Recent zbMATH articles in MSC 30C65https://zbmath.org/atom/cc/30C652023-03-23T18:28:47.107421ZWerkzeugDimension compression and expansion under homeomorphisms with exponentially integrable distortionhttps://zbmath.org/1503.300512023-03-23T18:28:47.107421Z"Hitruhin, Lauri"https://zbmath.org/authors/?q=ai:hitruhin.lauriSummary: We improve both dimension compression and expansion bounds for homeomorphisms with \(p\)-exponentially integrable distortion. To the first direction, we also introduce estimates for the compression multifractal spectra, which will be used to estimate compression of dimension, and for the rotational multifractal spectra. For establishing the expansion case, we use the multifractal spectra of the inverse mapping and construct examples proving sharpness.On the Hölder continuity of ring \(Q\)-homeomorphismshttps://zbmath.org/1503.300552023-03-23T18:28:47.107421Z"Arsenović, Miloš"https://zbmath.org/authors/?q=ai:arsenovic.milos"Mateljević, Miodrag"https://zbmath.org/authors/?q=ai:mateljevic.miodrag-sSummary: We prove the Hölder continuity of a homeomorphism \(f\) defined on a bounded domain \(\Omega\subset\mathbb{R}^n\) with Lipschitz boundary. The homeomorphism \(f\) is assumed to belong to a certain Orlicz-Sobolev class and to satisfy a distortion condition near the boundary.Cluster sets theorems on metric measure spaceshttps://zbmath.org/1503.300562023-03-23T18:28:47.107421Z"Cristea, Mihai"https://zbmath.org/authors/?q=ai:cristea.mihai.1|cristea.mihaiSummary: We generalize some theorems of Tsuji and Iversen concerning cluster sets of plane holomorphic mappings to the class of open, light mappings between metric measure spaces and satisfying generalized modular inequalities. We also generalize in this class some results of Vuorinen concerning asymptotic values and extension of continuity for quasiregular mappings.Metrics and quasimetrics induced by point pair functionhttps://zbmath.org/1503.300572023-03-23T18:28:47.107421Z"Dautova, Dina"https://zbmath.org/authors/?q=ai:dautova.dina"Nasyrov, Semen"https://zbmath.org/authors/?q=ai:nasyrov.semen-r"Rainio, Oona"https://zbmath.org/authors/?q=ai:rainio.oona"Vuorinen, Matti"https://zbmath.org/authors/?q=ai:vuorinen.matti-kSummary: We study the point pair function in subdomains \(G\) of \({\mathbb{R}}^n\). We prove that, for every domain \(G\subsetneq{\mathbb{R}}^n\), this function is a quasi-metric with the constant less than or equal to \(\sqrt{5}/2\). Moreover, we show that it is a metric in the domain \(G={\mathbb{R}}^n{\backslash}\{0\}\) with \(n\ge 1\). We also consider generalized versions of the point pair function, depending on a parameter \(\alpha >0\), and show that in some domains these generalizations are metrics if and only if \(\alpha \le 12\).On equicontinuity of the families of mappings with one normalization condition in terms of prime endshttps://zbmath.org/1503.300582023-03-23T18:28:47.107421Z"Ilkevych, N. S."https://zbmath.org/authors/?q=ai:ilkevych.n-s"Sevost'yanov, E. A."https://zbmath.org/authors/?q=ai:sevostyanov.evgenySummary: We study mappings with branching satisfying certain conditions of distortion for the modulus of families of paths. Under the conditions that the domain of definition of mappings has a weakly flat boundary, the mapped domain is regular, and the majorant responsible for the distortion of the modulus of families of paths is integrable, it is proved that the families of all specified mappings with one normalization condition are equicontinuous in the closure of a given domain.Two problems on homogenization in geometryhttps://zbmath.org/1503.300592023-03-23T18:28:47.107421Z"Ivrii, Oleg"https://zbmath.org/authors/?q=ai:ivrii.oleg-v"Marković, Vladimir"https://zbmath.org/authors/?q=ai:markovic.vladimir|markovic.vladimir-mFor the entire collection see [Zbl 1478.00019].Three-dimensional quasiconformal mappings and axisymmetric problemshttps://zbmath.org/1503.300602023-03-23T18:28:47.107421Z"Shevelev, Yu. D."https://zbmath.org/authors/?q=ai:shevelev.yu-dSummary: Quasiconformal mappings of axisymmetric domains are considered as a special case of three-dimensional transformations. For a three-dimensional steady irrotational flow of an inviscid incompressible fluid, two stream functions are introduced along with the velocity potential. Any solenoidal vector can be represented as the cross product of the gradients of two stream functions. As a result, a relationship between the velocity components and the stream functions