Diophantine approximation on Veech surfaces

Hubert, Pascal; Schmidt, Thomas A.

Bulletin de la Société Mathématique de France, Tome 140 (2012) no. 4, pp. 551-568.

Nous montrons que les fractions continues generalisées $Z$ de Y. Cheung s’adaptent pour exprimer l’approximation par vecteurs de connexion de selles sur n’importe quelle surface de translation compacte. C’est-à-dire, nous démontrons la finitude de la constant de Minkowski pour chaque surface de translation compacte. De plus, pour une surface de Veech en forme standard, nous montrons que chaque composant de n’importe quel vecteur de connexion de selle domine, dans un sens approprié, ses conjugués. Les fractions continues de connexions de selle permettent de reconnaître certaines directions transcendantales par leur développement.

We show that Y. Cheung’s general $Z$-continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates in an appropriate sense. The saddle connection continued fractions then allow one to recognize certain transcendental directions by their developments.

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