Skip to main content
SpringerLink
Log in

Entropy Spectrum of Lyapunov Exponents for Nonhyperbolic Step Skew-Products and Elliptic Cocycles

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

Abstract

We study the fiber Lyapunov exponents of step skew-product maps over a complete shift of N, \({N\ge2}\), symbols and with C1 diffeomorphisms of the circle as fiber maps. The systems we study are transitive and genuinely nonhyperbolic, exhibiting simultaneously ergodic measures with positive, negative, and zero exponents. Examples of such systems arise from the projective action of \({2\times 2}\) matrix cocycles and our results apply to an open and dense subset of elliptic \({\mathrm{SL}(2,\mathbb{R})}\) cocycles. We derive a multifractal analysis for the topological entropy of the level sets of Lyapunov exponent. The results are formulated in terms of Legendre–Fenchel transforms of restricted variational pressures, considering hyperbolic ergodic measures only, as well as in terms of restricted variational principles of entropies of ergodic measures with a given exponent. We show that the entropy of the level sets is a continuous function of the Lyapunov exponent. The level set of the zero exponent has positive, but not maximal, topological entropy. Under the additional assumption of proximality, as for example for skew-products arising from certain matrix cocycles, there exist two unique ergodic measures of maximal entropy, one with negative and one with positive fiber Lyapunov exponent.

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.: Density of positive Lyapunov exponents for \({{\rm SL}(2,\mathbb{R})}\)-cocycles. J. Am. Math. Soc. 24(4), 999–1014 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Avila A., Bochi J., Yoccoz J.-C.: Uniformly hyperbolic finite-valued \({{\rm SL}(2,\mathbb{R})}\)-cocycles. Comment. Math. Helv. 85(4), 813–884 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Avila A., Viana M.: Simplicity of Lyapunov spectra: a sufficient criterion. Port. Math. (N.S.) 64(3), 311–376 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barreira L., Saussol B.: Variational principles and mixed multifractal spectra. Trans. Am. Math. Soc. 353(10), 3919–3944 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bochi J.: Genericity of zero Lyapunov exponents. Ergod. Theory Dyn. Syst. 22(6), 1667–1696 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bochi J., Bonatti C., Díaz L.J.: Robust criterion for the existence of nonhyperbolic ergodic measures. Commun. Math. Phys. 344(3), 751–795 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Bochi J., Bonatti C., Gelfert K.: Dominated Pesin theory: convex sum of hyperbolic measures. Isr. J. Math. 226(1), 387–417 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bochi J., Rams M.: The entropy of Lyapunov-optimizing measures of some matrix cocycles. J. Mod. Dyn. 10, 255–286 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bochi J., Viana M.: Uniform (projective) hyperbolicity or no hyperbolicity: a dichotomy for generic conservative maps. Ann. Inst. Henri Poincaré Anal. Non Linéaire 19(1), 113–123 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Bonatti C., Díaz L.J, Ures R.: Minimality of strong stable and unstable foliations for partially hyperbolic diffeomorphisms. J. Inst. Math. Jussieu 1(4), 513–541 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bowen R.: Topological entropy for noncompact sets. Trans. Am. Math. Soc. 184, 125–136 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bowen, R.: Some systems with unique equilibrium states. Math. Syst. Theory 8(3), 193–202 (1974/1975)

  13. Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, volume 470 of Lecture Notes in Mathematics, revised edition. Springer, Berlin (2008). With a preface by David Ruelle, Edited by Jean-René Chazottes

  14. Burns K., Gelfert K.: Lyapunov spectrum for geodesic flows of rank 1 surfaces. Discrete Contin. Dyn. Syst. 34(5), 1841–1872 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cowieson W., Young L.-S.: SRB measures as zero-noise limits. Ergod. Theory Dyn. Syst. 25(4), 1115–1138 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Crauel H.: Extremal exponents of random dynamical systems do not vanish. J. Dyn. Differ. Equ. 2(3), 245–291 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  17. Damanik D.: Schrödinger operators with dynamically defined potentials. Ergod. Theory Dyn. Syst. 37(6), 1681–1764 (2017)

