TY - JOUR
T1 - Rel leaves of the Arnoux–Yoccoz surfaces
AU - Hooper, W. Patrick
AU - Weiss, Barak
N1 - Publisher Copyright:
© 2017, Springer International Publishing AG, part of Springer Nature.
PY - 2018/4/1
Y1 - 2018/4/1
N2 - 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 Xn- Xn-1- ⋯ - X- 1 has no totally real subfields other than Q.
AB - 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 Xn- Xn-1- ⋯ - X- 1 has no totally real subfields other than Q.
KW - 37Exx
UR - http://www.scopus.com/inward/record.url?scp=85037622401&partnerID=8YFLogxK
U2 - 10.1007/s00029-017-0367-x
DO - 10.1007/s00029-017-0367-x
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AN - SCOPUS:85037622401
VL - 24
SP - 875
EP - 934
JO - Selecta Mathematica, New Series
JF - Selecta Mathematica, New Series
SN - 1022-1824
IS - 2
ER -