Veech surfaces with non-periodic directions in the trace field

TL;DR

This paper demonstrates that certain Veech surfaces with trace fields of degree greater than two have non-periodic directions with zero SAF-invariant, and provides explicit examples of pseudo-Anosov diffeomorphisms with this property.

Contribution

It establishes the existence of non-periodic directions with zero SAF-invariant in Veech surfaces with higher degree trace fields and constructs explicit pseudo-Anosov examples.

Findings

Veech surfaces with trace field degree > 2 have non-periodic directions with zero SAF-invariant

Explicit pseudo-Anosov diffeomorphisms with zero SAF-invariant in their contracting directions

Extension of known examples to higher degree trace fields

Abstract

We show that each of Veech's original examples of translation surfaces with ``optimal dynamics'' whose trace field is of degree greater than two has non-periodic directions of vanishing SAF-invariant. Furthermore, we give explicit examples of pseudo-Anosov diffeomorphisms whose contracting direction has zero SAF-invariant.