Algebraically periodic translation surfaces

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dc.creatorCalta, Kariane
dc.creatorSmillie, John
dc.date2007-03-20
dc.date2007-10-24
dc.date.accessioned2026-07-07T08:38:13Z
dc.date.available2026-07-07T08:38:13Z
dc.descriptionAlgebraically periodic directions on translation surfaces were introduced by Calta in her study of genus two translation surfaces. We say that a translation surface with three or more algebraically periodic directions is an algebraically periodic surface. We show that for an algebraically periodic surface the slopes of the algebraically periodic directions are given by a number field which we call the periodic direction field. We show that translation surfaces with pseudo-Anosov automorphisms provide examples. In this case the periodic direction field is the holonomy field. We show that every algebraic field arises as the periodic direction field of a translation surface arising from a right-angled billiard table. The J-invariant of a translation surface was introduced by Kenyon and Smillie. We analyze the $J$ invariants of algebraically periodic surfaces and show that in some cases they are determined by the periodic direction field. We give explicit formulas for $J$ invariants in these cases. The Homological Affine Group was introduced by McMullen in his study of translation surfaces in genus two. We calculate this group for many algebraically periodic surfaces and relate it to the automorphism group of the J-invariant. We show that surfaces which admit certain decompositions into squares have totally real periodic direction field. This is related to a result of Hubert and Lanneau.
dc.descriptiontypos and errors corrected
dc.identifierhttps://arxiv.org/abs/math/0703567
dc.identifierhttp://arxiv.org/abs/math/0703567
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/140651
dc.subjectDynamical Systems
dc.subjectGeneral Topology
dc.subject37D50; 57M50
dc.titleAlgebraically periodic translation surfaces
dc.typetext

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