Skip to main content
SpringerLink
Log in

Diabolical Entropy

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Milnor and Thurston’s famous paper proved monotonicity of the topological entropy for the real quadratic family. Guckenheimer showed that it is Hölder continuous. We obtain a precise formula for the Hölder exponent at almost every quadratic parameter. Furthermore, the entropy at most parameters is proven to be in a set of Hausdorff dimension smaller than one, while most values of the entropy arise from a set of parameters of dimension smaller than one.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Avila, A., Moreira, C.G.: Statistical properties of unimodal maps: physical measures, periodic orbits and pathological laminations. Publ. Math. Inst. Hautes Études Sci. (101), 1–67 (2005)

  2. Avila A., Moreira C.G.: Statistical properties of unimodal maps: the quadratic family. Ann. Math. (2) 161(2), 831–881 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bandtlow, O.F., Rugh, H.H.: Entropy continuity for interval maps with holes. Ergodic Theory and Dynamical Systems, 1–26 (2017)

  4. Benedicks M., Carleson L.: The dynamics of the H énon map. Ann. Math. (2) 133(1), 73–169 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bishop, C.J., Peres, Y.: Fractals in probability and analysis, volume 162 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2017)

  6. Blokh A.M., Lyubich M.Y.: Measurable dynamics of S-unimodal maps of the interval. Ann. Sci. École Norm. Sup. (4) 24(5), 545–573 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brucks K., Misiurewicz M.: The trajectory of the turning point is dense for almost all tent maps. Ergod. Theory Dyn. Syst. 16(6), 1173–1183 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bruin H.: Non-monotonicity of entropy of interval maps. Phys. Lett. A 202(5-6), 359–362 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Bruin H.: For almost every tent map, the turning point is typical. Fund. Math. 155(3), 215–235 (1998)

    MathSciNet  MATH  Google Scholar 

  10. Bruin H., Holland M., Nicol M.: Livšic regularity for Markov systems. Ergod. Theory Dyn. Syst. 25(6), 1739–1765 (2005)

    Article  MATH  Google Scholar 

  11. Bruin H., Holland M., Nicol M.: Livšic regularity for Markov systems. Ergod. Theory Dyn. Syst. 25(6), 1739–1765 (2005)

    Article  MATH  Google Scholar 

  12. Bruin H., van Strien S.: Monotonicity of entropy for real multimodal maps. J. Am. Math. Soc. 28(1), 1–61 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Carminati C., Tiozzo G.: The local Hölder exponent for the dimension of invariant subsets of the circle. Ergod. Theory Dyn. Syst. 37(6), 1825–1840 (2017)

    Article  MATH  Google Scholar 

  14. Collet P., Eckmann J.-P.: On the abundance of aperiodic behaviour for maps on the interval. Commun. Math. Phys. 73(2), 115–160 (1980)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. de Melo, W., van Strien, S.: One-dimensional dynamics, volume 25 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer, Berlin (1993)

  16. Dobbs N.: Visible measures of maximal entropy in dimension one. Bull. Lond. Math. Soc. 39(3), 366–376 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dobbs N.: On cusps and flat tops. Ann. Inst. Fourier (Grenoble) 64(2), 571–605 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dobbs, N., Todd M.: Free energy jumps up. Preprint arXiv:1512.09245, (2015)

  19. Douady, A.: Topological entropy of unimodal maps: monotonicity for quadratic polynomials. In: Real and complex dynamical systems (Hillerød, 1993), volume 464 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pp. 65–87. Kluwer Academic Publisher, Dordrecht, (1995)

  20. Dudko, D., Schleicher, D.: Core entropy of quadratic polymonials. Preprint arXiv:1412.8760, (2014)

  21. Eckmann J.-P., Wittwer P.: A complete proof of the Feigenbaum conjectures. J. Stat. Phys. 46(3-4), 455–475 (1987)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Freitas J.M.: Continuity of SRB measure and entropy for Benedicks–Carleson quadratic maps. Nonlinearity 18(2), 831–854 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Graczyk J., Świa̧tek Grzegorz: Generic hyperbolicity in the logistic family. Ann. Math. (2) 146(1), 1–52 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  24. Guckenheimer, J.: The growth of topological entropy for one-dimensional maps. In: Global Theory of Dynamical Systems (Proceedings of Internatinal Conference, Northwestern University, Evanston, Ill., 1979), volume 819 of Lecture Notes in Math., pp. 216–223. Springer, Berlin (1980)

  25. Hofbauer F., Raith P.: The Hausdorff dimension of an ergodic invariant measure for a piecewise monotonic map of the interval. Can. Math. Bull. 35(1), 84–98 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Inou H.: Extending local analytic conjugacies. Trans. Am. Math. Soc. 363(1), 331–343 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Isola S., Politi A.: Universal encoding for unimodal maps. J. Stat. Phys. 61(1-2), 263–291 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  28. Jakobson M.V.: Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys. 81(1), 39–88 (1981)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Jiang, Y.: Renormalization and geometry in one-dimensional and complex dynamics, volume 10 of Advanced Series in Nonlinear Dynamics. World Scientific Publishing Co., Inc., River Edge, NJ, (1996)

