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Quasi-multiplicativity of Typical Cocycles

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Abstract

We show that typical [in the sense of Bonatti and Viana (Ergod Theory Dyn Syst 24(5):1295–1330, 2004) and Avila and Viana (Port Math 64:311–376, 2007)] Hölder and fiber-bunched \(\text {GL}_d(\mathbb {R})\)-valued cocycles over subshifts of finite type are uniformly quasi-multiplicative with respect to all singular value potentials. We prove the continuity of the singular value pressure and its corresponding (necessarily unique) equilibrium state for such cocycles, and apply this result to repellers. Moreover, we show that the pointwise Lyapunov spectrum is closed and convex, and establish partial multifractal analysis on the level sets of pointwise Lyapunov exponents for such cocycles.

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References

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

    MathSciNet  MATH  Google Scholar 

  2. Barreira, L.M.: A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems. Ergod. Theory Dyn. Syst. 16(5), 871–927 (1996)

    MathSciNet  MATH  Google Scholar 

  3. Barreira, L.: Dimension estimates in nonconformal hyperbolic dynamics. Nonlinearity 16(5), 1657–1672 (2003)

    ADS  MathSciNet  MATH  Google Scholar 

  4. Backes, L., Brown, A.W., Butler, C.: Continuity of Lyapunov exponents for cocycles with invariant holonomies. J. Mod. Dyn. 12, 223–260 (2018)

    MathSciNet  MATH  Google Scholar 

  5. Ban, J., Cao, Y., Huyi, H.: The dimensions of a non-conformal repeller and an average conformal repeller. Trans. Am. Math. Soc. 362(2), 727–751 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Barreira, L., Gelfert, K.: Multifractal analysis for Lyapunov exponents on nonconformal repellers. Commun. Math. Phys. 267(2), 393–418 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  8. Bárány, B., Hochman, M., Rapaport, A.: Hausdorff dimension of planar self-affine sets and measures. Invent. Math. 216(3), 601–659 (2019)

    ADS  MathSciNet  MATH  Google Scholar 

  9. Bochi, J., Morris, I.D.: Equilibrium states of generalised singular value potentials and applications to affine iterated function systems. Geom. Funct. Anal. 28(4), 995–1028 (2018)

    MathSciNet  MATH  Google Scholar 

  10. Bowen, R.: Markov partitions for axiom a diffeomorphisms. Am. J. Math. 92(3), 725–747 (1970)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics, vol. 470. Springer, Berlin (1975)

    MATH  Google Scholar 

  14. Bowen, R.: Hausdorff dimension of quasi-circles. Publ. Math. Inst. Hautes Études Sci. 50(1), 11–25 (1979)

    MATH  Google Scholar 

  15. Barreira, L., Pesin, Y.B.: Lyapunov Exponents and Smooth Ergodic Theory, vol. 23. American Mathematical Society, Philadelphia (2002)

    MATH  Google Scholar 

  16. Barreira, L., Pesin, Y., Schmeling, J.: On a general concept of multifractality: multifractal spectra for dimensions, entropies, and Lyapunov exponents. multifractal rigidity. Chaos Interdiscip. J. Nonlinear Sci. 7(1), 27–38 (1997)

    MathSciNet  MATH  Google Scholar 

  17. Backes, L., Poletti, M., Varandas, P., Lima, Y.: Simplicity of Lyapunov spectrum for linear cocycles over non-uniformly hyperbolic systems. Ergod. Theory Dyn. Syst. (2019). https://doi.org/10.1017/etds.2019.22

  18. Breuillard, E., Sert, C.: The joint spectrum. arXiv preprint arXiv:1809.02404 (2018)

  19. Bonatti, C., Viana, M.: Lyapunov exponents with multiplicity 1 for deterministic products of matrices. Ergod. Theory Dyn. Syst. 24(5), 1295–1330 (2004)

    MathSciNet  MATH  Google Scholar 

  20. Cao, Y., Feng, D., Huang, W.: The thermodynamic formalism for sub-additive potentials. Discrete Contin. Dyn. Syst. 20(3), 639–657 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Climenhaga, V.: Multifractal formalism derived from thermodynamics. Electron. Res. Announc. 17, 1–11 (2010)

    MathSciNet  MATH  Google Scholar 

  22. Climenhaga, V.: The thermodynamic approach to multifractal analysis. Ergod. Theory Dyn. Syst. 34(5), 1409–1450 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Chen, J., Pesin, Y.: Dimension of non-conformal repellers: a survey. Nonlinearity 23(4), R93–R114 (2010)

    ADS  MathSciNet  MATH  Google Scholar 

  24. Cao, Y., Pesin, Y., Zhao, Y.: Dimension estimates for non-conformal repellers and continuity of sub-additive topological pressure. Geom. Funct. Anal. 29, 1325–1368 (2019)

    MathSciNet  MATH  Google Scholar 

  25. Díaz, L.J., Gelfert, K., Rams, M.: Entropy spectrum of Lyapunov exponents for nonhyperbolic step skew-products and elliptic cocycles. Commun. Math. Phys. 367(2), 351–416 (2019)

    ADS  MathSciNet  MATH  Google Scholar 

  26. Falconer, K.J.: The Hausdorff Dimension of Self-Affine Fractals. Mathematical Proceedings of the Cambridge Philosophical Society, vol. 103, pp. 339–350. Cambridge University Press, Cambridge (1988)

