Mapping Class Research Articles

Double Johnson filtrations for mapping class groups

In this paper, we first develop a general theory of Johnson filtrations and Johnson homomorphisms for a group [Formula: see text] acting on another group [Formula: see text] equipped with a filtration indexed by a “good” ordered commutative monoid. Then, specializing it to the case where the monoid is the additive monoid [Formula: see text] of pairs on non-negative integers, we obtain a theory of double Johnson filtrations and homomorphisms. We apply this theory to the mapping class group [Formula: see text] of a surface [Formula: see text] with one boundary component, equipped with the normal subgroups [Formula: see text], [Formula: see text] of [Formula: see text] associated to a standard Heegaard splitting of the [Formula: see text]-sphere. We also consider the case where the group [Formula: see text] is the automorphism group of a free group.

Generating the level 2 subgroup by involutions

We obtain a minimal generating set of involutions for the level 2 subgroup of the mapping class group of a closed nonorientable surface.

Graphs of curves and arcs quasi-isometric to big mapping class groups

Graphs of curves and arcs quasi-isometric to big mapping class groups

Finite presentations for the balanced superelliptic mapping class groups

The balanced superelliptic mapping class group is the normalizer of the transformation group of the balanced superelliptic covering space in the mapping class group of the total surface. We give finite presentations for the balanced superelliptic mapping class groups of closed surfaces, surfaces with one marked point, and surfaces with one boundary component. To give these presentations, we construct finite presentations for corresponding liftable mapping class groups in a different generating set from Ghaswala–Winarski’s presentation in [9].

Homological representations of low genus mapping class groups

Given a finite group G acting on a surface S, the centralizer of G in the mapping class group Mod(S) has a natural representation given by its action on the homology H1(S;Q). We consider the question of whether this representation has arithmetic image. Several authors have given positive and negative answers to this question. We give a complete answer when S has genus at most 3.

Intersection results for general classes of maps

Intersection results for general classes of maps

A fixed point result for DND-convex-power condensing mappings

In this paper, based on the so-called degree of nondensifiability (DND), we introduce the concept of DND-convex-power condensing mapping which generalizes that of the DND-condensing one. A fixed point result for this new class of mappings is proved, and with some examples we evidence the differences between it and other fixed point theorems of the same type. As application of our results, we prove under suitable conditions the existence of fixed point for the superposition operator, defined in the Banach spaces of the continuous functions from [Formula: see text] and with values in a Banach space.

On a new approach of enriched operators

We establish the existence and uniqueness of fixed points of generalized contractions in the setting of Banach spaces and prove the convergence of Mann iteration for this general class of mappings. Also, we show the existence of fixed points and the convergence of Mann iteration as well for generalized nonexpansive mappings. Last but not least, we provide two applications, one from the field of numerical analysis of linear systems and another one dealing with functional equations. This new approach significantly extends the classes of enriched contractions and enriched nonexpansive mappings, and allows the use of Mann iteration as opposed to all papers on the subject, which necessarily have to rely on Krasnoselskij iteration.

Ahlfors type p-valent conditions for biharmonic functions

In 1973, Ahlfors established a sufficient condition for an analytic function to be univalent in the unit disk |z|<1 and has a quasiconformal extension. Using his result, many known conditions for univalence and quasiconformal extendibility of analytic functions in the unit disk were deduced. Interestingly, his result was generalized to the class of harmonic mappings. The main aim of this paper is to present another generalization of Ahlfors's result by establishing p-valent conditions for biharmonic mappings defined in the unit disk and exterior. Moreover, we also determine conditions for a harmonic mapping of the unit disk to be univalent and has a quasiconformal extension.

