Algebraically periodic translation surfaces

Calta, Kariane; Smillie, John

doi:10.3934/jmd.2008.2.209

Cited by 23 publications

(25 citation statements)

References 19 publications

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“…The surfaces obtained from Thurston-Veech construction are completely algebraically periodic in the sense of Calta and Smillie [2]. In particular, if such a surface admits a special splitting .T 1 ; T 2 ; T 3 ; v 1 ; v 2 /, then v 2 must be parallel to a vector in the lattice associated to T 3 .…”

Section: Thurston-veech Surface With Cubic Trace Fieldmentioning

confidence: 99%

Parallelogram decompositions and generic surfaces in H^hyp(4)

Nguyen

2011

Geom. Topol.

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The space H hyp .4/ is the moduli space of pairs .M; !/, where M is a hyperelliptic Riemann surface of genus 3 and ! is a holomorphic 1-form having only one zero. In this paper, we first show that every surface in H hyp .4/ admits a decomposition into parallelograms and simple cylinders following a unique model. We then show that if this decomposition satisfies some irrational condition, then the GL C .2; R/-orbit of the surface is dense in H hyp .4/; such surfaces are called generic. Using this criterion, we prove that there are generic surfaces in H hyp .4/ with coordinates in any quadratic field, and there are Thurston-Veech surfaces with trace field of degree three over Q which are generic. 51H25; 37B05

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Section: Thurston-veech Surface With Cubic Trace Fieldmentioning

confidence: 99%

Parallelogram decompositions and generic surfaces in H^hyp(4)

Nguyen

2011

Geom. Topol.

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“…The applicability of this theorem is strongly related to the results of [5], but we give self contained simple proofs. Theorem 19 applies to regular polygons since we can write the vertices of any regular polygon as (cos(2πk/d), sin(2πk/d)), with k = 0, .…”

Section: Applicationsmentioning

confidence: 95%

λ-stability of periodic billiard orbits

Gaivão

Troubetzkoy

2018

Nonlinearity

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Abstract. We introduce a new notion of stability for periodic orbits in polygonal billiards. We say that a periodic orbit of a polygonal billiard is λ-stable if there is a periodic orbit for the corresponding pinball billiard which converges to it as λ → 1. This notion of stability is unrelated to the notion introduced by Galperin, Stepin and Vorobets. We give sufficient and necessary conditions for a periodic orbit to be λ-stable and prove that the set of d-gons having at most finite number of λ-stable periodic orbits is dense is the space of d-gons. Moreover, we also determine completely the λ-stable periodic orbits in integrable polygons.

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“…Note that all surfaces in the eigenform loci that are given by the Thurston-Veech construction (see [McM06a,Section 4] for a concise survey) are completely algebraically periodic in the sense of [CS08]. Hence all these surfaces have zero flux and we cannot use the flux to rule out the existence of Teichmüller curves.…”

Section: Prym Eigenform Locimentioning

confidence: 99%

Non-Existence and Finiteness Results for Teichmüller Curves in Prym Loci

Lanneau

Möller

2019

Experimental Mathematics

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The minimal stratum in Prym loci have been the first source of infinitely many primitive, but not algebraically primitive Teichmüller curves. We show that the stratum Prym(2,1,1) contains no such Teichmüller curve and the stratum Prym(2,2) at most 92 such Teichmüller curves.This complements the recent progress establishing general -but non-effective -methods to prove finiteness results for Teichmüller curves and serves as proof of concept how to use the torsion condition in the non-algebraically primitive case.

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Algebraically periodic translation surfaces

Cited by 23 publications

References 19 publications

Parallelogram decompositions and generic surfaces in H^hyp(4)

Parallelogram decompositions and generic surfaces in H^hyp(4)

λ-stability of periodic billiard orbits

Non-Existence and Finiteness Results for Teichmüller Curves in Prym Loci

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Algebraically periodic translation surfaces

Cited by 23 publications

References 19 publications

Parallelogram decompositions and generic surfaces in Hhyp(4)

Parallelogram decompositions and generic surfaces in Hhyp(4)

λ-stability of periodic billiard orbits

Non-Existence and Finiteness Results for Teichmüller Curves in Prym Loci

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Product

Resources

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Parallelogram decompositions and generic surfaces in H^hyp(4)

Parallelogram decompositions and generic surfaces in H^hyp(4)