poster: Counting poblem and Siegel-Veech constants on wind-tree models
speaker: Angel Pardo (Institut Fourier, Université Grenoble Alpes)abstract: We study periodic wind-tree models, billiards in the plane endowed with \(\mathbb{Z}^2\)-periodic identical symmetric right-angled obstacles, that is, the variant introduced by Delecroix-Zorich (2015), using the approach introduced by Delacroix-Hubert-Lelièvre (2011-13). Following ideas developed by Eskin-Masur (2001), Eskin-Masur-Zorich (2003) and Athreya-Eskin-Zorich (2012) we show that for almost all wind-tree model, with respect to a natural Lebesgue-type measure, the number of (homotopy classes of) periodic billiard trajectories (up to \(\mathbb{Z}^2\)-translations) of bounded length has quadratic asymptotic growth rate and we compute explicitly, for generic wind-tree billiards, the associated coefficient. More precisely, we prove the following.
\(Theorem\) For almost every (Delacroix-Zorich) wind-tree billiard \(\Pi\) with \(4m\) right-angled corners, the number \(N(L,\Pi)\) of (homotopy classes of) closed billiard trajectories of length at most \(L\) in \(\Pi\) has quadratic asymptotic growth rate \[ N(L,\Pi) \sim c(m) \cdot \frac{\pi L^2}{\mathrm{Area}\left(\Pi/\mathbb{Z}^2\right)},\] where \[c(m) = \left(20m^2 - 95m - 78 + 78\cdot 4^m\frac{(m!)^2}{(2m)!}\right)\frac{1}{6\pi^2}.\]
The constant \(c(m)\) corresponds to Siegel-Veech constants (the constant in a Siegel-type formula introduced by Veech (1998), the Siegel-Veech formula) of some particular configurations of cylinders of closed geodesics on compact flat surfaces associated to (generic) wind-tree billiards.
The proof relies heavily on the \(\mathrm{SL}(2,\mathbb{R})\)-action and the Teichmüller flow on moduli space of translation surfaces, which provide an effective renormalization scheme for translation surfaces and billiards in rational polygons.
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