Mapping Class Research Articles

Approximating fixed points of weak enriched contractions using Kirk’s iteration scheme of higher order

In this paper, we introduce two types of weak enriched contractions, namely weak enriched F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{F}$\\end{document}-contraction, weak enriched F′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{F^{\\prime}}$\\end{document}-contraction, and k-fold averaged mapping based on Kirk’s iterative algorithm of order k. The types of contractions introduced herein unify, extend, and generalize several existing classes of enriched and weak enriched contraction mappings. Moreover, K-fold averaged mappings can be viewed as a generalization of the averaged mappings and double averaged mappings. We then prove the existence of a unique fixed point of the k-fold averaged mapping associated with weak enriched contractions introduced herein. We study necessary conditions that guarantee the equality of the sets of fixed points of the k-fold averaged mapping and weak enriched contractions. We show that an appropriate Kirk’s iterative algorithm can be used to approximate a fixed point of a k-fold averaged mapping and of the two weak enriched contractions. We also study the well-posedness, limit shadowing property, and Ulam–Hyers stability of the k-fold averaged mapping. We provide necessary conditions that ensure the periodic point property of each illustrated weak enriched contraction. Some examples are presented to show that our results are a potential generalization of the comparable results in the existing literature.

Big mapping class groups with uncountable integral homology

Big mapping class groups with uncountable integral homology

The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover

The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover

A few remarks on the Poincaré metric on a singular holomorphic foliation

Let F be a Riemann surface foliation on M∖E, where M is a complex manifold and E⊂M is a closed set. Assume that F is hyperbolic, i.e. all the leaves of the foliation F are hyperbolic Riemann surfaces. Fix a Hermitian metric g on M. We consider the Verjovsky modulus of uniformization map η, which measures the largest possible derivative in the class of holomorphic maps from the unit disc into the leaves of F. Various results are known to ensure the continuity of the map η along the transverse directions, with suitable conditions on M, F and E. For a domain U⊂M, let FU be the holomorphic foliation given by the restriction of F to the domain U, i.e. F|U. We consider the modulus of uniformization map ηU corresponding to the foliation FU and study its variation when the corresponding domain U varies in the Caratheodory kernel sense, motivated by the work of Lins Neto and Canille Martins.

Automatic Continuity for Homeomorphism Groups and Big Mapping Class Groups

We show that for any manifold M, compact or homeomorphic to the interior of a compact manifold, the homeomorphism group of M has automatic continuity. The same is true for the relative homeomorphism group Homeo(M,X) where X is homeomorphic to the union of a Cantor set and a (possibly, empty) finite set, and the big mapping class group Homeo(M,X)/Homeo0(M,X). In other words, the algebraic structure of these groups is extremely rigid and determines their topology in a very strong way.

Curvature of new Kähler metrics on the total space of Griffiths negative vector bundle and quasi-Fuchsian space

We study Kähler metrics on the total space of Griffiths negative holomorphic vector bundles over Kähler manifolds. As an application, we construct mapping class group invariant Kähler metrics on [Formula: see text], the holomorphic tangent bundle of Teichmüller space of a closed surface [Formula: see text]. Consequently,we obtain a new mapping class group invariant Kähler metric on the quasi-Fuchsian space [Formula: see text], which extends the Weil–Petersson metric on the Teichmüller space [Formula: see text]. We also calculate its curvature and prove non-positivity for the curvature along the tautological directions.

The balanced superelliptic mapping class groups are generated by three elements

Abstract The balanced superelliptic mapping class group is the normalizer of the transformation group of the balanced superelliptic covering in the mapping class group of the total surface. We prove that the balanced superelliptic mapping class groups with either one marked point, one boundary component, or no marked points and boundary are generated by three elements. To prove this, we also show that its liftable mapping class groups are also generated by three elements. These generating sets are minimal except for several cases of closed surfaces.

