Counting problem on wind-tree models

Abstract

We study periodic wind-tree models, billiards in the plane endowed with Z 2 -periodically located identical connected symmetric right-angled obstacles. We show asymptotic formulas for the number of (isotopy classes of) closed billiard trajectories (up to Z 2 -translations) on the wind-tree billiard. We also compute explicitly the associated Siegel-Veech constant for generic wind-tree billiards depending on the number of corners on the obstacle.

Quantitative error term in the counting problem on Veech wind-tree models

“…A cylinder C is (F, f )-bad if and only if pr F X γ C = ±f . In fact, since F is symplectic and two dimensional, C is an (F, f )-bad cylinder if and only if pr F X γ C = 0 is colinear to f (see [Pa,Remark 3.1]). Moreover, the action of SL(2, R) on homology (that is, the Kontsevich-Zorich cocycle) is by integer matrices, then, this is equivalent to say that pr F X γ C = ±f .…”

“…Then, thanks to the symmetries of X, there are two equivariant subbundles F (h) and F (v) of H 1 defined over M, such that h ∈ F (h) and v ∈ F (v) (see [Pa] for more details). Furthermore, we have the following (see [Pa,Corollary 5]).…”

A remark on $\mathbb{Z}^d$-covers of Veech surfaces

“…Periodic wind-tree models -both the classical model and the Delecroix-Zorich variant-yields Z 2 -periodic translation surfaces defined by cocycles lying in 2dimensional subspaces (see, e.g., [DHL14,DZ15,Par18]). It follows, by Corollary 3, that when the underlying compact surface is a Veech surface, the Veech group of the infinite surface is of the first kind.…”

A Non-varying Phenomenon with an Application to the Wind-Tree Model

“…In fact, by topological considerations, one can easily show that C has trivial monodromy if and only if r ij (C) ∈ {0, 2} (see e.g. [21,Lemma 6.2]), as in Figure 3.…”

“…The author [21] studied the counting problem on wind-tree models proving that the number of periodic trajectories has quadratic asymptotic growth rate and computed, in the generic case, the Siegel-Veech constants for the classical wind-tree model as well as for the Delecroix-Zorich variant. In this work we prove that, for the classical wind-tree model, this constant does not depend on the dimensions of the obstacles, exhibiting a non-varying phenomenon analogous to the one described above.…”