Equidistribution of saddle connections on translation surfaces

Dozier, Benjamin

doi:10.3934/jmd.2019004

Cited by 7 publications

(12 citation statements)

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“…It is a recurrence-type result which controls the length of the shortest saddle connection, on average, over translation surfaces on a large "circle" centered at any X. Here we deduce the proposition directly from a more general result proved in [Doz17]; in that paper the more general result is used to study the distribution of angles of saddle connections.…”

Section: Recurrence Results For the Proofsmentioning

confidence: 69%

“…Proof of Proposition 1.1. This is a special case of Proposition 2.1 in [Doz17]. That Proposition involves integrating over any subinterval I ⊂ [0, 2π]; the above is simply the case when I equals [0, 2π].…”

Section: Recurrence Results For the Proofsmentioning

confidence: 98%

“…If we keep track of the c max coming from the proof of Theorem 1.2, we find it grows at most exponentially in the genus, but it seems hard to do better than exponential using our method. The exponential nature of the method arises from an induction on the complexity of certain "complexes" of saddle connections in the proof of the generalization of Proposition 1.1 given in [Doz17].…”

Section: Open Questions On Extremal Siegel-veech Constantsmentioning

confidence: 99%

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Convergence of Siegel–Veech constants

Dozier

2018

Geom Dedicata

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We show that for any weakly convergent sequence of ergodic SL2(R)-invariant probability measures on a stratum of unit-area translation surfaces, the corresponding Siegel-Veech constants converge to the Siegel-Veech constant of the limit measure. Together with a measure equidistribution result due to Eskin-Mirzakhani-Mohammadi, this yields the (previously conjectured) convergence of sequences of Siegel-Veech constants associated to Teichmüller curves in genus two.The proof uses a recurrence result closely related to techniques developed by Eskin-Masur. We also use this recurrence result to get an asymptotic quadratic upper bound, with a uniform constant depending only on the stratum, for the number of saddle connections of length at most R on a unit-area translation surface.

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Section: Recurrence Results For the Proofsmentioning

confidence: 69%

Section: Recurrence Results For the Proofsmentioning

confidence: 98%

Section: Open Questions On Extremal Siegel-veech Constantsmentioning

confidence: 99%

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Convergence of Siegel–Veech constants

Dozier

2018

Geom Dedicata

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“…for m almost every ω. Dozier [5] proved analogues of these results for saddle connections with holonomies in certain sectors of directions. Forni [9] has proved for every flat surface ω that the convergence (2) holds in density: there is a set Z ⊂ R of zero upper-density such that…”

Section: Introductionmentioning

confidence: 84%

Uniform distribution of saddle connection lengths (with an appendix by Daniel El-Baz and Bingrong Huang)

Chaika¹,

Robertson²

2019

Journal of Modern Dynamics

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“…Construction and circle averages of logsmooth functions. The main goal of this section is to prove Corollary 4.6, which extends the statements of [Doz19] (giving averages over intervals) to include so-called logsmooth functions from [Ath06] (which gives averages over the full circle) Definition 5. A complex K in ω is a closed subset of X whose boundary ∂K consists of a union of disjoint (in the interior) saddle connections such that if ∂K contains three saddle connections bounding a triangle, then the interior of that triangle is in K. Given a complex K the complexity of K is the number of saddle connections needed to triangulate K. For any δ > 0 and k ∈ N, if M is the complexity of ω,…”

Section: And and The Trivial Boundsmentioning

confidence: 99%

Shrinking rates of horizontal gaps for generic translation surfaces

Chaika¹,

Fairchild²

2022

Preprint

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A translation surface is given by polygons in the plane, with sides identified by translations to create a closed Riemann surface with a flat structure away from finitely many singular points. Understanding geodesic flow on a surface involves understanding saddle connections. Saddle connections are the geodesics starting and ending at these singular points and are associated to a discrete subset of the plane. To measure the behavior of saddle connections of length at most R, we obtain precise decay rates as R → ∞ for the difference in angle between two almost horizontal saddle connections.Remark 1.2. Note that the choice of a horizontal gap is a convenience. Apply a rotation to the full measure set in Theorem 1.1, and we obtain the same result in a different direction. Consider a countable subset D n = {θ n } n∈N ⊆ [0, 2π). Since a countable union of measure 0 subsets is still measure zero, we obtain a natural corollary that the smallest gap of any of the directions in D n has the same decay rate as given in Theorem 1.1.We first provide an explanation for the scaling factor of R 2 . Masur ([Mas88, Mas90]) showed |Λ ω (R)| has quadratic growth in the sense that for each ω there exist constants c 1 , c 2 so thatfor all large enough R. There are at most 4g − 4 saddle connections in the same direction, so |Θ ω (R)| also has quadratic growth. This result explains the scaling factor of R 2 in Theorem 1.1. The quadratic growth of saddle connections was subsequently built on in [Vee98, EM01, Vor05, EMM15, NRW20].For almost every translation surface, Theorem 1.1 gives partial information on how Θ ω (R) is distributed in [−π, π). For every translation surface, [Mas86] ), then by density the adjacent differences θ j+1 − θ j → 0 as R → ∞ for every ω. Thus ζ ω (R) → 0 as R → ∞ for every ω. However, we cannot expect Theorem 1.1 to hold for every translation surface. Indeed when ω is a lattice surface (see [Mas22, Sections 5 and 7] for a definition), [AC12] showed for every unbounded ψ we have lim infThe results of Theorem 1.1 considers the behavior of a single gap. There is also substantial work done on studying the behavior of the family of gaps. Namely one can study the entire set Θ ω (R). The distribution of normalized gaps exists for almost every ω by [AC12]. In many cases, the distribution has also been computed [ACL15, BMMM + 21, KSW21, UW16, San21]. Considering the behavior of a single gap, which is the focus of the current paper, is orthogonal because the behavior of a single gap does not affect the distribution of gaps.1.1. Outline of proof. The proof follows the now standard strategy of relating a problem about the geometry of a translation surface ω to the orbit of ω under Teichmüller geodesic flow, g t = e t 0 0 e −t . In Section 2 we reduce our problem about gaps to a shrinking target problem for g t . The shrinking targets are sets A t obtained by relating Theorem 1.1 to whether or not g t ω ∈ A t for arbitrarily large t. To prove Theorem 1.1 (2) we use independence results for the sets g −t A t . A key tool to ...

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Equidistribution of saddle connections on translation surfaces

Cited by 7 publications

References 13 publications

Convergence of Siegel–Veech constants

Convergence of Siegel–Veech constants

Uniform distribution of saddle connection lengths (with an appendix by Daniel El-Baz and Bingrong Huang)

Shrinking rates of horizontal gaps for generic translation surfaces

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