Algebraically periodic translation surfaces
Kariane Calta John Smillie
Journal of Modern Dynamics 2008, 2(2): 209-248 doi: 10.3934/jmd.2008.2.209
We develop an algebraic framework for studying translation surfaces. We study the Sah--Arnoux--Fathi-invariant and the collection of directions in which it vanishes. We show that these directions are described by a number field which we call the periodic direction field. We study the $J$-invariant of a translation surface, introduced by Kenyon and Smillie and used by Calta. We relate the $J$-invariant to the periodic direction field. For every number field $K\subset\ \mathbb R$ we show that there is a translation surface for which the periodic direction field is $K$. We study automorphism groups associated to a translation surface and relate them to the $J$-invariant. We relate the existence of decompositions of translation surfaces into squares with the total reality of the periodic direction field.
keywords: SAF-invariant. translation surfaces J-invariant algebraic periodicity
Infinitely many lattice surfaces with special pseudo-Anosov maps
Kariane Calta Thomas A. Schmidt
Journal of Modern Dynamics 2013, 7(2): 239-254 doi: 10.3934/jmd.2013.7.239
We give explicit pseudo-Anosov homeomorphisms with vanishing Sah-Arnoux-Fathi invariant. Any translation surface whose Veech group is commensurable to any of a large class of triangle groups is shown to have an affine pseudo-Anosov homeomorphism of this type. We also apply a reduction to finite triangle groups and thereby show the existence of nonparabolic elements in the periodic field of certain translation surfaces.
keywords: translation surface flux Veech group. Pseudo-Anosov SAF invariant

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