Algebraic intersection for a family of Veech surfaces

Boulanger, Julien; Lanneau, Erwan; Massart, Daniel

Annales Henri Lebesgue, Tome 7 (2024), pp. 787-821

We study some properties of the function $\mathrm{KVol}$ defined by

$$\text{KVol}(X,\omega ):=\mathrm{Vol}(X,\omega )\underset{\alpha ,\beta }{sup}\frac{\mathrm{Int}(\alpha ,\beta )}{l_g\left(\alpha \right)l_g\left(\beta \right)}$$

on the moduli space of translation surfaces. For the Teichmüller discs ${𝒯}_n$ of the original Veech surfaces arising from the right-angled triangles $(\pi /2,\pi /n,(n-2)\pi /2n)$ for odd $n\ge 5$, we establish the first known explicit formula for $\mathrm{KVol}$ (beyond the case of the moduli space of flat tori).

On étudie la fonction $\mathrm{KVol}$ sur les espaces de modules de surfaces de translation définie par

$$\text{KVol}(X,\omega ):=\mathrm{Vol}(X,\omega )\underset{\alpha ,\beta }{sup}\frac{\mathrm{Int}(\alpha ,\beta )}{l_g\left(\alpha \right)l_g\left(\beta \right)}.$$

En particulier, sur les disques de Teichmüller ${𝒯}_n$ des surfaces de Veech associées au billard dans les triangles $(\pi /2,\pi /n,(n-2)\pi /2n)$ pour $n$ impair $\ge 5$, nous donnons la première formule explicite connue pour $\mathrm{KVol}$ (en dehors du cas des espaces de modules de tores plats).

Classification : 37E05, 37D40
Keywords: Lyapunov exponents, translation surface, Teichmüller curve, algebraic intersection

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