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Topology and topological sequence entropy

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Abstract

Let X be a compact metric space and T: XX be continuous. Let h* (T) be the supremum of topological sequence entropies of T over all subsequences of ℤ+ and S(X) be the set of the values h*(T) for all continuous maps T on X. It is known that {0} ⊆ S(X) ⊆ {0, log 2, log 3, …} ∪ {∞}. Only three possibilities for S(X) have been observed so far, namely S(X) = {0}, S(X) = {0, log 2, ∞} and S(X) = {0, log 2, log 3, …}∪{∞}.

In this paper we completely solve the problem of finding all possibilities for S(X) by showing that in fact for every set {0} ⊆ A ⊆ {0, log 2, log 3, …} ∪ {∞} there exists a one-dimensional continuum XA with S(XA) = A. In the construction of XA we use Cook continua. This is apparently the first application of these very rigid continua in dynamics.

We further show that the same result is true if one considers only homeomorphisms rather than continuous maps. The problem for group actions is also addressed. For some class of group actions (by homeomorphisms) we provide an analogous result, but in full generality this problem remains open.

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References

  1. Adler R L, Konheim A G, McAndrew M H. Topological entropy. Trans Amer Math Soc, 1965, 114: 309–319

    MathSciNet  MATH  Google Scholar 

  2. Akin E, Rautio J. Chain transitive homeomorphisms on a space: All or none. Pacific J Math, 2017, 291: 1–49

    MathSciNet  MATH  Google Scholar 

  3. Alseda L, Llibre J, Misiurewicz M. Combinatorial Dynamics and Entropy in Dimension One, 2nd ed. Advanced Series in Nonlinear Dynamics, vol. 5. Singapore: World Scientific, 2000

    MATH  Google Scholar 

  4. Blanchard F. A disjointness theorem involving topological entropy. Bull Soc Math France, 1993, 121: 465–478

    MathSciNet  MATH  Google Scholar 

  5. Blanchard F, Host B, Maass A. Topological complexity. Ergodic Theory Dynam Systems, 2000, 20: 641–662

    MathSciNet  MATH  Google Scholar 

  6. Blanchard F, Huang W, Snoha L’. Topological size of scrambled sets. Colloq Math, 2008, 110: 293–361

    MathSciNet  MATH  Google Scholar 

  7. Borsuk K. Theory of Retracts. Monografie Matematyczne, vol. 44. Warsaw: Państwowe Wydawnictwo Naukowe, 1967

    Google Scholar 

  8. Cánovas J. S. On topological sequence entropy of circle maps. Appl Gen Topol, 2001, 2: 1–7

    MathSciNet  MATH  Google Scholar 

  9. Cánovas J S. Topological sequence entropy of interval maps. Nonlinearity, 2004, 17: 49–56

    MathSciNet  MATH  Google Scholar 

  10. Cánovas J S. A guide to topological sequence entropy. In: Progress in Mathematical Biology Research. New York: Nova Science Publishers, 2008, 101–139

    Google Scholar 

  11. Christopher M. Entropy of shift maps of the pseudo-arc. Topology Appl, 2012, 159: 34–39

    MathSciNet  MATH  Google Scholar 

  12. Cook H. Continua which admit only the identity mapping onto non-degenerate subcontinua. Fund Math, 1967, 60: 241–249

    MathSciNet  MATH  Google Scholar 

  13. de Groot J. Groups represented by homeomorphism groups. Math Ann, 1959, 138: 80–102

    MathSciNet  MATH  Google Scholar 

  14. de Groot J, Wille R J. Rigid continua and topological group-pictures. Arch Math, 1958, 9: 441–446

    MathSciNet  MATH  Google Scholar 

  15. Downarowicz T, Snoha L, Tywoniuk D. Minimal spaces with cyclic group of homeomorphisms. J Dynam Differential Equations, 2017, 29: 243–257

    MathSciNet  MATH  Google Scholar 

  16. Franzová N, Smítal J. Positive sequence topological entropy characterizes chaotic maps. Proc Amer Math Soc, 1991, 112: 1083–1086

    MathSciNet  MATH  Google Scholar 

  17. García-Ramos F. Weak forms of topological and measure-theoretical equicontinuity: Relationships with discrete spectrum and sequence entropy. Ergodic Theory Dynam Systems, 2017, 37: 1211–1237

    MathSciNet  MATH  Google Scholar 

  18. Glasner E. The structure of tame minimal dynamical systems for general groups. Invent Math, 2018, 211: 213–244

    MathSciNet  MATH  Google Scholar 

  19. Glasner E, Weiss B. Quasi-factors of zero-entropy systems. J Amer Math Soc, 1995, 8: 665–686

    MathSciNet  MATH  Google Scholar 

  20. Goodman TNT. Topological sequence entropy. Proc Lond Math Soc (3), 1974, 29: 331–350

    MathSciNet  MATH  Google Scholar 

  21. Handel M. A pathological area preserving C diffeomorphism of the plane. Proc Amer Math Soc, 1982, 86: 163–168

    MathSciNet  MATH  Google Scholar 

  22. Hocking J G, Young G S. Topology. Reading-London: Addison-Wesley Publishing, 1961

    MATH  Google Scholar 

  23. Huang W. Tame systems and scrambled pairs under an Abelian group action. Ergodic Theory Dynam Systems, 2006, 26: 1549–1567

    MathSciNet  MATH  Google Scholar 

  24. Huang W, Li S, Shao S, et al. Null systems and sequence entropy pairs. Ergodic Theory Dynam Systems, 2003, 23: 1505–1523

