Symbolic extensions in intermediate smoothness on surfaces

David Burguet

Annales scientifiques de l'École Normale Supérieure (2012)

  • Volume: 45, Issue: 2, page 337-362
  • ISSN: 0012-9593

Abstract

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We prove that 𝒞 r maps with r > 1 on a compact surface have symbolic extensions, i.e., topological extensions which are subshifts over a finite alphabet. More precisely we give a sharp upper bound on the so-called symbolic extension entropy, which is the infimum of the topological entropies of all the symbolic extensions. This answers positively a conjecture of S. Newhouse and T. Downarowicz in dimension two and improves a previous result of the author [11].

How to cite

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Burguet, David. "Symbolic extensions in intermediate smoothness on surfaces." Annales scientifiques de l'École Normale Supérieure 45.2 (2012): 337-362. <http://eudml.org/doc/272179>.

@article{Burguet2012,
abstract = {We prove that $\mathcal \{C\}^r$ maps with $r&gt;1$ on a compact surface have symbolic extensions, i.e., topological extensions which are subshifts over a finite alphabet. More precisely we give a sharp upper bound on the so-called symbolic extension entropy, which is the infimum of the topological entropies of all the symbolic extensions. This answers positively a conjecture of S. Newhouse and T. Downarowicz in dimension two and improves a previous result of the author [11].},
author = {Burguet, David},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {entropy structure; symbolic extension; Yomdin’s theory},
language = {eng},
number = {2},
pages = {337-362},
publisher = {Société mathématique de France},
title = {Symbolic extensions in intermediate smoothness on surfaces},
url = {http://eudml.org/doc/272179},
volume = {45},
year = {2012},
}

TY - JOUR
AU - Burguet, David
TI - Symbolic extensions in intermediate smoothness on surfaces
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2012
PB - Société mathématique de France
VL - 45
IS - 2
SP - 337
EP - 362
AB - We prove that $\mathcal {C}^r$ maps with $r&gt;1$ on a compact surface have symbolic extensions, i.e., topological extensions which are subshifts over a finite alphabet. More precisely we give a sharp upper bound on the so-called symbolic extension entropy, which is the infimum of the topological entropies of all the symbolic extensions. This answers positively a conjecture of S. Newhouse and T. Downarowicz in dimension two and improves a previous result of the author [11].
LA - eng
KW - entropy structure; symbolic extension; Yomdin’s theory
UR - http://eudml.org/doc/272179
ER -