New Version of Fractional Pachpatte-Type Integral Inequalities via Coordinated ℏ-Convexity via Left and Right Order Relation

In particular, the fractional forms of Hermite–Hadamard inequalities for the newly defined class of convex mappings proposed that are known as coordinated left and right ℏ-convexity (LR-ℏ-convexity) over interval-valued codomain. We exploit the use of double Riemann–Liouville fractional integral to derive the major results of the research. We also examine the key results’ numerical validations that examples are nontrivial. By taking the product of two left and right coordinated ℏ-convexity, some new versions of fractional integral inequalities are also obtained. Moreover, some new and classical exceptional cases are also discussed by taking some restrictions on endpoint functions of interval-valued functions that can be seen as applications of these new outcomes.

Constructing new high-order polynomial chaotic maps and application in pseudorandom number generator

The chaotic map have been widely applied in fields such as pseudorandom number generation(PRNG) and image encryption due to its excellent chaotic performance. In this paper, two classes of high-order polynomial chaotic maps of special form are proposed based on the Li-Yorke theorem. The dynamic behavior of the proposed maps is numerically analyzed, including bifurcation and Lyapunov exponent, and the analysis results prove the validity of the proposed conclusions. The proposed polynomial chaotic maps have a larger parameters and chaotic range, as well as a more stable Lyapunov exponent. Furthermore, based on the coupling chaotic systems, we design a pseudorandom number generator(PRNG), and the number of chaos parameters are expanded through the coupling control parameters in a PRNG. Then the performance of the pseudorandom sequence generated by the PRNG is tested and analysed. The test and analysis results show that the pseudorandom sequence has favorabble security, structural complexity, and randomness. Especially, the information entropy of 7.9998 and the key space size of 2208 exceed the recently reported pseudorandom number generators(PRNGs). In comparison with other PRNGs based on chaotic maps in the recent literature, this paper provides comprehensive performance test and analysis of the proposed PRNG and demonstrates its potential for cryptographic applications.

Topological entanglement entropy for torus-knot bipartitions and the Verlinde-like formulas

The topological Rényi and entanglement entropies depend on the bipartition of the manifold and the choice of the ground states. However, these entanglement quantities remain invariant under a coordinate transformation when the bipartition also undergoes the identical transformation. In topological quantum field theories (TQFTs), these coordinate transformations reduce to representations of the mapping class group on the manifold of the Hilbert space. We employ this invariant property of the Rényi and entanglement entropies under coordinate transformations for TQFTs in (2 + 1) dimensions on a torus with various bipartitions. By utilizing the replica trick and the surgery method to compute the topological Rényi and entanglement entropies, the invariant property results in Verlinde-like formulas. Furthermore, for the bipartition with interfaces as two non-intersecting torus knots, an SL(2, ℤ) transformation can untwist the torus knots, leading to a simple bipartition with an effective ground state. This invariant property allows us to demonstrate that the topological entanglement entropy has a lower bound −2 ln D, where D is the total quantum dimensions of the system.

Stable laws for random dynamical systems

Abstract In this paper, we consider random dynamical systems formed by concatenating maps acting on the unit interval $[0,1]$ in an independent and identically distributed (i.i.d.) fashion. Considered as a stationary Markov process, the random dynamical system possesses a unique stationary measure $\nu $ . We consider a class of non-square-integrable observables $\phi $ , mostly of form $\phi (x)=d(x,x_0)^{-{1}/{\alpha }}$ , where $x_0$ is a non-recurrent point (in particular a non-periodic point) satisfying some other genericity conditions and, more generally, regularly varying observables with index $\alpha \in (0,2)$ . The two types of maps we concatenate are a class of piecewise $C^2$ expanding maps and a class of intermittent maps possessing an indifferent fixed point at the origin. Under conditions on the dynamics and $\alpha $ , we establish Poisson limit laws, convergence of scaled Birkhoff sums to a stable limit law, and functional stable limit laws in both the annealed and quenched case. The scaling constants for the limit laws for almost every quenched realization are the same as those of the annealed case and determined by $\nu $ . This is in contrast to the scalings in quenched central limit theorems where the centering constants depend in a critical way upon the realization and are not the same for almost every realization.

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.