Rel leaves of the Arnoux-Yoccoz surfaces

TL;DR

This paper studies the dense and divergent properties of rel leaves in Arnoux-Yoccoz translation surfaces, revealing their ergodic and periodic foliations, and includes a field theory result crucial for understanding leaf denseness.

Contribution

It demonstrates the density of rel leaves in certain strata for all genus g ≥ 3 and establishes divergence of the imaginary-rel trajectory, with a new field theoretic proof for leaf denseness.

Findings

Rel leaves are dense in the connected component of the stratum for all g ≥ 3.

The imaginary-rel trajectory of the surface diverges.

Horizontal foliation is uniquely ergodic for the Arnoux-Yoccoz surface, but periodic for others.

Abstract

We analyze the rel leaves of the Arnoux-Yoccoz translation surfaces. We show that for any genus $g \geq 3$, the leaf is dense in the connected component of the stratum $H(g -1 , g -1)$ to which it belongs, and the one-sided imaginary-rel trajectory of the surface is divergent. For one surface on this trajectory, namely the Arnoux-Yoccoz surface itself, the horizontal foliation is invariant under a pseudo-Anosov map (and in particular is uniquely ergodic), but for all other surfaces, the horizontal foliation is completely periodic. The appendix proves a field theoretic result needed for denseness of the leaf: for any $n \geq 3$, the field extension of the rationals obtained by adjoining a root of $X^n-X^{n-1}-\ldots-X-1$ has no totally real subfields other than the rationals.