Infinitely many lattice surfaces with special pseudo-Anosov maps

TL;DR

This paper constructs explicit pseudo-Anosov homeomorphisms with zero Sah-Arnoux-Fathi invariant on a broad class of translation surfaces, revealing new examples with special dynamical properties.

Contribution

It introduces explicit pseudo-Anosov maps with vanishing invariant for surfaces related to triangle groups, expanding known examples in the field.

Findings

Existence of infinitely many such surfaces with special pseudo-Anosov maps.

Identification of non-parabolic elements in the periodic field of certain translation surfaces.

Connection between Veech groups and the construction of these pseudo-Anosov maps.

Abstract

We give explicit pseudo-Anosov homeomorphisms with vanishing Sah-Arnoux-Fathi invariant. Any translation surface whose Veech group is commensurable to any of a large class of triangle groups is shown to have an affine pseudo-Anosov homeomorphism of this type. We also apply a reduction to finite triangle groups and thereby show the existence of non-parabolic elements in the periodic field of certain translation surfaces.