Quadratic Differentials Research Articles

Effective count of square-tiled surfaces with prescribed real and imaginary foliations in connected components of strata

Abstract We prove an effective estimate with a power saving error term for the number of square-tiled surfaces in a connected component of a stratum of quadratic differentials whose vertical and horizontal foliations belong to prescribed mapping class group orbits and which have at most L squares. This result strengthens asymptotic counting formulas in the work of Delecroix, Goujard, Zograf, Zorich, and the author.

Correction to: Limit Cycles in a Class of Planar Discontinuous Piecewise Quadratic Differential Systems with a Non-regular Line of Discontinuity (II)

Correction to: Limit Cycles in a Class of Planar Discontinuous Piecewise Quadratic Differential Systems with a Non-regular Line of Discontinuity (II)

Limit Cycles of the Three-dimensional Quadratic Differential System via Hopf Bifurcation

تم في هذه الدراسة النظر في النظام التفاضلي التربيعي ثلاثي الأبعاد، حيث يصبح نقطة الأصل الإحداثيات نقطة توازن هوبف. تم دراسة وجود واستقرار الدورات النهائية التي تنبثق من نقطة هوبف. يتم حساب معاملات ليبانوف المرتبطة بنقطة هوبف باستخدام طريقة الإسقاط. أولاً، تم تحديد أربع عائلات من شروط المعلمات التي من خلالها يمكن للنظام التفاضلي التربيعي ثلاثي الأبعاد أن يظهر البعد الثالث لتشعب هوبف. تم إعطاء الدليل التحليلي لكل مجموعة من الحالات المعلمية عن طريق حساب معاملات ليبانوف، تصفير لقيمة معاملات ليبانوف الأولى و الثاني و غيرالصفري لمعاملات ليبانوف الثالثة. تم تقديم الشروط الواضحة لوجودية واستقرارية ثلاث دورات النهائية ناشئة عن كل عائلة من تشعبات هوبف. يُظهر ناتج الوجود نقطة هوبف مستقرة (غير مستقرة)، مصحوبة بظهور دورتين حديتين مستقرتين (غير مستقرة) جنبًا إلى جنب مع دورة حدية واحدة غير مستقرة (مستقرة) في جوار النقطة الأصل غير المستقر (المستقر) لنظام المعادلة التربيعية ثلاثية الأبعاد. بالإضافة إلى ذلك، يتم استخدام النتيجة لاستكشاف الدورات النهائية لنظام الجذب الفوضوي n-scroll، والذي له العديد من الاستخدامات العملية، بما في ذلك الاتصال الآمن، والتشفير، وتوليد الأرقام العشوائية، والروبوتات المتنقلة المستقلة. تم اشتقاق الشروط التي بموجبها تصبح نقطة الأصل لهذا النظام هي نقطة هوبف، ويمكن أن توجد ثلاث دورات نهائية حول نقطة هوبف. وأخيرًا، توضح العروض العددية أن النظام يخضع لتشعب هوبف فوق الحرج، مما يؤدي إلى دورتين حديتين مستقرتين وواحدة غير مستقرة. وعلاوة على ذلك، تم التحقق من كافة النتائج.

Limit Cycles in a Class of Planar Discontinuous Piecewise Quadratic Differential Systems with a Non-regular Line of Discontinuity (II)

Limit Cycles in a Class of Planar Discontinuous Piecewise Quadratic Differential Systems with a Non-regular Line of Discontinuity (II)

Extensibility of Solutions of Nonautonomous Systems of Quadratic Differential Equations and Their Application in Optimal Control Problems

Extensibility of Solutions of Nonautonomous Systems of Quadratic Differential Equations and Their Application in Optimal Control Problems

Meanders, hyperelliptic pillowcase covers, and the Johnson filtration

We provide minimal constructions of meanders with particular combinatorics. Using these meanders, we give minimal constructions of hyperelliptic pillowcase covers with a single horizontal cylinder and simultaneously a single vertical cylinder so that one or both of the core curves are separating curves on the underlying surface. In the case where both of the core curves are separating, we use these surfaces in a construction of Aougab–Taylor in order to prove that for any hyperelliptic connected component of the moduli space of quadratic differentials with no poles there exist ratio-optimising pseudo-Anosovs lying arbitrarily deep in the Johnson filtration and stabilising the Teichmüller disk of a quadratic differential lying in this connected component.

