Length spectra and degeneration of flat metrics

Duchin, Moon; Leininger, Christopher J.; Rafi, Kasra

doi:10.1007/s00222-010-0262-y

Cited by 75 publications

(120 citation statements)

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“…More precisely, fixing for the sake of concreteness a distance d : C K (Σ) × C K (Σ) → R + inducing the topology of C K (Σ) (see [6] for a concrete choice), we prove:…”

Section: Measured Laminationsmentioning

confidence: 98%

“…In view of Corollary 4.4, it follows from Mirzakhani's result [14] on the existence of limit (1.1) that the limit (1.3) also exists for any possible filling current for any arbitrary hyperbolic surface of finite type. For instance, it follows that the analogue of the limit (1.1) also exists if we measure lengths with respect to an arbitrary metric of negative curvature [15], or with respect to a singular flat structure [6]. All this might be worth noting because Mirzakhani's arguments, using trace relations, may be hard to apply directly in these situations.…”

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confidence: 99%

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Counting curves in hyperbolic surfaces

Erlandsson

Souto

2016

Geom. Funct. Anal.

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Abstract. Let Σ be a hyperbolic surface. We study the set of curves on Σ of a given type, i.e. in the mapping class group orbit of some fixed but otherwise arbitrary γ0. For example, in the particular case that Σ is a once-punctured torus, we prove that the cardinality of the set of curves of type γ0 and of at most length L is asymptotic to L 2 times a constant. 1.Throughout this paper we let Σ be a complete hyperbolic surface of finite area, with genus g and r punctures, and distinct from a thrice punctured sphere. By an immersed multicurve, or simply multicurve, in Σ we will mean an immersed compact 1-dimensional submanifold of Σ each of whose components represents (the conjugacy class of) a primitive non-peripheral element in π 1 (Σ). Two multicurves γ, γ are of the same type if they belong to the same mapping class orbit, meaning that there is a diffeomorphism φ of Σ such that γ and φ(γ ) are isotopic as immersed submanifolds. In general, isotopic geodesics are considered to be equivalent. For instance, every multicurve γ is isotopic to a geodesic multicurve and the length Σ (γ) is the length of the latter.In this paper we study the set S γ 0 = Map(Σ) · γ 0 of (isotopy classes of) multicurves of some given type γ 0 . More precisely, we are interested in the behavior, when L tends to infinity, of the number |{γ ∈ S γ 0 | Σ (γ) ≤ L}| of multicurves in Σ of type γ 0 and of at most length L. Since this number grows coarsely like a polynomial of degree 6g − 6 + 2r (see [18] for the case that γ 0 is simple and [20,21] or Corollary 3.6 below for the general case), the perhaps most grappling question is whether the limitexists. Our main result is that it does if Σ is a once-punctured torus: Theorem 1.1. Let Σ be a complete hyperbolic surface of finite volume homeomorphic to a once punctured torus and let γ 0 ⊂ Σ be a multicurve. The limit (1.1) exists and moreover we have where µ Thu is the Thurston measure on the space of measured laminations ML(Σ) and C γ 0 > 0 depends only on γ 0 .In the case of simple multicurves Theorem 1.1 is due to . Also, for simple multicurves, Mirzakhani [13] proved that the limit (1.1) exists for all g and r. Building on the work of Mirzakhani, Rivin [19] established the existence of the limit (1.1) for multicurves with a single self-intersection.Remark. Recently, and independently of our work, Mirzakhani [14] has established the existence of (1.1) in complete generality. Her argument and ours are different in nature and in some sense complementary. See the remarks following the statement of Corollary 4.4 in this introduction for more on the relation between Mirzakhani's result and ours.Still in the setting of simple multicurves, the case of the torus is much more treatable than the general one because any two simple multicurves in the torus are of the same type as long as they have the same number of components. This means that, in the case of the torus, counting simple multicurves of some fixed type basically reduces to counting all simple multicurves, a much simpler problem. In fact,...

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“…More precisely, fixing for the sake of concreteness a distance d : C K (Σ) × C K (Σ) → R + inducing the topology of C K (Σ) (see [6] for a concrete choice), we prove:…”

Section: Measured Laminationsmentioning

confidence: 98%

mentioning

confidence: 99%

Counting curves in hyperbolic surfaces

Erlandsson

Souto

2016

Geom. Funct. Anal.

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“…Here, a mapping ω : (X, d X ) → (Y, d Y ) between metric spaces is said to be a (K, D)-rough homothety if (6) |d Y (ω(x 1 ), ω(x 2 )) − Kd X (x 1 , x 2 )| ≤ D for x 1 , x 2 ∈ X (cf. Chapter 7 of [5]).…”

Section: Corollary 12 (Criterion For Parallelism)mentioning

confidence: 99%

“…Let y n = R Gn,x (t n ). Let L F,yn be the geodesic current associated to the singular flat structure defined as Q n := J F,yn / J F,yn given by Duchin, Leininger and Rafi in [6]. Suppose on the contrary that i(a, F ) = 0.…”

Section: Theorem 72 (Uniqueness Of the Underlying Foliations) For Anymentioning

confidence: 99%

Geometry of the Gromov product: Geometry at infinity of Teichmüller space

Miyachi¹

2017

J. Math. Soc. Japan

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Abstract. This paper is devoted to study of transformations on metric spaces. It is done in an effort to produce qualitative version of quasi-isometries which takes into account the asymptotic behavior of the Gromov product in hyperbolic spaces. We characterize a quotient semigroup of such transformations on Teichmüller space by use of simplicial automorphisms of the complex of curves, and we will see that such transformation is recognized as a "coarsification" of isometries on Teichmüller space which is rigid at infinity. We also show a hyperbolic characteristic that any finite dimensional Teichmüller space does not admit (quasi)-invertible rough-homothety.

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“…The geometry of such flat metrics and their relationship to the hyperbolic metric in the same conformal class have been investigated in [DLR10] and [Raf07]. We are interested in the asymptotic behavior of geodesics on S.…”

Section: Introductionmentioning

confidence: 99%

Typical geodesics on flat surfaces

Dankwart¹

2011

Conform. Geom. Dyn.

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Abstract. We investigate typical behavior of geodesics on a closed flat surface S of genus g ≥ 2. We compare the length quotient of long arcs in the same homotopy class with fixed endpoints for the flat and the hyperbolic metric in the same conformal class. This quotient is asymptotically constant F a.e. We show that F is bounded from below by the inverse of the volume entropy e(S). Moreover, we construct a geodesic flow together with a measure on S which is induced by the Hausdorff measure of the Gromov boundary of the universal cover. Denote by e(S) the volume entropy of S and let c be a compact geodesic arc which connects singularities. We show that a typical geodesic passes through c with frequency that is comparable to exp(−e(S)l(c)). Thus a typical bi-infinite geodesic contains infinitely many singularities, and each geodesic between singularities c appears infinitely often with a frequency proportional to exp(−e(S)l(c)).

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Length spectra and degeneration of flat metrics

Cited by 75 publications

References 25 publications

Counting curves in hyperbolic surfaces

Counting curves in hyperbolic surfaces

Geometry of the Gromov product: Geometry at infinity of Teichmüller space

Typical geodesics on flat surfaces

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