is obtained for determining the velocity potential. On the one hand, these transformations underlie Lavrentiev-harmonic mappings. On the other hand, these conditions can be treated as a generalization of the Cauchy-Riemann conditions to the three-dimensional case. In this work, the generalized three-dimensional Cauchy-Riemann conditions for harmonic mappings are reduced to the usual Cauchy-Riemann conditions in polar coordinates of complex variable functions. Lavrentiev-harmonic conditions are used to construct an analogue of quasiconformal mapping of axisymmetric domains and to generalize mappings of axisymmetric domains to arbitrary domains. Examples of visualization of quasiconformal mappings of axisymmetric domains and their generalizations are given.Gromov hyperbolicity in the free quasiworld. Ihttps://zbmath.org/1503.300612023-03-23T18:28:47.107421Z"Zhou, Qingshan"https://zbmath.org/authors/?q=ai:zhou.qingshan"Ponnusamy, Saminathan"https://zbmath.org/authors/?q=ai:ponnusamy.saminathanSummary: With the aid of a Gromov hyperbolic characterization of uniform domains, we first give an affirmative answer to an open question arisen by Väisälä under weaker assumption. Next, we show that the three-point condition introduced by Väisälä is necessary to obtain quasisymmetry for quasimöbius maps between bounded connected spaces in a quantitative way. Based on these two results, we investigate the boundary behavior of freely quasiconformal and quasihyperbolic mappings on uniform domains of Banach spaces and partially answer another question raised by Väisälä in different ways.Geometric characterizations of Gromov hyperbolic Hölder domainshttps://zbmath.org/1503.300622023-03-23T18:28:47.107421Z"Zhou, Qingshan"https://zbmath.org/authors/?q=ai:zhou.qingshan"Rasila, Antti"https://zbmath.org/authors/?q=ai:rasila.anttiSummary: In this paper, we investigate Hölder continuity of quasiconformal mappings in \(\mathbb{R}^n\) from the points of view of quasihyperbolic geometry and the theory of Gromov hyperbolic spaces. We establish several characterizations of Gromov hyperbolic domains satisfying the Gehring-Martio-type quasihyperbolic boundary conditions. As applications, we generalize certain results concerning Hölder continuity of conformal mappings, establishing counterparts of results of Becker and Pommerenke, Smith and Stegenga, and Näkki and Palka in higher-dimensional Euclidean spaces.Homeomorphisms of finite metric distortion between Riemannian manifoldshttps://zbmath.org/1503.301332023-03-23T18:28:47.107421Z"Afanas'eva, Elena"https://zbmath.org/authors/?q=ai:afanaseva.elena-s"Golberg, Anatoly"https://zbmath.org/authors/?q=ai:golberg.anatolySummary: The theory of multidimensional quasiconformal mappings employs three main approaches: analytic, geometric (modulus) and metric ones. In this paper, we use the last approach and establish the relationship between homeomorphisms of finite metric distortion (FMD-homeomorphisms), finitely bi-Lipschitz, quasisymmetric and quasiconformal mappings on Riemannian manifolds. One of the main results shows that FMD-homeomorphisms are lower \(Q\)-homeomorphisms. As an application, there are obtained some sufficient conditions for boundary extensions of FMD-homeomorphisms. These conditions are illustrated by several examples of FMD-homeomorphisms.Permutable quasiregular mapshttps://zbmath.org/1503.370602023-03-23T18:28:47.107421Z"Tsantaris, Athanasios"https://zbmath.org/authors/?q=ai:tsantaris.athanasiosLet \(f\) and \(g\) be two functions, rational, transcendental or quasi-regular. They are commuting if \(f\circ g=g\circ f\). It is known since the pioneering works of Fatou and Julia that commuting rational maps on the Riemann sphere have the same Julia sets. But it is still unclear if two commuting transcendental entire functions also enjoy this property.
The present author studies commuting quasi-regular maps and generalizes a result of \textit{W. Bergweiler} and \textit{A. Hinkkanen} [Math. Proc. Camb. Philos. Soc. 126, No. 3, 565--574 (1999; Zbl 0939.30019)]
to quasi-regular maps of transcendental type (see Theorem 1.2). Some other results in this setting are also obtained. For instance, commuting quasi-regular maps whose Julia sets have positive capacity and satisfy an additional condition also have same Julia sets. Moreover, two quasi-meromorphic functions of transcendental type for which \(\infty\) is not an exceptional value are shown to have same Julia sets.
Reviewer: Weiwei Cui (Lund)