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arXiv · 1508.05363

Rel leaves of the Arnoux-Yoccoz surfaces

W. Patrick Hooper·Barak Weiss
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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.

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W. Patrick Hooper, Barak Weiss. 2015-08-21. Rel leaves of the Arnoux-Yoccoz surfaces. https://doi.org/10.1007/s00029-017-0367-x

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