Zygmund vector fields, Hilbert transform and Fourier coefficients in shear coordinates

Šarić, Dragomir

doi:10.1353/ajm.2013.0056

Cited by 4 publications

(4 citation statements)

References 27 publications

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“…The forward implication was proved in [Ro09] and we provide a proof of the converse in Appendix B. We note that this description of the topology of the augmented Teichmüller space was used recently by D. Šarić [Sa11] to obtain a hyperbolic geometric proof of H. Masur's result [Ma76] about the extensions of the Weil-Petersson metric to this augmentation.…”

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confidence: 70%

Ptolemy groupoids, shear coordinates and the augmented Teichmuller space

Roger

2012

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We start by describing how ideal triangulations on a surface degenerate under pinching of a multicurve. We use this process to construct a homomorphism from the Ptolemy groupoid of a surface to that of a pinched surface which is natural with respect to the action of the mapping class group. We then apply this construction to the study of shear coordinates and their extension to the augmented Teichmüller space. In particular, we give an explicit description of the action of the mapping class group on the augmented Teichmüller space in terms of shear coordinates.

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confidence: 70%

Ptolemy groupoids, shear coordinates and the augmented Teichmuller space

Roger

2012

Preprint

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“…Finally, we prove that Γ v (l a ,b , l c ,d ) can be replaced by Γ v ([a, b], [c, d]) in (21). Note that (21)…”

Section: Proofs Of Lemmas 31 and 32mentioning

confidence: 84%

“…Adopting an approach of Bonahon [2] for closed surfaces to arbitrary hyperbolic surfaces, the universal Teichmüller space T (D) embeds into the space of geodesic currents of D when geodesic currents are equipped with the uniform weak * topology (cf. [3,15,[18][19][20][21] and Section 2) and this embedding is real analytic (cf. Otal [16]).…”

Section: Limit Points Of Teichmüller Geodesic Raysmentioning

confidence: 98%

Limits of Teichmüller geodesics in the Universal Teichmüller space

Hakobyan

Šarić

2018

Proc. London Math. Soc.

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A Thurston boundary of the universal Teichmüller space T(D) is the set of projective bounded measured laminations PMLbddfalse(double-struckDfalse) of double-struckD. We prove that each Teichmüller geodesic ray in T(D) converges to a unique limit point in the Thurston boundary of T(D) in the weak∗ topology. In particular, there is an open and dense set of geodesic rays, which have unique (weak*‐)limits in the Thurston boundary. We also show that the main result is sharp by providing an example of a Teichmüller geodesic, which does not converge in the uniform weak∗ topology.

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“…In the following, we use the upper half plane H as a model for the hyperbolic plane and recall the Farey tesselation and the shear map, which were introduced by Penner [19] and furthered studied by Saric [21] [22].…”

Section: The Farey Tesselation Shear Maps and Proof Of Theoremmentioning

confidence: 99%

Characterization of the asymptotic Teichmüller space of the open unit disk through shears

Fan

2014

Pure and Applied Mathematics Quarterly

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We give a parametrization to the asymptotic Teichmüller space AT (D) of the open unit disk D through equivalent classes of shear functions induced by quasisymmetric homeomorphisms on the Farey tesselation of D. Then using the parametrization, we define a new metric on AT (D). Two other related metrics are also introduced on AT (D) by using cross-ratio distortions or quadrilateral dilatations under the boundary maps on degenerating sequences of quadruples or quadrilaterals. We show that the topologies induced by the three metrics are equivalent to the one induced by the Teichmüller metric on AT (D). Before proving our main results, we revisit and rectify a mistake in the proof in [21] on the characterization of quasisymmetric homeomorphisms in terms of shear functions.2000 Mathematics Subject Classification. 30C75, 30F60.

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Zygmund vector fields, Hilbert transform and Fourier coefficients in shear coordinates

Cited by 4 publications

References 27 publications

Ptolemy groupoids, shear coordinates and the augmented Teichmuller space

Ptolemy groupoids, shear coordinates and the augmented Teichmuller space

Limits of Teichmüller geodesics in the Universal Teichmüller space

Characterization of the asymptotic Teichmüller space of the open unit disk through shears

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