  30. Jonker L., Rand D.: Bifurcations in one dimension. II. A versal model for bifurcations. Invent. Math. 63(1), 1–15 (1981)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Ledrappier F.: Some properties of absolutely continuous invariant measures on an interval. Ergod. Theory Dyn. Syst. 1(1), 77–93 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  32. Levin G.: On an analytic approach to the Fatou conjecture. Fund. Math. 171(2), 177–196 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lyubich, M.: Dynamics of quadratic polynomials. I, II. Acta Math. 178(2):185–247, 247–297 (1997)

  34. Lyubich M.: Almost every real quadratic map is either regular or stochastic. Ann. Math. (2) 156(1), 1–78 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. Milnor, J., Thurston W.: On iterated maps of the interval. In: Dynamical systems (College Park, MD, 1986–87), volume 1342 of Lecture Notes in Math., pp. 465–563. Springer, Berlin, (1988)

  36. Misiurewicz, M.: Absolutely continuous measures for certain maps of an interval. Inst. Hautes Études Sci. Publ. Math. (53):17–51 (1981)

  37. Misiurewicz M.: Jumps of entropy in one dimension. Fund. Math. 132(3), 215–226 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  38. Misiurewicz, M., Szlenk W.: Entropy of piecewise monotone mappings. In: Dynamical systems, Vol. II—Warsaw, pp. 299–310. Astérisque, No. 50. Soc. Math. France, Paris, (1977)

  39. Nowicki T.: Some dynamical properties of S-unimodal maps. Fund. Math. 142(1), 45–57 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  40. Nowicki T., Sands D.: Non-uniform hyperbolicity and universal bounds for S-unimodal maps. Invent. Math. 132(3), 633–680 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Perera, M., Perrier A.: Sur l’absolue continuité de l’entropie dans la famille quadratique. Master’s thesis, (2012)

  42. Popovici I., Volberg A.: Rigidity of harmonic measure. Fund. Math. 150(3), 237–244 (1996)

    MathSciNet  MATH  Google Scholar 

  43. Przytycki F.: Accessibility of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps. Fund. Math. 144(3), 259–278 (1994)

    MathSciNet  MATH  Google Scholar 

  44. Przytycki, F., Rivera-Letelier, J.: Geometric pressure for multimodal maps of the interval. Accepted, Memoirs of the AMS. Preprint arXiv:1405.2443, (2014)

  45. Raith P.: Continuity of the measure of maximal entropy for unimodal maps on the interval. Qual. Theory Dyn. Syst. 4(1), 67–76 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  46. Ruelle D.: An inequality for the entropy of differentiable maps. Bol. Soc. Brasil. Mat. 9(1), 83–87 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  47. Sands, D.: Topological conditions for positive Lyapunov exponent in unimodal maps. Ph.D. thesis, University of Cambridge, Cambridge (1993)

  48. Sands D.: Misiurewicz maps are rare. Commun. Math. Phys. 197(1), 109–129 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. Singer D.: Stable orbits and bifurcation of maps of the interval. SIAM J. Appl. Math. 35(2), 260–267 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  50. Sullivan, D.: Bounds, quadratic differentials, and renormalization conjectures. In: American Mathematical Society centennial publications, vol. II (Providence, RI, 1988), pp. 417–466. Amer. Math. Soc., Providence, RI, (1992)

  51. Tiozzo G.: Continuity of core entropy of quadratic polynomials. Invent. Math. 203(3), 891–921 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. Tiozzo, G.: The local Hölder exponent for the entropy of real unimodal maps. arXiv preprint arXiv:1707.01575, 2017

  53. Tsujii M.: Positive Lyapunov exponents in families of one-dimensional dynamical systems. Invent. Math. 111(1), 113–137 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  54. Tsujii M.: On continuity of Bowen-Ruelle-Sinai measures in families of one-dimensional maps. Commun. Math. Phys. 177(1), 1–11 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. Ulam S.M., von Neumann J.: On combination of stochastic and deterministic processes. Bull. A.M.S. 53(11), 1120 (1947)

    Google Scholar 

  56. Zdunik A.: Harmonic measure on the Julia set for polynomial-like maps. Invent. Math. 128(2), 303–327 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank Magnus Aspenberg, Viviane Baladi, Michael Benedicks, Davoud Cheraghi, Jean-Pierre Eckmann, Jacek Graczyk, Hans Koch and Masato Tsujii for helpful comments and conversations and the referees for insightful reports.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 14, Ireland

    Neil Dobbs

  2. Université Paris Est - Créteil, 61 avenue du Général de Gaulle, 94010, Créteil Cedex, France

    Nicolae Mihalache

Authors
  1. Neil Dobbs
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicolae Mihalache
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Neil Dobbs.

Additional information

Communicated by C. Liverani

In memory of Tan Lei.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

N.D. was supported by the ERC Bridges grant while at the University of Geneva.

N.M. was supported by the ERC AG COMPAS grant, CNRS semester and ANR LAMBDA.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dobbs, N., Mihalache, N. Diabolical Entropy. Commun. Math. Phys. 365, 1091–1123 (2019). https://doi.org/10.1007/s00220-019-03293-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-019-03293-y

Search

Navigation