The Teichmüller space of a surface is equipped with Thurston's asymmetric metric. Stretch lines are oriented geodesics for this metric on . We give the asymptotic behavior of the lengths of the measured geodesic laminations as one follows a stretch line in the positive direction.
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
168,657 papers · 148 categories
Trend · papers per month
48 results for “stretch lines”
A cylindrical stretch line is a stretch line, in the sense of Thurston, whose horocyclic lamination is a weighted multicurve. In this paper, we show that two correctly parameterized cylindrical lines are parallel if and only if these lines converge towards the same point in Thurston's boundary of Teichmüller space.
In 1986 William P. Thurston introduced the celebrated (asymmetric) Lipschitz distance on the Teichmueller space of a (closed or punctured) surface. In this paper we extend his work to the Teichmueller space of a surface with boundary endowed the arc distance. In this new setting we construct a large family of geodesics…
Describes envelopes of Thurston metric on Teichmüller space.
problem Characterizing the shape and properties of envelopes in Teichmüller space.
method Using harmonic stretch lines and topological invariants, the shape and properties of envelopes are described.
result Envelopes are contractible and vary continuously with endpoints.
In the Teichmüller space of a hyperbolic surface of finite type, we construct geodesic lines for Thurston's asymmetric metric having the property that when they are traversed in the reverse direction, they are also geodesic lines (up to reparametrization). The lines we construct are special stretch lines in the sense o…
Abstract notes on hyperbolic surfaces and Teichmüller spaces.
problem Understanding the geometry of surfaces and Teichmüller spaces.
method Survey of results on stretch lines and Thurston's metric.
result Analogies between Thurston's metric and Teichmüller's metric.
There are many industrial situations where rods are used to stir a fluid, or where rods repeatedly stretch a material such as bread dough or taffy. The goal in these applications is to stretch either material lines (in a fluid) or the material itself (for dough or taffy) as rapidly as possible. The growth rate of mater…
New stretch maps minimize distortion in geometric group theory.
problem Finding optimal maps in geometric group theory.
method Proving minimizers using modulus of curve families and MSP.
result Stretch maps are minimizers of mean quasiconformal distortion.
Finsler metrics with relatively non-negative (non-positive, respectively), constant and isotropic stretch curvatures are investigated in this paper. In particular, it is proved that every non-Riemannian -metric with a nonzero constant flag curvature and a non-zero relatively isotropic stretch curvature over a m…
Every weak Perron number is realized as a stretch factor of a homeomorphism on a surface.
problem Finding stretch factors for weak Perron numbers.
method Constructing an end-periodic homeomorphism on a surface.
result Every weak Perron number is an end-periodic stretch factor.
Lower bound on stretch factor for periodic maps.
problem Finding a lower bound on stretch factors for periodic maps.
method Using core characteristic of end-periodic homeomorphisms, we derive a lower bound on the Handel-Miller stretch factor.
result The derived bound is sharp and measures topological complexity.
In 1974, Thurston proved that, up to isotopy, every automorphism of closed orientable surface is either periodic, reducible, or pseudo-Anosov. The latter case has lead to a rich theory with applications ranging from dynamical systems to low dimensional topology. Associated with every pseudo-Anosov map is a real number …
Study stretch laminations in hyperbolic 3-manifolds via circle-valued maps.
problem Characterize stretch laminations in hyperbolic 3-manifolds.
method Use Thurston norm and Dehn filling slope length to determine stretch laminations as unions of core curves.
result Show existence of infinitely many examples with fibration and only closed leaves.
Study finds how periodic surfaces can bend without stretching.
problem Understanding isometric deformations of periodic surfaces.
method Characterization of isometric deformations using a constraint derived from Gauss theorem.
result Relates surface stretching to bending and twisting.
Minimal stretch factor for non-orientable surfaces is small.
problem Finding the minimal stretch factor for pseudo-Anosov homeomorphisms on non-orientable surfaces.
method Adapting Thurston's theory of fibered faces for non-orientable 3-manifolds.
result The minimal stretch factor is asymptotically on the order of 1/g.
The paper studies maximal stretch and Lipschitz maps on negatively curved manifolds.
problem Investigating maximal stretch and Lipschitz maps on negatively curved manifolds.
method Defined maximal stretch for negatively curved manifolds and connected it to best Lipschitz maps.
result The Mather set may not be lifts of geodesic laminations but shares similar features.
Quadratic-time algorithm computes stretch factors and foliations for pseudo-Anosov mapping classes.
problem Computing stretch factors and foliations for pseudo-Anosov mapping classes efficiently.
method Quadratic-time algorithm using input word and length as complexity measure.
result First algorithm to compute stretch factors and foliations in sub-exponential time.
