Rel leaves of the Arnoux-Yoccoz surfaces

Abstract

We analyze the rel leaves of the Arnoux-Yoccoz translation surfaces. We show that for any genus $g\ge 3$, 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 $n\ge 3$, the field extension of the rationals obtained by adjoining a root of $X^n-X^{n-1}-\dots -X-1$ has no totally real subfields other than the rationals.

Appendix by Lior Bary-Soroker, Mark Shusterman and Umberto Zannier. The prior version was published, but had errors in §6. Erroneous statements have been indicated and an erratum was included as Appendix B which corrects the errors. Main results are still correct. 70 pages, 9 figures. arXiv admin note: text overlap with arXiv:1506.06773

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