The rel leaf and real-rel ray of the Arnoux-Yoccoz surface in genus 3

TL;DR

This paper studies the behavior of the rel leaf and real-rel trajectories of the Arnoux-Yoccoz surface in genus 3, revealing density in a specific stratum and divergence of trajectories, with implications for foliation invariance and ergodicity.

Contribution

It demonstrates the density of the rel leaf in a particular stratum and shows the divergence of the real-rel trajectory, highlighting unique foliation properties.

Findings

Rel leaf is dense in alH(2,2)^{ ext{odd}}

Real-rel trajectory diverges

Vertical foliation invariance under pseudo-Anosov map

Abstract

We analyze the rel leaf of the Arnoux-Yoccoz translation surface in genus 3. We show that the leaf is dense in the stratum $\mathcal{H}(2,2)^{\mathrm{odd}}$ but that the real-rel trajectory of the surface is divergent. On this real-rel trajectory, the vertical foliation of one surface is invariant under a pseudo-Anosov map (and in particular is uniquely ergodic), but the vertical foliations on all other surfaces are completely periodic.