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The entropy theory of symbolic extensions

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Abstract

Fix a topological system (X,T), with its space K(X,T) of T-invariant Borel probabilities. If (Y,S) is a symbolic system (subshift) and ϕ:(Y,S)→(X,T) is a topological extension (factor map), then the function h ϕ ext on K(X,T) which assigns to each μ the maximal entropy of a measure ν on Y mapping to μ is called the extension entropy function of ϕ. The infimum of such functions over all symbolic extensions is called the symbolic extension entropy function and is denoted by h sex. In this paper we completely characterize these functions in terms of functional analytic properties of an entropy structure on (X,T). The entropy structure ℋ is a sequence of entropy functions h k defined with respect to a refining sequence of partitions of X (or of X×Z, for some auxiliary system (Z,R) with simple dynamics) whose boundaries have measure zero for all the invariant Borel probabilities. We develop the functional analysis and computational techniques to produce many dynamical examples; for instance, we resolve in the negative the question of whether the infimum of the topological entropies of symbolic extensions of (X,T) must always be attained, and we show that the maximum value of h sex need not be achieved at an ergodic measure. We exhibit several characterizations of the asymptotically h-expansive systems of Misiurewicz, which emerge as a fundamental natural class in the context of the entropy structure. The results of this paper are required for the Downarowicz-Newhouse results [DN] on smooth dynamical systems.

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References

  1. Asimow, L., Ellis, A.: Convexity Theory and its Applications in Functional Analysis. Lond. Math. Soc. Monogr. 16. Academic Press 1980

  2. Boyle, M., Fiebig, D., Fiebig, U.: Residual entropy, conditional entropy and subshift covers. Forum Math. 14, 713–757 (2002)

    MathSciNet  MATH  Google Scholar 

  3. Blanchard, F., Glasner, E., Host, B.: A variation on the variational principle and applications to entropy pairs. Ergodic Theory Dyn. Syst. 17, 29–43 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lect. Notes Math. 470. Springer 1975

  5. Buzzi, J.: Intrinsic ergodicity of smooth interval maps. Isr. J. Math. 100, 125–161 (1997)

    MathSciNet  MATH  Google Scholar 

  6. Choquet, G.: Lectures on Analysis, Vol. II: Representation Theory. W.A. Benjamin, Inc. 1969

  7. Denker, M., Grillenberger, C., Sigmund, K.: Ergodic Theory on Compact Spaces. Lect. Notes Math. 527. Springer 1976

  8. Downarowicz, T.: The Choquet simplex of invariant measures for minimal flows Isr. J. Math. 74, 241–256 (1991)

    MathSciNet  MATH  Google Scholar 

  9. Downarowicz, T.: Entropy of a symbolic extension of a totally disconnected dynamical system. Ergodic Theory Dyn. Syst. 21, 1051–1070 (2001)

    MathSciNet  MATH  Google Scholar 

  10. Downarowicz, T.: Entropy Structure. Preprint (2003)

  11. Downarowicz, T., Newhouse, S.: Symbolic extensions in smooth dynamical systems. Preprint (2002)

  12. Downarowicz, T., Serafin, J.: Fiber entropy and conditional variational principles in compact non-metrizable spaces. Fundam. Math. 172, 217–247 (2002)

    MathSciNet  MATH  Google Scholar 

  13. Downarowicz, T., Serafin, J.: Possible entropy functions. Isr. J. Math. 135, 221–251 (2003)

    MathSciNet  Google Scholar 

  14. Jacobs, K.,Keane, M.: 0-1 sequences of Toeplitz type. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 13, 123–131 (1969)

    MATH  Google Scholar 

  15. Kifer, Y., Weiss, B.: Generating partitions for random transformations. Ergodic Theory Dyn. Syst. 22, 1813–1830 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kulesza, J.: Zero-dimensional covers of finite-dimensional dynamical systems. Ergodic Theory Dyn. Syst. 15, 939–950 (1995)

    MathSciNet  MATH  Google Scholar 

  17. Ledrappier, F.: A variational principle for the topological conditional entropy. Lect. Notes Math. 729, 78–88. Springer 1979

    Google Scholar 

  18. Ledrappier, F., Walters, P.: A relativised variational principle for continuous transformations. J. Lond. Math. Soc. 16, 568–576 (1977)

    MATH  Google Scholar 

  19. Lindenstrauss, E.: Mean dimension, small entropy factors and an embedding theorem. Publ. Math., Inst. Hautes Étud. Sci. 89, 227–262 (1999)

    Google Scholar 

  20. Lindenstrauss, E., Weiss, B.: Mean topological dimension. Isr. J. Math 115, 1–24 (2000)

    MathSciNet  MATH  Google Scholar 

  21. Misiurewicz, M.: Diffeomorphism without any measure with maximal entropy. Bull. Acad. Polon. Sci., Sér. Sci. Math., Astr. et Phys. 21, 903–910 (1973)

    Google Scholar 

  22. Misiurewicz, M.: Topological conditional entropy. Stud. Math. 55, 175–200 (1976)

    MATH  Google Scholar 

  23. Newhouse, S.: Continuity properties of entropy. Ann. Math. 129, 215–235 (1989)

    MathSciNet  MATH  Google Scholar 

  24. Ormes, N.: Strong orbit realization for minimal homeomorphisms. J. Anal. Math. 71, 103–133 (1997)

    MathSciNet  MATH  Google Scholar 

  25. Parry, W.: Entropy and generators in ergodic theory. New York, Amsterdam: W.A. Benjamin, Inc. 1969

  26. Phelps, R.R.: Lectures on Choquet’s Theorem, second edition. Lect. Notes Math. 1757. Springer 2001

  27. Phelps, R.R.: Personal communication (2001)

  28. Tong, H.: Some characterizations of normal and perfectly normal spaces. Duke Math. J. 19, 289–292 (1952)

    MathSciNet  MATH  Google Scholar 

  29. Walters, P.: An introduction to ergodic theory. Springer 1982

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Authors and Affiliations

  1. Department of Mathematics, University of Maryland, 20742-4015, College Park, MD, USA

    Mike Boyle

  2. Institute of Mathematics, Technical University, Wybrzeze Wyspiańskiego 27, 50-370, Wrocław, Poland

    Tomasz Downarowicz

Authors
  1. Mike Boyle
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  2. Tomasz Downarowicz
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Corresponding authors

Correspondence to Mike Boyle or Tomasz Downarowicz.

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Mathematics Subject Classification (2000)

Primary: 37B10; Secondary: 37B40, 37C40, 37C45, 37C99, 37D35

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Boyle, M., Downarowicz, T. The entropy theory of symbolic extensions. Invent. math. 156, 119–161 (2004). https://doi.org/10.1007/s00222-003-0335-2

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  • DOI: https://doi.org/10.1007/s00222-003-0335-2

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