    MathSciNet  MATH  Google Scholar 

  25. Huang W, Lu P, Ye X. Measure-theoretical sensitivity and equicontinuity. Israel J Math, 2011, 183: 233–283

    MathSciNet  MATH  Google Scholar 

  26. Huang W, Maass A, Ye X. Sequence entropy pairs and complexity pairs for a measure. Ann Inst Fourier (Grenoble), 2004, 54: 1005–1028

    MathSciNet  MATH  Google Scholar 

  27. Huang W, Ye X. A local variational relation and applications. Israel J Math, 2006, 151: 237–279

    MathSciNet  MATH  Google Scholar 

  28. Huang W, Ye X. Combinatorial lemmas and applications to dynamics. Adv Math, 2009, 220: 1689–1716

    MathSciNet  MATH  Google Scholar 

  29. Hurewicz W, Wallman H. Dimension Theory. Princeton: Princeton University Press, 1941

    MATH  Google Scholar 

  30. Kerr D, Li H F. Independence in topological and C*-dynamics. Math Ann, 2007, 338: 869–926

    MathSciNet  MATH  Google Scholar 

  31. Krzempek J. Finite-to-one maps and dimension. Fund Math, 2004, 182: 95–106

    MathSciNet  MATH  Google Scholar 

  32. Krzempek J. Some examples of higher-dimensional rigid continua. Houston J Math, 2009, 35: 485–500

    MathSciNet  MATH  Google Scholar 

  33. Kushnirenko A G. On metric invariants of entropy type (in Russian). Uspekhi Mat Nauk, 1967, 22: 57–65; English translation: Russian Math Surveys, 1967, 22: 53–61

    MathSciNet  Google Scholar 

  34. Lewis W. Continuum theory and dynamics problems. In: Continuum Theory and Dynamical Systems. Contemp Math, vol. 117. Providence: Amer Math Soc, 1991, 99–101

    Google Scholar 

  35. Li H F, Rong Z. Null actions and RIM non-open extensions of strongly proximal actions. ArXiv:1807.00423, 2018

  36. Maass A, Shao S. Structure of bounded topological-sequence-entropy minimal systems. J Lond Math Soc (2), 2007, 76: 702–718

    MathSciNet  MATH  Google Scholar 

  37. Maćkowiak T. Singular arc-like continua. Dissertationes Math, 1986, 257: 1–40

    MathSciNet  MATH  Google Scholar 

  38. Nadler S B Jr. Continuum Theory: An Introduction. New York: Marcel Dekker, 1992

    MATH  Google Scholar 

  39. Pultr A, Trnková V. Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories. Amsterdam-New York: North-Holland Publishing, 1980

    MATH  Google Scholar 

  40. Sharkovsky A N. Coexistence of cycles of a continuous mapping of the line into itself (in Russian). Ukrain Math Zh, 1964, 16: 61–71

    Google Scholar 

  41. Smítal J. Chaotic functions with zero topological entropy. Trans Amer Math Soc, 1986, 297: 269–282

    MathSciNet  MATH  Google Scholar 

  42. Tan F. The set of sequence entropies for graph maps. Topology Appl, 2011, 158: 533–541

    MathSciNet  MATH  Google Scholar 

  43. Tan F, Ye X, Zhang R F. The set of sequence entropies for a given space. Nonlinearity, 2010, 23: 159–178

    MathSciNet  MATH  Google Scholar 

  44. Trnková V. Non-constant continuous mappings of metric or compact Hausdorff spaces. Comment Math Univ Carolin, 1972, 13: 283–295

    MathSciNet  MATH  Google Scholar 

  45. Trnková V. All small categories are representable by continuous nonconstant mappings of bicompacta. Soviet Math Dokl, 1976, 17: 1403–1406

    MATH  Google Scholar 

  46. Whyburn G T. Analytic Topology. AMS Collections of Publications, vol. 28. Providence: Amer Math Soc, 1942

    Google Scholar 

  47. Ye X, Zhang R F. Countable compacta admitting homeomorphisms with positive sequence entropy. J Dynam Differential Equations, 2008, 20: 867–882

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The first author was supported by the Slovak Research and Development Agency (Grant No. APVV-15-0439) and by VEGA (Grant No. 1/0786/15). The second author was supported by National Natural Science Foundation of China (Grant Nos. 11371339 and 11431012). The third author was supported by National Natural Science Foundation of China (Grant Nos. 11871188 and 11671094). This project was started when the first two authors visited Max Planck Institute for Mathematics, Bonn, in 2009. Much of the work on the project was done also when the first author visited University of Science and Technology of China, Hefei in 2011 and 2016 and when the third author visited Matej Bel University in 2018. The authors thank all these institutions for the warm hospitality and financial supports. The authors thank Jerzy Krzempek for providing them with the reference [37], where the existence of Cook continua in the plane is proved. This enabled the authors to simplify the geometry of their constructions. Sincere thanks of the authors go to Hanfeng Li for his comments on the preliminary version of the paper, which resulted in the homeomorphism case in Subsection 8.2, the group actions case in Subsection 8.3, and an open problem in Subsection 9.6. The authors also thank the anonymous referees for their helpful comments and suggestions that improved the manuscript.

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Correspondence to Ruifeng Zhang.

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Snoha, L., Ye, X. & Zhang, R. Topology and topological sequence entropy. Sci. China Math. 63, 205–296 (2020). https://doi.org/10.1007/s11425-019-9536-7

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