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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. [AV07]

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

  2. [Bar96]

    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)

  3. [Bar03]

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

  4. [BBB18]

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

  5. [BCH10]

    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)

  6. [BG06]

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

  7. [BG14]

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

  8. [BHR19]

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

  9. [BM18]

    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)

  10. [Bow70]

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

  11. [Bow73]

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

  12. [Bow74]

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

  13. [Bow75]

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

  14. [Bow79]

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

  15. [BP02]

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

  16. [BPS97]

    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)

  17. [BPVL19]

    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. [BS18]

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

  19. [BV04]

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

  20. [CFH08]

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

  21. [Cli10]

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

  22. [Cli14]

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

  23. [CP10]

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

  24. [CPZ19]

    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)

  25. [DGR19]

    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)

  26. [Fal88a]

    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)

  27. [Fal88b]

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

  28. [Fal94]

    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)

  29. [Fen03]

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

  30. [Fen09]

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

  31. [Fen11]

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

  32. [FFW01]

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

  33. [FH10]

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

  34. [FK11]

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

  35. [FS14]

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

  36. [Fur63]

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

  37. [Hut81]

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

  38. [Kel98]

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

  39. [KS13]

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

  40. [Kä04]

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

  41. [LW77]

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

  42. [PW01]

    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. [Roh61]

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

  44. [Rue82]

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

  45. [Rue89]

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

  46. [Sin68]

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

  47. [Sol98]

    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)

  48. [Wal00]

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

  49. [Zha97]

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

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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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Correspondence to Kiho Park.

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

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