Ergodic Theory of the Space of Measured Laminations

Lindenstrauss, Elon; Mirzakhani, Maryam

doi:10.1093/imrn/rnm126

Cited by 24 publications

(49 citation statements)

References 22 publications

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“…We prove next that any such limit ν is supported by the space ML of measured laminations (Lemma 4.2), and that it gives measure 0 to the set of non-filling laminations (Lemma 4.4). It then follows from Lindenstrauss-Mirzakhani's classification of locally finite mapping class group invariant measures on ML [12,11] that the limit ν is a multiple c · µ Thu of the Thurston measure (Lemma 4.5). All that remains is to show that c = m(α) m Σ .…”

Section: Introductionmentioning

confidence: 96%

Geodesic Currents and Counting Problems

Rafi

Souto

2019

Geom. Funct. Anal.

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Section: Introductionmentioning

confidence: 96%

Geodesic Currents and Counting Problems

Rafi

Souto

2019

Geom. Funct. Anal.

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“…h : QT g,n → ML g,n × ML g,n − ∆ which assigns to every marked non-zero, integrable, holomorphic quadratic differential on S g,n its real and imaginary foliations interpreted as measured geodesic laminations on S g,n . Following the conventions of Lindenstrauss and Mirzakhani in [LM08], we refer to this map as the Hubbard-Masur map.…”

Section: Introductionmentioning

confidence: 99%

Counting square-tiled surfaces with prescribed real and imaginary foliations and connections to Mirzakhani's asymptotics for simple closed hyperbolic geodesics

Arana-Herrera¹

2020

Journal of Modern Dynamics

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We show that the number of square-tiled surfaces of genus g, with n marked points, with one or both of its horizontal and vertical foliations belonging to fixed mapping class group orbits, and having at most L squares, is asymptotic to L 6g−6+2n times a product of constants appearing in Mirzakhani's count of simple closed hyperbolic geodesics. Many of the results in this paper reflect recent discoveries of Delecroix, Goujard, Zograf, and Zorich, but the approach considered here is very different from theirs. We follow conceptual and geometric methods inspired by Mirzakhani's work.

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“…After this paper was posted on the arXiv I received the preprint [LM07] of Lindenstrauss and Mirkzakhani which contains another proof of the above theorem.…”

Section: Introductionmentioning

confidence: 99%