Phase transitions on the Markov and Lagrange dynamical spectra

Lima, Davi; Moreira, Carlos Gustavo

doi:10.1016/j.anihpc.2020.11.007

Lima, Davi ¹ ; Moreira, Carlos Gustavo ^2,³

¹ a Instituto de Matemática, Universidade Federal de Alagoas, Av. Lourival Melo Mota s/n, 57072-900, Maceió, Brazil
² b School of Mathematical Sciences, Nankai University, Tianjin 300071, PR China
³ c IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil

Annales de l'I.H.P. Analyse non linéaire, septembre – octobre 2021, Tome 38 (2021) no. 5, pp. 1429-1459.

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Résumé

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 a and $\tilde{a}$ in the Markov and Lagrange spectra respectively, such that for any $δ > 0$ , the intersection of the Markov spectrum with $(- \infty, a - δ)$ has Hausdorff dimension smaller than one, while the intersection of the Markov spectrum with $(a, a + δ)$ has non-empty interior. Similarly, the intersection of the Lagrange spectrum with $(- \infty, \tilde{a} - δ)$ has Hausdorff dimension smaller than one, while the intersection of the Lagrange spectrum with $(\tilde{a}, \tilde{a} + δ)$ has non-empty interior. We give an open set of examples where $a \neq \tilde{a}$ and we prove that, in the conservative case, generically, $a = \tilde{a}$ and, for any $δ > 0$ , the intersection of the Lagrange spectrum with $(a - δ, a)$ has Hausdorff dimension one.

Reçu le : 2018-01-11
Révisé le : 2020-02-14
Accepté le : 2020-11-25

MR Zbl

DOI : 10.1016/j.anihpc.2020.11.007

Mots-clés : Fractal geometry, Markov and Lagrange dynamical spectra, Lagrange dynamical spectrum, Horseshoes and Regular Cantor sets, Hyperbolic dynamics, Diophantine approximations

Affiliations des auteurs :

Lima, Davi ¹ ; Moreira, Carlos Gustavo ^{2, 3}

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     title = {Phase transitions on the {Markov} and {Lagrange} dynamical spectra},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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Lima, Davi; Moreira, Carlos Gustavo. Phase transitions on the Markov and Lagrange dynamical spectra. Annales de l'I.H.P. Analyse non linéaire, septembre – octobre 2021, Tome 38 (2021) no. 5, pp. 1429-1459. doi : 10.1016/j.anihpc.2020.11.007. http://www.numdam.org/articles/10.1016/j.anihpc.2020.11.007/

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