

Copyright information

© The Hebrew University Magnes Press 2006

Authors and Affiliations

  • Wen Huang
    • 1
  • Xiangdong Ye
    • 1
  1. 1.Department of MathematicsUniversity of Science and Technology of ChinaHefei, AnhuiP.R. China

Personalised recommendations