Quadratic differentials and foliations on infinite Riemann surfaces

Quadratic differentials and foliations on infinite Riemann surfaces

Bounded differentials on the unit disk and the associated geometry

For a harmonic diffeomorphism between the Poincaré disks, Wan [J. Differential Geom. 35 (1992), pp. 643–657] showed the equivalence between the boundedness of the Hopf differential and the quasi-conformality. In this paper, we will generalize this result from quadratic differentials to r r -differentials. We study the relationship between bounded holomorphic r r -differentials/ ( r − 1 ) (r-1) -differential and the induced curvature of the associated harmonic maps from the unit disk to the symmetric space S L ( r , R ) / S O ( r ) SL(r,\mathbb R)/SO(r) arising from cyclic/subcyclic Higgs bundles. Also, we show the equivalence between the boundedness of holomorphic differentials and having a negative upper bound of the induced curvature on hyperbolic affine spheres in R 3 \mathbb {R}^3 , maximal surfaces in H 2 , n \mathbb {H}^{2,n} and J J -holomorphic curves in H 4 , 2 \mathbb {H}^{4,2} . Benoist-Hulin and Labourie-Toulisse have previously obtained some of these equivalences using different methods.

Reconstructing orbit closures from their boundaries

We introduce and study diamonds of G L + ( 2 , R ) \mathrm {GL}^+(2, \mathbb {R}) -invariant subvarieties of Abelian and quadratic differentials, which allow us to recover information on an invariant subvariety by simultaneously considering two degenerations, and which provide a new tool for the classification of invariant subvarieties. We classify a surprisingly rich collection of diamonds where the two degenerations are contained in “trivial” invariant subvarieties. Our main results have been applied to classify large collections of invariant subvarieties; the statement of those results do not involve diamonds, but their proofs rely on them.

Delay Painlevé-I equation, associated polynomials and Masur-Veech volumes

We study a delay-differential analogue of the first Painlevé equation obtained as a delay periodic reduction of Shabat's dressing chain. We construct formal entire solutions to this equation and introduce a new family of polynomials (called Bernoulli-Catalan polynomials), which are defined by a nonlinear recurrence of Catalan type, and which share properties with Bernoulli and Euler polynomials. We also discuss meromorphic solutions and describe the singularity structure of this delay Painlevé-I equation in terms of an affine Weyl group of type A1(1). As an application we demonstrate the link with the problem of calculation of the Masur-Veech volumes of the moduli spaces of meromorphic quadratic differentials by re-deriving some of the known formulas.

The Liouville current of holomorphic quadratic differential metrics

In this paper we study the Liouville current of flat cone metrics coming from a holomorphic quadratic differential. Anja Bankovic and Christopher J. Leininger proved in Bankovic and Leininger (Trans Am Math Soc 370:1867–1884, 2018) that for a fixed closed surface, there is an injection map from the space of flat cone metrics to the space of geodesic currents. We manage to show that metrics coming from holomorphic quadratic differentials can be distinguished from other flat metrics by just looking at the geodesic currents. The key idea is to analyze the support of Liouville current, which is a topological invariant independent of the metric, and get information about cone angles and holonomy. The holonomy part involves some subtlety of relationship between singular foliation and geodesic lamination. We also obtain a new proof of a classical result that almost all simple geodesics of a quadratic differential metric will be dense in the surface. Furthermore, for other flat cone metrics, there is no simple dense geodesic.

Characterizing 4-string contact interaction using machine learning

The geometry of 4-string contact interaction of closed string field theory is characterized using machine learning. We obtain Strebel quadratic differentials on 4-punctured spheres as a neural network by performing unsupervised learning with a custom-built loss function. This allows us to solve for local coordinates and compute their associated mapping radii numerically. We also train a neural network distinguishing vertex from Feynman region. As a check, 4-tachyon contact term in the tachyon potential is computed and a good agreement with the results in the literature is observed. We argue that our algorithm is manifestly independent of number of punctures and scaling it to characterize the geometry of n-string contact interaction is feasible.

Global Phase Portraits of Piecewise Quadratic Differential Systems with a Pseudo-Center

This paper deals with the global dynamics of planar piecewise smooth differential systems constituted by two different vector fields separated by one straight line that passes through the origin. From a quasi-canonical family of piecewise quadratic differential systems with a pseudo-focus point at the origin, which has six parameters, we investigate the subfamilies where the origin is indeed a pseudo-center. For such subfamilies, we classify its global phase portraits in the Poincaré disk and the associated bifurcation sets.

