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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
##### Keywords:
Periodic Lorentz gas; average first exit time
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