Long hitting time for translation flows and L-shaped billiards

Abstract

We consider the flow in direction $ \theta $ on a translation surface and we study the asymptotic behavior for $ r\to 0 $ of the time needed by orbits to hit the $ r $-neighborhood of a prescribed point, or more precisely the exponent of the corresponding power law, which is known as hitting time. For flat tori the limsup of hitting time is equal to the Diophantine type of the direction $ \theta $. In higher genus, we consider a generalized geometric notion of Diophantine type of a direction $ \theta $ and we seek for relations with hitting time. For genus two surfaces with just one conical singularity we prove that the limsup of hitting time is always less or equal to the square of the Diophantine type. For any square-tiled surface with the same topology the Diophantine type itself is a lower bound, and any value between the two bounds can be realized, moreover this holds also for a larger class of origamis satisfying a specific topological assumption. Finally, for the so-called Eierlegende Wollmilchsau origami, the equality between limsup of hitting time and Diophantine type subsists. Our results apply to L-shaped billiards.