Monodromy Kernels for Strata of Translation Surfaces

Abstract The non-hyperelliptic connected components of the strata of translation surfaces are conjectured to be orbifold classifying spaces for some groups commensurable to some mapping class groups. The topological monodromy map of the non-hyperelliptic components projects naturally to the mapping class group of the underlying punctured surface and is an obvious candidate to test commensurability. In the present article, we prove that for the components $\mathcal {H}(3,1)$ and $\mathcal {H}^{nh}(4)$ in genus 3 the monodromy map fails to demonstrate the conjectured commensurability. In particular, building on the work of Wajnryb, we prove that the kernels of the monodromy maps for $\mathcal {H}(3,1)$ and $\mathcal {H}^{nh}(4)$ are large, as they contain a non-abelian free group of rank $2$.

Landau's theorem for harmonic Bloch functions on simply connected domains

For α ∈ ( 0 , ∞ ) , let B H , Ω ( α ) denote the class of harmonic α-Bloch mappings on a proper simply connected domain Ω ⊆ C . In this article, we study coefficient estimates for harmonic α-Bloch mappings on a proper simply connected domain containing the unit disk. Furthermore, we establish the Landau's theorem and coefficient estimates for the harmonic α-Bloch space B H , Ω γ ( α ) on the shifted disk Ω γ , where Ω γ := { z ∈ C : | z + γ 1 − γ | < 1 1 − γ } and 0 ≤ γ < 1 .

Applications of quantum representations of mapping class groups

Applications of quantum representations of mapping class groups

Bloch-type theorems for meromorphic harmonic mappings

In this paper, we first derive a Landau-type theorem and a Bloch-type theorem for the class of meromorphic harmonic mappings. Then we establish two new versions of Bloch-type theorems for certain K-quasiregular meromorphic harmonic mappings in the unit disk.

Strong convergent algorithm for finding minimum-norm solutions of quasimonotone variational inequalities with fixed point constraint and application

The class of quasimonotone mappings are known to be more general and applicable than the classes of pseudomonotone and monotone mappings. However, only very few results can be found in the literature on quasimonotone variational inequality problems and most of these results are on weak convergent algorithms. In this paper, we study the quasimonotone variational inequality problem (VIP) with constraint of fixed point problem (FPP) of quasi-pseudocontractive mappings. We introduce a new inertial Tseng’s extragradient method with self-adaptive step size for approximating the minimum-norm solutions of the aforementioned problem in the framework of Hilbert spaces. We prove that the sequence generated by the proposed method converges strongly to a common (minimum-norm) solution of the quasimonotone VIP and FPP of quasi-pseudocontractive mappings without the knowledge of the Lipschitz constant of the cost operator. We provide several numerical experiments for the proposed method in comparison with existing methods in the literature. Finally, we applied our result to image restoration problem. Our result improves, extends and generalizes several of the recently announced results in this direction.

A Large Class of Dendrite Maps for Which Möbius Disjointness Property of Sarnak is Fulfilled

A Large Class of Dendrite Maps for Which Möbius Disjointness Property of Sarnak is Fulfilled

Transitivity of normal subgroups of the mapping class groups on character varieties

Transitivity of normal subgroups of the mapping class groups on character varieties

Optimal pair of fixed points for a new class of noncyclic mappings under a $ (\varphi, \mathcal{R}^t) $-enriched contraction condition

&lt;abstract&gt;&lt;p&gt;In the present study, we commenced by presenting a new class of maps, termed noncyclic $ (\varphi, \mathcal{R}^t) $-enriched quasi-contractions within metric spaces equipped with a transitive relation $ \mathcal{R}^t $. Subsequently, we identified the conditions for the existence of an optimal pair of fixed points pertaining to these mappings, thereby extending and refining a selection of contemporary findings documented in some articles. Specifically, our analysis will encompass the outcomes pertinent to reflexive and strictly convex Banach spaces.&lt;/p&gt;&lt;/abstract&gt;

Solution approximation of fractional boundary value problems and convergence analysis using AA-iterative scheme

