Abstract
We remove a small disc of radius ε >0 from the flat
torus T^{2} and consider a pointlike particle that
starts moving from the center of the disk with linear trajectory
under angle ω. Let \tildeτ_{ε} (ω)
denote the first exit time of the particle. For any interval
I⊂ [0,2\pi), any r>0, and any δ >0, we estimate
the moments of \tildeτ_{ε} on I and prove the
asymptotic formula
∫_{I} \tildeτ^{r}_{ε} (ω) dω = c_{r} I ε^{r} +O_{δ}
(ε^{r+(1/8)δ}) as
ε → 0^{+},
where c_{r} is the constant
(12/\pi^{2}) ∫_{0}^{1/2}(x(x^{r1}+(1x)^{r1})
+(1(1x)^{r})/(rx(1x))  (1(1x)^{r+1})/((r+1)x(1x)))dx.
A similar estimate is obtained for the moments of the number of
reflections in the side cushions when T^{2} is
identified with [0,1)^{2}.

Author information
F. P. Boca:
Institute of Mathematics of the Romanian Academy, P.O.Box 1764, RO014700, Bucharest, Romania
fboca@math.uiuc.edu
R. N. Gologan:
Institute of Mathematics of the Romanian Academy, P.O.Box 1764, RO014700 Bucharest, Romania
Radu.Gologan@imar.ro
A. Zaharescu:
Institute of Mathematics of the Romanian Academy, P.O.Box 1764, RO014700, Bucharest, Romania
zaharesc@math.uiuc.edu