    MATH  Google Scholar 

  27. Falconer, K.J.: A subadditive thermodynamic formalism for mixing repellers. J. Phys. A Math. Gen. 21(14), L737–L742 (1988)

    ADS  MathSciNet  MATH  Google Scholar 

  28. Falconer, K.J.: Bounded Distortion and Dimension for Non-conformal Repellers. Mathematical Proceedings of the Cambridge Philosophical Society, vol. 115, pp. 315–334. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  29. Feng, D.-J.: Lyapunov exponents for products of matrices and multifractal analysis. Part i: positive matrices. Isr. J. Math. 138(1), 353–376 (2003)

    MATH  Google Scholar 

  30. Feng, D.-J.: Lyapunov exponents for products of matrices and multifractal analysis. Part ii: general matrices. Isr. J. Math. 170(1), 355–394 (2009)

    MATH  Google Scholar 

  31. Feng, D.-J.: Equilibrium states for factor maps between subshifts. Adv. Math. 226(3), 2470–2502 (2011)

    MathSciNet  MATH  Google Scholar 

  32. Fan, A.-H., Feng, D.-J., Jun, W.: Recurrence, dimension and entropy. J. Lond. Math. Soc. 64(1), 229–244 (2001)

    MathSciNet  MATH  Google Scholar 

  33. Feng, D.-J., Huang, W.: Lyapunov spectrum of asymptotically sub-additive potentials. Commun. Math. Phys. 297(1), 1–43 (2010)

    ADS  MathSciNet  MATH  Google Scholar 

  34. Feng, D.-J., Käenmäki, A.: Equilibrium states of the pressure function for products of matrices. Discrete Contin. Dyn. Syst. 30(3), 699–708 (2011)

    MathSciNet  MATH  Google Scholar 

  35. Feng, D.-J., Shmerkin, P.: Non-conformal repellers and the continuity of pressure for matrix cocycles. Geom. Funct. Anal. 24(4), 1101–1128 (2014)

    MathSciNet  MATH  Google Scholar 

  36. Furstenberg, H.: Noncommuting random products. Trans. Am. Math. Soc. 108(3), 377–428 (1963)

    MathSciNet  MATH  Google Scholar 

  37. Hutchinson, J.E.: Fractals and self similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981)

    MathSciNet  MATH  Google Scholar 

  38. Keller, G.: Equilibrium States in Ergodic Theory, vol. 42. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  39. Kalinin, B., Sadovskaya, V.: Cocycles with one exponent over partially hyperbolic systems. Geom. Dedicata 167(1), 167–188 (2013)

    MathSciNet  MATH  Google Scholar 

  40. Käenmäki, A.: On natural invariant measures on generalised iterated function systems. Ann. Acad. Sci. Fenn. Math. 29, 419–458 (2004)

    MathSciNet  MATH  Google Scholar 

  41. Ledrappier, F., Walters, P.: A relativised variational principle for continuous transformations. J. Lond. Math. Soc. 2(3), 568–576 (1977)

    MathSciNet  MATH  Google Scholar 

  42. Pesin, Y., Weiss, H.: The multifractal analysis of Birkhoff averages and large deviations. In: Broer, H., Krauskopf, B., Vegter, G. (eds.) Global Analysis of Dynamical Systems, pp. 419–431. IoP Publishing, Bristol (2001)

  43. Rohlin, V.A.: Exact endomorphisms of lebesgue spaces. Izv. Akad. Nauk SSSR Ser. Mat. 25, 499–530 (1961)

    MathSciNet  Google Scholar 

  44. Ruelle, D.: Repellers for real analytic maps. Ergod. Theory Dyn. Syst. 2(1), 99–107 (1982)

    MathSciNet  MATH  Google Scholar 

  45. Ruelle, D.: The thermodynamic formalism for expanding maps. Commun. Math. Phys. 125(2), 239–262 (1989)

    ADS  MathSciNet  MATH  Google Scholar 

  46. Sinai, Y.G.: Markov partitions and c-diffeomorphisms. Funct. Anal. Appl. 2(1), 61–82 (1968)

    MathSciNet  MATH  Google Scholar 

  47. Solomyak, Boris: Measure and Dimension for Some Fractal Families. Mathematical Proceedings of the Cambridge Philosophical Society, vol. 124, pp. 531–546. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  48. Walters, P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics, vol. 79. Springer, Berlin (2000)

    MATH  Google Scholar 

  49. Zhang, Y.: Dynamical upper bounds for Hausdorff dimension of invariant sets. Ergod. Theory Dyn. Syst. 17(3), 739–756 (1997)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author is very grateful to his advisor, Amie Wilkinson, for her support and numerous helpful discussions. The author would also like to thank Clark Butler for sharing his insights and for pointing out an error in Sect. 3 of the original draft, and Aaron Brown for many helpful suggestions. Lastly, the author also thanks De-Jun Feng for his comments, Ping Ngai Chung for improving the readability of the paper, and anonymous referees for many useful comments that helped improve the paper.

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  1. Department of Mathematics, The University of Chicago, Chicago, IL, 60637, USA

    Kiho Park

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Communicated by C. Liverani.

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Park, K. Quasi-multiplicativity of Typical Cocycles. Commun. Math. Phys. 376, 1957–2004 (2020). https://doi.org/10.1007/s00220-020-03701-8

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