Growth of Reciprocal pseudo-Anosovs on Lattice Surfaces

Abstract

Motivated by number theory, Reciprocal geodesics were first introduced by Sarnak [23], who studied theirasymptotic growth on the modular curve. Erlandsson-Souto [7] gave a geometric interpretation and gener- alization of reciprocal geodesics and a dynamical proof of asymptotic counting results in the more general setting of hyperbolic orbifolds H2/Γ where Γ is a lattice. We introduce the notion of reciprocal pseudo-Anosov maps of translation surfaces and establish a correspondence between such maps and reciprocal geodesics. We then show how to apply the Erlandsson-Souto results to compute the asymptotic growth for particular families of highly symmetric surfaces known as lattice surfaces or Veech surfaces [26], and to in fact compute the constants for the asymptotic growth of pseudo-Anosov maps on certain families of lattices surfaces, called Bouw-M¨oller [5] and primitive square-tiled surfaces [24]