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 become \(r\)-dense, or more precisely the exponent of the corresponding power law, which is known as \emph{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 an extended 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, moreover, on any square-tiled surface, the bound is sharp for a big set of directions. Our results apply to L-shaped billiards.