Phase transitions on the Markov and Lagrange dynamical spectra

  • Davi Lima

    Instituto de Matemática, Universidade Federal de Alagoas, Av. Lourival Melo Mota s/n, 57072-900, Maceió, Brazil
  • Carlos Gustavo Moreira

    School of Mathematical Sciences, Nankai University, Tianjin 300071, PR China; IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil
Phase transitions on the Markov and Lagrange dynamical spectra cover

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Abstract

The Markov and Lagrange dynamical spectra were introduced by Moreira and share several geometric and topological aspects with the classical ones. However, some features of generic dynamical spectra associated to hyperbolic sets can be proved in the dynamical case and we do not know if they are true in classical case.

They can be a good source of natural conjectures about the classical spectra: it is natural to conjecture that some properties which hold for generic dynamical spectra associated to hyperbolic maps also hold for the classical Markov and Lagrange spectra.

In this paper, we show that, for generic dynamical spectra associated to horseshoes, there are transition points and in the Markov and Lagrange spectra respectively, such that for any , the intersection of the Markov spectrum with has Hausdorff dimension smaller than one, while the intersection of the Markov spectrum with has non-empty interior. Similarly, the intersection of the Lagrange spectrum with has Hausdorff dimension smaller than one, while the intersection of the Lagrange spectrum with has non-empty interior. We give an open set of examples where and we prove that, in the conservative case, generically, and, for any , the intersection of the Lagrange spectrum with has Hausdorff dimension one.

A correction to this paper is available.

Cite this article

Davi Lima, Carlos Gustavo Moreira, Phase transitions on the Markov and Lagrange dynamical spectra. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 5, pp. 1429–1459

DOI 10.1016/J.ANIHPC.2020.11.007