Dynamical Systems
The beginning of the Lagrange spectrum of certain origamis of genus two
Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 475-479.

The initial portion of the Lagrange spectrum ${L}_{B7}$ 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 ${L}_{B7}$ is an isolated point ${\varphi }_{1}$, but the second smallest value ${\varphi }_{2}$ of ${L}_{B7}$ is an accumulation point. Also, they conjectured that the portion ${L}_{B7}\cap \left[{\varphi }_{2},{\eta }_{1}\right)$ is a Cantor set for a specific value ${\eta }_{1}$ and they asked about the continuity properties of the Hausdorff dimension of ${L}_{B7}\cap \left(-\infty ,t\right)$ as a function of $t<{\eta }_{1}$. In this note, we solve affirmatively these problems.

Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.65
Carlos Matheus 1

1 CMLS, CNRS, École polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France
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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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.65/

[1] Pierre Arnoux Le codage du flot géodésique sur la surface modulaire, Enseign. Math., Volume 40 (1994) no. 1-2, pp. 29-48 | Zbl

[2] Aline Cerqueira; Carlos Matheus; Carlos G. Moreira Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra, J. Mod. Dyn., Volume 12 (2018), pp. 151-174 | DOI | MR | Zbl

[3] Pascal Hubert; Samuel Lelièvre Prime arithmetic Teichmüller discs in $ℋ\left(2\right)$, Isr. J. Math., Volume 151 (2006), pp. 281-321 | DOI | Zbl

[4] Pascal Hubert; Samuel Lelièvre; Luca Marchese; Corinna Ulcigrai The Lagrange spectrum of some square-tiled surfaces, Isr. J. Math., Volume 225 (2018) no. 2, pp. 553-607 | DOI | MR | Zbl

[5] Pascal Hubert; Luca Marchese; Corinna Ulcigrai Lagrange spectra in Teichmüller dynamics via renormalization, Geom. Funct. Anal., Volume 25 (2015) no. 1, pp. 180-255 | DOI | Zbl

[6] Carlos Matheus; Carlos G. Moreira $HD\left(M\setminus L\right)>0.353$, Acta Arith., Volume 188 (2019) no. 2, pp. 183-208 | Zbl

[7] Carlos G. Moreira Geometric properties of the Markov and Lagrange spectra, Ann. Math., Volume 188 (2018) no. 1, pp. 145-170 | DOI | MR | Zbl

[8] Anton Zorich Flat surfaces, Frontiers in number theory, physics, and geometry I. On random matrices, zeta functions, and dynamical systems, Springer, 2006, pp. 403-437 | Zbl

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