We explicitly construct pseudo-Anosov maps on the closed surface of genus with orientable foliations whose stretch factor is a Salem number with algebraic degree . Using this result, we show that there is a pseudo-Anosov map whose stretch factor has algebraic degree , for each positive even integer s…
Minimal stretch factors for certain pseudo-Anosov maps are bounded.
problem Bounding the stretch factor of orientation-reversing pseudo-Anosov maps.
method Using the silver ratio and properties of puncture orbits to derive bounds.
result The bound is asymptotically sharp.
Study how large-scale flows align small-scale vortices in 3D Euler equations.
problem Understanding how large-scale flows align small-scale vortices in 3D Euler equations.
method Constructing a Lagrangian coordinate to identify when the Lie bracket is zero and investigating the locality of the pressure term.
result Clarified conditions under which small-scale vortices are aligned by large-scale flows.
Study on solvable Lie groups with specific Weyl connections.
problem Characterizing solvable Lie groups with invariant stretched non-positive Weyl connections.
method Analyzing structure and classification of solvable Lie groups.
result Classification of solvable Lie groups and compact solvmanifolds with invariant SNP connections.
In this paper, we study the Galois conjugates of stretch factors of pseudo-Anosov elements of the mapping class group of a surface. We show that - except in low-complexity cases - these conjugates are dense in the complex plane. For this, we use Penner's construction of pseudo-Anosov mapping classes. As a consequence, …
Study connects group invariants through outer automorphisms and polynomial relations.
problem Understanding polynomial invariants of free-by-cyclic groups.
method Introducing orientable fully irreducible outer automorphisms to relate McMullen polynomial and Alexander polynomial.
result Characterization of when homological stretch factor equals geometric stretch factor.
The paper examines properties of spherical Finsler metrics, proving their semi-C-reducibility and conditions for vanishing mean stretch curvature.
problem Analyzing curvature properties of spherical Finsler metrics.
method Proving semi-C-reducibility and finding conditions for vanishing mean stretch curvature.
result Conditions for a general spherically symmetric Finsler metric to have vanishing mean stretch curvature.
We determine the smallest stretch factor among pseudo-Anosov maps with an orientable invariant foliation on the closed nonorientable surfaces of genus 4, 5, 6, 7, 8, 10, 12, 14, 16, 18 and 20. We also determine the smallest stretch factor of an orientation-reversing pseudo-Anosov map with orientable invariant foliation…
Study reveals uniform difference in stretch factors between genus two handlebody group and outer automorphism group.
problem Analyzing the relationship between stretch factors in genus two handlebody group and outer automorphism group.
method Examined natural homomorphism from genus g handlebody group to outer automorphism group of free groups, focusing on pseudo-Anosov mapping classes and their stretch factors.
result Minimum stretch factor in genus two handlebody group is less than ten times the stretch factor of fully irreducible outer automorphism.
The motivation for this paper is to justify a remark of Thurston that the algebraic degree of stretch factors of pseudo-Anosov maps on a surface can be as high as the dimension of the Teichmüller space of . In addition to proving this, we completely determine the set of possible algebraic degrees of pseudo-Anoso…
New geometric invariant limits the number of semi-arithmetic groups.
problem Understanding the structure of semi-arithmetic Fuchsian groups.
method Introducing a new geometric invariant called stretch and using the arithmetic Margulis lemma.
result There exist only finitely many conjugacy classes of semi-arithmetic groups with bounded arithmetic dimension, stretch, and coarea.
We quantitatively relate the Patterson-Sullivant currents and generic stretching factors for free group automorphisms to the asymmetric Lipschitz metric on Outer space and to Guirardel's intersection number.
We consider the problem of distortion minimal morphing of -dimensional compact connected oriented smooth manifolds without boundary embedded in . Distortion involves bending and stretching. In this paper, minimal distortion (with respect to stretching) is defined as the infinitesimal relative change in vol…
Upper bound found for 2-systole in stretched S² x S² metrics.
problem Bounding the 2-systole in stretched S² x S² metrics.
method Using Gromov and Zhu's developments, derived an upper bound.
result Upper bound for 2-systole derived.
The paper compares inserting and stretching points for grid refinement near critical points.
problem Decreased accuracy of finite difference methods near discontinuities.
method Comparison of inserting and stretching points for grid refinement near critical points.
result Proposes a new fast and simple stretching function.
