The beginning of the Lagrange spectrum of certain origamis of genus two

Carlos Matheus

doi:10.5802/crmath.65

Dynamical Systems

The beginning of the Lagrange spectrum of certain origamis of genus two

Carlos Matheus ¹

¹CMLS, CNRS, École polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France

Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 475-479

Abstract

The initial portion of the Lagrange spectrum $L_{B 7}$ of certain square-tiled surfaces of genus two was described in details in the work of Hubert–Lelièvre–Marchese–Ulcigrai. In particular, they proved that the smallest element of $L_{B 7}$ is an isolated point $ϕ_{1}$ , but the second smallest value $ϕ_{2}$ of $L_{B 7}$ is an accumulation point. Also, they conjectured that the portion $L_{B 7} \cap [ϕ_{2}, η_{1})$ is a Cantor set for a specific value $η_{1}$ and they asked about the continuity properties of the Hausdorff dimension of $L_{B 7} \cap (- \infty, t)$ as a function of $t < η_{1}$ . In this note, we solve affirmatively these problems.

Article information
Cite this article

Received: 2020-04-22
Revised: 2020-04-27
Accepted: 2020-04-27
Published online: 2020-07-28

DOI: 10.5802/crmath.65

Author's affiliations:

Carlos Matheus ¹

¹ CMLS, CNRS, École polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

Carlos Matheus. The beginning of the Lagrange spectrum of certain origamis of genus two. Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 475-479. doi: 10.5802/crmath.65

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