Shrinking targets on square-tiled surfaces

Abstract

We prove a shrinking target property for the action of subgroups of Veech groups on square-tiled surfaces. Our main tool, of independent interest, is the construction of a Fourier-like transform that we use to show that the L 2 -norm of the Koopman operator for the action is equivalent to the L 2 -norm of a Markov operator related to a random walk on a subgroup. This generalizes work of V. Finkelshtein on the flat torus.