On the finite blocking property

Monteil, Thierry

Annales de l'Institut Fourier, Tome 55 (2005) no. 4, pp. 1195-1217.

On dit qu’un billard polygonal $𝒫$ a la propriété de blocage fini si pour tout couple de points $(O,A)$ de $𝒫$ il existe un nombre fini de points “bloquants” $B_1,\cdots ,B_n$ tels que toute trajectoire de billard de $O$ à $A$ rencontre l’un des $B_i$. En généralisant notre construction d’un contre exemple à un théorème de Hiemer et Snurnikov, nous montrons que les seuls polygones réguliers qui ont la propriété de blocage fini sont les carrés, le triangle équilateral et l’hexagone. Puis nous étendons ce résultat aux surfaces de translation. Nous prouvons que les seules surfaces de Veech jouissant de la propritété de blocage fini sont les revêtements ramifiés du tore. Nous donnons aussi une condition suffisante locale pour qu’une surface de translation ne jouisse pas de la propriété de blocage fini. Cela nous permet de donner une classification complète pour les surfaces en forme de L ainsi que d’obtenir un résultat de densité dans l’espace des surfaces de translation en tout genre $g\ge 2$.

A planar polygonal billiard $𝒫$ is said to have the finite blocking property if for every pair $(O,A)$ of points in $𝒫$ there exists a finite number of “blocking” points $B_1,\cdots ,B_n$ such that every billiard trajectory from $O$ to $A$ meets one of the $B_i$’s. Generalizing our construction of a counter-example to a theorem of Hiemer and Snurnikov, we show that the only regular polygons that have the finite blocking property are the square, the equilateral triangle and the hexagon. Then we extend this result to translation surfaces. We prove that the only Veech surfaces with the finite blocking property are the torus branched coverings. We also provide a local sufficient condition for a translation surface to fail the finite blocking property. This enables us to give a complete classification for the L-shaped surfaces as well as to obtain a density result in the space of translation surfaces in every genus $g\ge 2$.

Classification : 37E35, 37D50, 37D40, 37A10, 5199, 30F30
Keywords: Blocking property, polygonal billiards, regular polygons, translation surfaces, Veech surfaces, torus branched covering, illumination, quadratic differentials
Mot clés : propriété de blocage, billards polygonaux, polygones réguliers, surfaces de translation, surfaces de Veech, revêtement ramifié du tore, illumination, différentielles quadratiques

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