The study examines the stretch factors of outer automorphisms and their latent symmetry.
problem Understanding stretch factors of outer automorphisms in free groups.
method Analyzes the latent symmetry of graphs and uses it to bound stretch factors.
result A precise notion of latent symmetry provides a lower bound on the number of folds required.
New periodic solutions found in 2n-body problem, braids of pseudo-Anosov type with stretch factors as metallic ratios.
problem Periodic solutions of the 2n-body problem and their braid types.
method Analyzing braid types and stretch factors associated with pseudo-Anosov braids.
result Braids from new periodic solutions are of pseudo-Anosov type with stretch factors as metallic ratios.
We define metrics on Culler-Vogtmann space, which are an analogue of the Teichmuller metric and are constructed using stretching factors. In fact the metrics we study are related, one being a symmetrised version of the other. We investigate the basic properties of these metrics, showing the advantages and pathologies o…
Proves even degrees can be realized in Abelian differential strata.
problem Realizing even degrees as stretch factors in Abelian differentials.
method Analyzes Thurston-Veech stretch factors in moduli spaces.
result Every even degree is realized in Abelian differentials.
The period map for 4-manifolds is dense and surjective under certain conditions.
problem Understanding the space of metrics on 4-manifolds and their harmonic forms.
method Constructing families of metrics by stretching along hypersurfaces.
result The period map is dense and surjective for 4-manifolds with .
We use daily data on bilateral interbank exposures and monthly bank balance sheets to study network characteristics of the Russian interbank market over Aug 1998 - Oct 2004. Specifically, we examine the distributions of (un)directed (un)weighted degree, nodal attributes (bank assets, capital and capital-to-assets ratio…
Paper proposes a method to efficiently cluster stretched mixtures.
problem Clustering stretched elliptical mixtures using standard methods like PCA and k-means fails.
method Proposes a non-convex program to transform data into a one-dimensional point cloud.
result Efficient first-order algorithm achieves near-optimal statistical precision.
We consider the question of whether solutions of variants of Teichmüller harmonic map flow from surfaces to general targets can degenerate in finite time. For the original flow from closed surfaces of genus at least , as well as the flow from cylinders, we prove that such a finite-time degeneration must occur in…
How periodic surfaces bendmath.DG
Periodic surfaces have a limited number of bending modes, equal to their membrane modes.
problem Understanding the limitations of bending modes in periodic surfaces.
method Analyzing deformation modes of periodic, piecewise smooth, simply connected surfaces.
result Effective membrane modes and bending modes are orthogonal, limiting the total number of modes to 3.
We extend martingale transport results to weak martingale transport.
problem Applying martingale transport results to weak martingale transport.
method Change of numeraire for weak martingale transport.
result Established the correspondence between stretched Brownian motion and its geometric counterpart.
We introduce a construction of pseudo-Anosov homeomorphisms on n-times punctured spheres and surfaces with higher genus using only sufficiently many positive half-twists. These constructions can produce explicit examples of pseudo-Anosov maps with various number-theoretic properties associated to the stretch factors, i…
Study periodic solutions in N-body problem, revealing braids with complex dynamics.
problem Periodic solutions of the planar Newtonian N-body problem with equal masses.
method Analyze braid structures derived from periodic solutions, proving pseudo-Anosov types.
result Braids from Yu's periodic solutions are pseudo-Anosov, with stretch factors reflecting complexity.
Characterizes bi-Perron numbers with specific Galois conjugates.
problem Understanding bi-Perron numbers with real or unimodular conjugates.
method Characterization through power properties and spectral radii of transformations.
result Bi-Perron numbers with real or unimodular conjugates admit a power as stretch factors or spectral radii.
Estimation of Z-Thickness and XY-Anisotropy of Electron Microscopy Images using Gaussian Processeseess.IV
Serial section electron microscopy (ssEM) is a widely used technique for obtaining volumetric information of biological tissues at nanometer scale. However, accurate 3D reconstructions of identified cellular structures and volumetric quantifications require precise estimates of section thickness and anisotropy (or stre…
Real Lagrangian tori in are Hamiltonian isotopic to the Clifford torus.
problem Unknottedness of real Lagrangian tori in .
method Neck-stretching argument, Gromov's foliation theorem, Cieliebak-Schwingenheuer criterion.
result Real Lagrangian tori in are Hamiltonian isotopic to the Clifford torus.
We introduce a property of mutation loops, called the sign stability, with a focus on an asymptotic behavior of the iteration of the tropical -transformation. A sign-stable mutation loop has a numerical invariant which we call the cluster stretch factor, in analogy with that of a pseudo-Anosov mapping clas…