    Article  MATH  Google Scholar 

  18. Díaz L.J., Esteves S., Rocha J.: Skew product cycles with rich dynamics: from totally non-hyperbolic dynamics to fully prevalent hyperbolicity. Dyn. Syst. 31(1), 1–40 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Díaz L.J., Fisher T.: Symbolic extensions and partially hyperbolic diffeomorphisms. Discrete Contin. Dyn. Syst. 29(4), 1419–1441 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Díaz L.J., Gelfert K.: Porcupine-like horseshoes: transitivity, Lyapunov spectrum, and phase transitions. Fund. Math. 216(1), 55–100 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Díaz L.J., Gelfert K., Rams M.: Nonhyperbolic step skew-products: ergodic approximation. Ann. Inst. Henri Poincaré Anal. Non Linéaire 34(6), 1561–1598 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Díaz L.J., Gelfert K., Rams M.: Topological and ergodic aspects of partially hyperbolic diffeomorphisms and nonhyperbolic step skew products. Proc. Steklov Inst. Math. 297(1), 98–115 (2017)

    Article  MATH  Google Scholar 

  23. Duarte, P., Klein, S.: Lyapunov Exponents of Linear Cocycles: Continuity via Large Deviations, volume 3 of Atlantis Studies in Dynamical Systems. Atlantis Press, Paris (2016)

  24. Fan A., Liao L., Peyrière J.: Generic points in systems of specification and Banach valued Birkhoff ergodic average. Discrete Contin. Dyn. Syst. 21(4), 1103–1128 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Feng D.-J.: Lyapunov exponents for products of matrices and multifractal analysis. I. Positive matrices. Isr. J. Math. 138, 353–376 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Feng D.-J., Lau K.-S.: The pressure function for products of non-negative matrices. Math. Res. Lett. 9(2-3), 363–378 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Furstenberg H.: Noncommuting random products. Trans. Am. Math. Soc. 108, 377–428 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gelfert K., Kwietniak D.: On density of ergodic measures and generic points. Ergod. Theory Dyn. Syst. 38(5), 1745–1767 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gelfert K., Przytycki F., Rams M.: On the Lyapunov spectrum for rational maps. Math. Ann. 348(4), 965–1004 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Gelfert K., Rams M.: The Lyapunov spectrum of some parabolic systems. Ergod. Theory Dyn. Syst. 29(3), 919–940 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Gorodetski, A., Il’yashenko, Y.S.: Some new robust properties of invariant sets and attractors of dynamical systems. Funktsional. Anal. i Prilozhen. 33(2), 16–30, 95 (1999)

  32. Gorodetski, A., Il’yashenko, Y.S.: Some properties of skew products over a horseshoe and a solenoid. Tr. Mat. Inst. Steklova 231(Din. Sist., Avtom. i Beskon. Gruppy), 96–118 (2000)

  33. Gorodetski, A., Il’yashenko, Y.S., Kleptsyn, V., Nal’skií, M.B.: Nonremovability of zero Lyapunov exponents. Funktsional. Anal. i Prilozhen. 39(1), 27–38, 95 (2005)

  34. Gorodetski, A., Pesin, Y.: Path connectedness and entropy density of the space of hyperbolic ergodic measures. In: Katok, A., Pesin, Y., Rodriguez Hertz, F. (eds.) Modern Theory of Dynamical Systems, volume 692 of Contemporary Mathematics, pp. 111–121. American Mathematical Society, Providence (2017)

  35. Iommi G., Todd M.: Dimension theory for multimodal maps. Ann. Henri Poincaré 12(3), 591–620 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Jenkinson O.: Ergodic optimization. Discrete Contin. Dyn. Syst. 15(1), 197–224 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Knill, O.: The upper Lyapunov exponent of \({{\rm SL}(2, {\rm R})}\) cocycles: discontinuity and the problem of positivity. In: Arnold, L., Crauel, H., Eckmann, J.-P. (eds.) Lyapunov Exponents (Oberwolfach, 1990), volume 1486 of Lecture Notes in Mathematics, pp. 86–97. Springer, Berlin (1991)

  38. Leplaideur R., Oliveira K., Rios I.: Equilibrium states for partially hyperbolic horseshoes. Ergod. Theory Dyn. Syst. 31(1), 179–195 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Makarov N., Smirnov S.: On “thermodynamics” of rational maps. I. Negative spectrum. Commun. Math. Phys. 211(3), 705–743 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Malicet D.: Random walks on \({{\rm Homeo}(S^1)}\). Commun. Math. Phys. 356(3), 1083–1116 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, volume 44 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1995)

