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The average length of a trajectory in a certain billiard in a flat two-torus. (English) Zbl 1066.37021
Summary: We remove a small disc of radius \(\varepsilon > 0\) from the flat torus \({\mathbb{T}}^2\) and consider a point-like particle that starts moving from the center of the disk with linear trajectory under angle \(\omega\). Let \(\tilde{\tau}_\varepsilon (\omega)\) denote the first exit time of the particle. For any interval \(I\subseteq [0,2\pi)\), any \(r > 0\), and any \(\delta > 0\), we estimate the moments of \(\tilde{\tau}_\varepsilon\) on \(I\) and prove the asymptotic formula \[ \int_I \tilde{\tau}^r_\varepsilon (\omega)\, d\omega = c_r | I| \varepsilon^{-r} +O_\delta (\varepsilon^{-r+\frac{1}{8}-\delta}) \qquad \text{as} \;\varepsilon \rightarrow 0^+, \] where \(c_r\) is the constant \[ \frac{12}{\pi2} \int\limits_0^{1/2} \left( x(x^{r-1}+(1-x)^{r-1}) +\frac{1-(1-x)^r}{rx(1-x)} - \frac{1-(1-x)^{r+1}}{(r+1)x(1-x)} \right) dx. \] A similar estimate is obtained for the moments of the number of reflections in the side cushions when \({\mathbb{T}}^2\) is identified with \([0,1)^2\).

MSC:
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
11B57 Farey sequences; the sequences \(1^k, 2^k, \dots\)
11P21 Lattice points in specified regions
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