Quadratic differentials as stability conditions: Collapsing subsurfaces

Abstract We introduce a new class of triangulated categories, which are Verdier quotients of three-Calabi–Yau categories from (decorated) marked surfaces, and show that its spaces of stability conditions can be identified with moduli spaces of framed quadratic differentials on Riemann surfaces with arbitrary order zeros and arbitrary higher order poles. A main tool in our proof is a comparison of two exchange graphs, obtained by tilting hearts in the quotient categories and by flipping mixed angulations associated with the quadratic differentials.

New Insights on Non-integrability and Dynamics in a Simple Quadratic Differential System

This study focuses on the integrability and qualitative behaviors of a quadratic differential system x˙=a+yz,y˙=-y+x2,z˙=b-4x.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dot{x}=a+yz,\\quad\\dot{y}=-y + x^{2},\\quad\\dot{z}=b-4x.$$\\end{document}We provide some new perspectives on the system and reveal its diverse properties, including non-integrability in the sense of absence of first integrals, bifurcations of co-dimension one or two, Jacobi instability and dynamics at infinity.

Periodic orbits in the thin part of strata

Abstract Let S be a closed oriented surface of genus g ≥ 0 {g\geq 0} with n ≥ 0 {n\geq 0} punctures and 3 ⁢ g - 3 + n ≥ 5 {3g-3+n\geq 5} . Let 𝒬 {{\mathcal{Q}}} be a connected component of a stratum in the moduli space 𝒬 ⁢ ( S ) {{\mathcal{Q}}(S)} of area one meromorphic quadratic differentials on S with n simple poles at the punctures or in the moduli space ℋ ⁢ ( S ) {{\mathcal{H}}(S)} of abelian differentials on S if n = 0 {n=0} . For a compact subset K of 𝒬 ⁢ ( S ) {{\mathcal{Q}}(S)} or of ℋ ⁢ ( S ) {{\mathcal{H}}(S)} , we show that the asymptotic growth rate of the number of periodic orbits for the Teichmüller flow Φ t {\Phi^{t}} on 𝒬 {{\mathcal{Q}}} which are entirely contained in 𝒬 - K {{\mathcal{Q}}-K} is at least h ⁢ ( 𝒬 ) - 1 {h({\mathcal{Q}})-1} , where h ⁢ ( 𝒬 ) > 0 {h({\mathcal{Q}})>0} is the complex dimension of ℝ + ⁢ 𝒬 {\mathbb{R}^{+}{\mathcal{Q}}} .

An Error Quasi Quadratic Differential Based Event-triggered MPC for Continuous Perturbed Nonlinear Systems

An Error Quasi Quadratic Differential Based Event-triggered MPC for Continuous Perturbed Nonlinear Systems

The Harmonic Map Compactification of Teichmüller Spaces for Punctured Riemann Surfaces

In the paper [The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989), no. 2, 449–479], Wolf provided a global coordinate system of the Teichmüller space of a closed oriented surface S S with the vector space of holomorphic quadratic differentials on a Riemann surface X X homeomorphic to S S . This coordinate system is via harmonic maps from the Riemann surface X X to hyperbolic surfaces. Moreover, he gave a compactification of the Teichmüller space by adding a point at infinity to each endpoint of harmonic map rays starting from X X in the space. Wolf also showed this compactification coincides with the Thurston compactification. In this paper, we extend the harmonic map ray compactification to the case of punctured Riemann surfaces and show that it still coincides with the Thurston compactification.

Nonvarying, affine and extremal geometry of strata of differentials

AbstractWe prove that the nonvarying strata of abelian and quadratic differentials in low genus have trivial tautological rings and are affine varieties. We also prove that strata of k-differentials of infinite area are affine varieties for all k. Vanishing of homology in degree higher than the complex dimension follows as a consequence for these affine strata. Moreover we prove that the stratification of the Hodge bundle for abelian and quadratic differentials of finite area is extremal in the sense that merging two zeros in each stratum leads to an extremal effective divisor in the boundary. A common feature throughout these results is a relation of divisor classes in strata of differentials as well as its incarnation in Teichmüller dynamics.

A rigidity result for holomorphic quadratic differentials of finite norm in the unit disk

AbstractLet be a holomorphic quadratic differential of finite norm in the unit disk . Let be equipped with the metric . In this paper, we prove that a geodesic‐preserving map of the unit disk into itself is affine. A map is called geodesic‐preserving if the image of each geodesic is a geodesic. We make no assumptions on the continuity of such a map.