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Effective counting on translation surfaces. (English) Zbl 1451.37049

A translation surface \(\mathbf{x}\) is a compact oriented surface equipped with an atlas of planar charts. The collection of all translation surfaces of a fixed genus, fixed number of singular points, and fixed cone angle at each singular point is called a stratum. Each connected component of the subset of area one surfaces in a stratum is the support of a natural smooth probability measure, which is called the flat measure. A saddle connection on a translation surface \(\mathbf{x}\) is a segment connecting two singular points which is linear in each planar chart and contains non singular points in its interior. The holonomy vector of a saddle connection is the vector in the plane obtained by integrating the pullback of the planar form \((dx,dy)\), along the saddle connection. Denote by \(V(\mathbf{x})\) the collection of all holonomy vectors of saddle connections for \(\mathbf{x}\).
Let \(\mathcal{H}\) be a stratum of translation surfaces, let \(G=\mathrm{SL}_2(\mathbb{R})\) and let \(\mathcal{L} \subset \mathcal{H}\) be the closure of a \(G\)-orbit in \(\mathcal{H}\). In [Ergodic Theory Dyn. Syst. 21, No. 2, 443–478 (2001; Zbl 1096.37501)], A. Eskin and H. Masur proved that there is \(c>0\) such that for almost every translation surface \(\mathbf{x}\) (with respect to the flat measure) in \(\mathcal{L}\), the number \(N(T,\mathbf{x})=|V(\mathbf{x}) \cap B(0, T)|\) of saddle connection with holonomy vector of length at most \(T\), which satisfies \[N(T, \mathbf{x})=cT^2+o(T^2).\]
The main goal of this paper is to prove an effective version of a celebrated result of Eskin and Masur. That is to establish that there exists \(\kappa>0\) such that for almost every translation surface \(\mathbf{x}\) in \(\mathcal{L}\), one has \[N(T, \mathbf{x})=cT^2+O(T^{2(1-\kappa)}).\]
Let \(\varphi_1< \varphi_2\) with \(\varphi_2-\varphi_1 \leq 2\pi\) and let \(N(T, \mathbf{x}, \varphi_1, \varphi_2)\) denote the cardinality of the intersection of \(V(\mathbf{x})\) with the sector \[S_{T, \varphi_1, \varphi_2}=\{r(\cos\varphi, \sin \varphi):0 \leq r \leq T, \varphi_1 \leq \varphi \leq \varphi_2\} \subset \mathbb{R}^2.\] Y. Vorobets [Contemp. Math. 385, 205–258 (2005; Zbl 1130.37015)] showed that there is \(c>0\) such that for almost every \(\mathbf{x} \in \mathcal{H}\) (with respect to the flat measure on \(\mathcal{H}\)), \[N(T, \mathbf{x}, \varphi_1, \varphi_2)=c(\varphi_2-\varphi_1)T^2+o(T^2).\] In this paper, the authors also provided an effective version of counting in sectors, that is there is a constant \(\kappa>0\) such that such that for almost every \(\mathbf{x} \in \mathcal{H}\), one has \[N(T, \mathbf{x}, \varphi_1, \varphi_2)=\frac{c}{2}(\varphi_2-\varphi_1)T^2+O_{\mathbf{x}, \varphi_2-\varphi_1}(T^{2(1-\kappa)}).\]

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37C35 Orbit growth in dynamical systems
37A30 Ergodic theorems, spectral theory, Markov operators
22D40 Ergodic theory on groups
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References:

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