&lt;abstract&gt;&lt;p&gt;Addressing the boundary value problems of fractional-order differential equations hold significant importance due to their applications in various fields. The aim of this paper was to approximate solutions for a class of boundary value problems involving Caputo fractional-order differential equations employing the AA-iterative scheme. Moreover, the stability and data dependence results of the iterative scheme were given for a certain class of mappings. Finally, a numerical experiment was illustrated to support the results presented herein. The results presented in this paper extend and unify some well-known comparable results in the existing literature.&lt;/p&gt;&lt;/abstract&gt;

Applying fixed point techniques to solve fractional differential inclusions under new boundary conditions

&lt;abstract&gt;&lt;p&gt;Many scholars have lately explored fractional-order boundary value issues with a variety of conditions, including classical, nonlocal, multipoint, periodic/anti-periodic, fractional-order, and integral boundary conditions. In this manuscript, the existence and uniqueness of solutions to sequential fractional differential inclusions via a novel set of nonlocal boundary conditions were investigated. The existence results were presented under a new class of nonlocal boundary conditions, Carathéodory functions, and Lipschitz mappings. Further, fixed-point techniques have been applied to study the existence of results under convex and non-convex multi-valued mappings. Ultimately, to support our findings, we analyzed an illustrative example.&lt;/p&gt;&lt;/abstract&gt;

Fixed Point Theorems for a New Nonlinear Mapping in Hilbert Spaces

This study presents a new class of nonlinear mappings in Hilbert space. We establish demiclosed principles and fixed point theorems for this nonlinear mapping. Some works of literature are improved on and expanded upon by the results that this study presents.

The existence of multiple topologically distinct solutions to $\sigma_{2,p}$-energy

Let ${\An} \subset \R^n$ be a bounded Lipschitz domain and consider the $\sigmap$-energy functional \begin{equation*} {{\mathbb F}_{\sigmap}}[u; {\An}] := \int_{\An} \big|{\wedge}^2 \nabla u\big|^p dx, \end{equation*} with $p\in\mathopen]1, \infty]$ over the space of measure preserving maps \begin{equation*} {\mathcal A}_p(\An) =\big\{u \in W^{1,2p}\big(\An, \R^n\big) : u|_{\partial \An} = {x},\ \det \nabla u =1 \mbox{ for ${\mathcal L}^n$-a.e.\ in $\An$} \big\}. \end{equation*} In this article we address the question of multiplicity {\it versus} uniqueness for {\it extremals} and {\it strong} local minimizers of the $\sigmap$-energy funcional $\mathbb F_{\sigmap}[\cdot; {\An}]$ in ${\mathcal A}_p({\An})$. We use a topological class of maps referred to as {\it generalised} twists and examine them in connection with the Euler-Lagrange equations associated with $\sigmap$-energy functional over ${\mathcal A}_p({\An})$. Most notably, we prove the existence of a countably infinite of topologically distinct twisting solutions to the later system in all {\it even} dimensions by linking the system to a set of nonlinear isotropic ODEs on the Lie group ${\rm SO}(n)$. In sharp contrast in {\it odd} dimensions the only solution is the map $u\equiv x$. The result relies on a careful analysis of the {\it full} versus the {\it restricted} Euler-Lagrange equations. Indeed, an analysis of curl-free vector fields generated by symmetric matrix fields plays a pivotal role.

Density of periodic measures and large deviation principle for generalised mod one transformations

We introduce a new class of piecewise monotonic maps, called generalised mod one transformations, which include all (α,β) and generalised β -transformations, and prove that all transitive generalised mod one transformations with positive topological entropy satisfy the level-2 large deviation principle with a unique measure of maximal entropy. This is obtained by our result on the density of periodic measures for more general class of piecewise mototonic maps, which is a partial answer to the open problem posed by Hofbauer and Raith (1998 Fund. Math. 157 221–34); Raith (www.math.iupui.edu/~mmisiure/open/PR1.pdf).