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

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.

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Correspondence to L. J. Díaz.

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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.

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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

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