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Entropy Spectrum

In: Lyapunov Exponents

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  • Luís Barreira

    (Universidade de Lisboa, Instituto Superior Técnico)

Abstract

This chapter is an introduction to the multifractal analysis of Lyapunov exponents. We concentrate on repellers, for which the exposition can be simplified to some extent, avoiding the additional technicalities when there is both contraction and expansion. Nevertheless, the difficulties are analogous. We start with a pragmatic review of some basic notions and results from ergodic theory and the thermodynamic formalism as well as from the coding of a repeller in terms of a topological Markov chain. We then describe the entropy spectrum for the Lyapunov exponent on a conformal repeller. In particular, we give an optimal cohomological assumption under which the entropy spectrum is analytic and strictly concave. A nontrivial consequence is that the Lyapunov exponent takes uncountably many values, with each of them attained in a dense set of positive topological entropy. We also briefly describe a corresponding entropy spectrum for the Lyapunov exponent on a conformal hyperbolic set.

Suggested Citation

  • Luís Barreira, 2017. "Entropy Spectrum," Springer Books, in: Lyapunov Exponents, chapter 0, pages 239-252, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-71261-1_12
    DOI: 10.1007/978-3-319-71261-1_12
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