Published online by Cambridge University Press: 01 June 2009
A translation surface 𝒮 is said to have the finite blocking property if for every pair (O,A) of points in 𝒮 there exists a finite number of ‘blocking’ points B1,…,Bn such that every geodesic from O to A meets one of the Bis. 𝒮 is said to be purely periodic if the directional flow is periodic in each direction whose directional flow contains a periodic trajectory (this implies that 𝒮 admits a cylinder decomposition in such directions). We will prove that the finite blocking property implies pure periodicity. We will also classify the surfaces that have the finite blocking property in genus two: such surfaces are exactly the torus branched coverings. Moreover, we prove that in every stratum such surfaces form a set of null measure. In Appendix A, we prove that completely periodic translation surfaces form a set of null measure in every stratum.