References

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  1. [1] M. Asaoka, Hyperbolic sets exhibiting C 1 -persistent homoclinic tangency for higher dimensions, Proc. Amer. Math. Soc.136 (2008), 677–686. Zbl1131.37028MR2358509
  2. [2] M. Boyle & T. Downarowicz, The entropy theory of symbolic extensions, Invent. Math.156 (2004), 119–161. Zbl1216.37004MR2047659
  3. [3] M. Boyle, D. Fiebig & U. Fiebig, Residual entropy, conditional entropy and subshift covers, Forum Math.14 (2002), 713–757. Zbl1030.37012MR1924775
  4. [4] D. Burguet, Entropy and local complexity of differentiable dynamical sytems, thèse de doctorat, École polytechnique, 2008. 
  5. [5] D. Burguet, A proof of Yomdin-Gromov’s algebraic lemma, Israel J. Math.168 (2008), 291–316. Zbl1169.14038MR2448063
  6. [6] D. Burguet, A direct proof of the tail variational principle and its extension to maps, Ergodic Theory Dynam. Systems29 (2009), 357–369. Zbl1160.37320MR2486774
  7. [7] D. Burguet, Examples of C r interval map with large symbolic extension entropy, Discrete Contin. Dyn. Syst.26 (2010), 873–899. Zbl1193.37049MR2600721
  8. [8] D. Burguet, Symbolic extensions for nonuniformly entropy expanding maps, Colloq. Math.121 (2010), 129–151. Zbl1277.37049MR2725708
  9. [9] D. Burguet, 𝒞 2 surface diffeomorphisms have symbolic extensions, Invent. Math.186 (2011), 191–236. Zbl1263.37065MR2836054
  10. [10] D. Burguet, Symbolic extensions and continuity properties of the entropy, Arch. Math. (Basel) 96 (2011), 387–400. Zbl1235.37007MR2794094
  11. [11] D. Burguet & K. McGoff, Orders of accumulation of entropy, Fund. Math.216 (2012), 1–53. Zbl1252.37008MR2864449
  12. [12] J. Buzzi, Intrinsic ergodicity of smooth interval maps, Israel J. Math.100 (1997), 125–161. Zbl0889.28009MR1469107
  13. [13] T. Catalan, A generic condition for existence of symbolic extension of volume preserving diffeomorphisms, preprint arXiv:1109.3080. Zbl1336.37013MR2997705
  14. [14] T. Catalan & A. Tahzibi, A lower bound for topological entropy of generic non Anosov diffeomorphisms, preprint arXiv:1011.2441. Zbl1317.37023
  15. [15] R. A. DeVore & G. G. Lorentz, Constructive approximation, Grund. Math. Wiss. 303, Springer, 1993. Zbl0797.41016MR1261635
  16. [16] L. J. Díaz & T. Fisher, Symbolic extensions and partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst.29 (2011), 1419–1441. Zbl1220.37016MR2773191
  17. [17] T. Downarowicz, Entropy structure, J. Anal. Math.96 (2005), 57–116. Zbl1151.37020MR2177182
  18. [18] T. Downarowicz, Entropy in dynamical systems, New Mathematical Monographs 18, Cambridge Univ. Press, 2011. Zbl1220.37001MR2809170
  19. [19] T. Downarowicz & A. Maass, Smooth interval maps have symbolic extensions: the antarctic theorem, Invent. Math.176 (2009), 617–636. Zbl1185.37100MR2501298
  20. [20] T. Downarowicz & S. E. Newhouse, Symbolic extensions and smooth dynamical systems, Invent. Math.160 (2005), 453–499. Zbl1067.37018MR2178700
  21. [21] J. Flum & M. Grohe, Parameterized complexity theory, Texts in Theoretical Computer Science. An EATCS Series, Springer, 2006. Zbl1143.68016MR2238686
  22. [22] M. Gromov, Entropy, homology and semialgebraic geometry, Séminaire Bourbaki, vol. 1985/86, exp. no 663, Astérisque 145–146 (1987), 225–240. Zbl0611.58041MR880035
  23. [23] K. McGoff, Orders of accumulation of entropy on manifolds, J. d’Anal. math. 114 (2011), 157–206. Zbl1248.54012MR2837084
  24. [24] M. Misiurewicz, Diffeomorphism without any measure with maximal entropy, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.21 (1973), 903–910. Zbl0272.28013MR336764
  25. [25] M. Misiurewicz, Topological conditional entropy, Studia Math.55 (1976), 175–200. Zbl0355.54035MR415587
  26. [26] L. Neder, Abschätzungen für die Ableitungen einer reellen Funktion eines reellen Arguments, Math. Z.31 (1930), 356–365. Zbl55.0133.03MR1545119JFM55.0133.03
  27. [27] S. E. Newhouse, Continuity properties of entropy, Ann. of Math.129 (1989), 215–235. Zbl0676.58039MR986792
  28. [28] V. I. Osedelets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (1968), 197–231. Zbl0236.93034
  29. [29] S. Ruette, Mixing C r maps of the interval without maximal measure, Israel J. Math.127 (2002), 253–277. Zbl1187.37057MR1900702
  30. [30] P. Walters, An introduction to ergodic theory, Graduate Texts in Math. 79, Springer, 1982. Zbl0475.28009MR648108

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