Some real Rel trajectories in $\mathcal{H}(1,1)$ that are not recurrent

TL;DR

This paper investigates the behavior of real Rel orbits in the stratum (1,1) of translation surfaces, identifying conditions for recurrence and providing explicit examples of non-recurrent trajectories.

Contribution

It establishes necessary and sufficient conditions for recurrence of real Rel orbits in (1,1) and constructs explicit non-recurrent examples.

Findings

Identifies conditions for recurrence of real Rel orbits.

Provides explicit examples of non-recurrent real Rel trajectories.

Analyzes tremors of branched double covers of tori.

Abstract

We study the real Rel orbits of some translation surfaces in the stratum $\mathcal{H}(1,1)$. Specifically, surfaces that are tremors of the locus of branched double covers of tori. We give necessary and sufficient conditions on tremors of a surface so that the real Rel orbit is recurrent. As a consequence, we are able to provide explicit examples of trajectories of real Rel that are not recurrent.