Rel leaves of the Arnoux–Yoccoz surfaces

We analyze the rel leaves of the Arnoux–Yoccoz translation surfaces. We show that for any genus g ⩾ 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...

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Bibliographic Details
Published in:Selecta mathematica (Basel, Switzerland) Switzerland), 2018-04, Vol.24 (2), p.875-934
Main Authors: Hooper, W. Patrick, Weiss, Barak
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
Trajectories
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Summary:We analyze the rel leaves of the Arnoux–Yoccoz translation surfaces. We show that for any genus g ⩾ 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 ⩾ 3 , the field extension of Q obtained by adjoining a root of X n - X n - 1 - ⋯ - X - 1 has no totally real subfields other than Q .
ISSN:1022-1824
1420-9020
DOI:10.1007/s00029-017-0367-x