  42. Navas, A.: Groups of Circle Diffeomorphisms. Chicago Lectures in Mathematics, Spanish edition. University of Chicago Press, Chicago (2011)

  43. Olsen L.: A multifractal formalism. Adv. Math. 116(1), 82–196 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  44. Pesin Y., Weiss H.: The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples. Chaos 7(1), 89–106 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Pfister C.-E., Sullivan W.G.: On the topological entropy of saturated sets. Ergod. Theory Dyn. Syst. 27(3), 929–956 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  46. Przytycki, F., Rivera-Letelier, J.: Geometric pressure for multimodal maps of the interval. Preprint arXiv:1405.2443v1. To appear in Memoirs of the American Mathematical Society

  47. Przytycki F., Rivera-Letelier J., Smirnov S.: Equality of pressures for rational functions. Ergod. Theory Dyn. Syst. 24(3), 891–914 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  48. Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. Studies in Advanced Mathematics. CRC Press, Boca Raton (1995)

  49. Rodriguez Hertz F., Rodriguez Hertz M., Tahzibi A., Ures R.: Maximizing measures for partially hyperbolic systems with compact center leaves. Ergod. Theory Dyn. Syst. 32(2), 825–839 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  50. Ruelle, D.: Thermodynamic Formalism: The Mathematical Structures of Equilibrium Statistical Mechanics. Cambridge Mathematical Library, 2nd edn. Cambridge University Press, Cambridge (2004)

  51. Sigmund K.: On dynamical systems with the specification property. Trans. Am. Math. Soc. 190, 285–299 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  52. Tahzibi, A., Yang, J.: Invariance principle and rigidity of high entropy measures. Trans. Amer. Math. Soc. 371(2), 1231–1251 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  53. Takens F., Verbitskiy E.: On the variational principle for the topological entropy of certain non-compact sets. Ergod. Theory Dyn. Syst. 23(1), 317–348 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  54. Viana, M.: Lectures on Lyapunov Exponents, volume 145 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2014)

  55. Walters, P.: An Introduction to Ergodic Theory, volume 79 of Graduate Texts in Mathematics. Springer, New York (1982)

  56. Wijsman R.A.: Convergence of sequences of convex sets, cones and functions. II. Trans. Am. Math. Soc. 123, 32–45 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  57. Yoccoz, J.-C.: Some questions and remarks about \({{\rm SL}(2,{\rm R})}\) cocycles. In: Brin,M., Hasselblatt, B., Pesin, Y. (eds.) Modern Dynamical Systems and Applications, pp. 447–458. Cambridge University Press, Cambridge (2004)

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, PUC-Rio, Marquês de São Vicente 225, Gávea, Rio de Janeiro, 22451-900, Brazil

    L. J. Díaz

  2. Instituto de Matemática, Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos 149, Cidade Universitária - Ilha do Fundão, Rio de Janeiro, 21945-909, Brazil

    K. Gelfert

  3. Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656, Warszaw, Poland

    M. Rams

Authors
  1. L. J. Díaz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Gelfert
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Rams
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. J. Díaz.

Additional information

Communicated by C. Liverani

Publisher’s Note

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

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. This research has been supported [in part] by CNE-Faperj, CNPq-Grants (Brazil), EUMarie-Curie IRSES “Brazilian–European partnership in Dynamical Systems” (FP7-PEOPLE-2012-IRSES 318999 BREUDS), and National Science Centre Grant 2014/13/B/ST1/01033 (Poland). The authors acknowledge the hospitality of IMPAN, IM-UFRJ, and PUC-Rio and thank Anton Gorodetski, Yali Liang, and Silvius Klein for their comments. They are very thankful to two anonymous referees for their useful comments.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Díaz, L.J., Gelfert, K. & Rams, M. Entropy Spectrum of Lyapunov Exponents for Nonhyperbolic Step Skew-Products and Elliptic Cocycles. Commun. Math. Phys. 367, 351–416 (2019). https://doi.org/10.1007/s00220-019-03412-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-019-03412-9

Search

Navigation