Related papers: Dynamics of mappings with constant dilatation

We investigate the rate of convergence of the iterates of an n-dimensional quasiregular mapping within the basin of attraction of a fixed point of high local index. A key tool is a refinement of a result that gives bounds on the distortion…

For $A\in M^{2\times 2}$ let $S(A)=\sqrt{A^T A}$, i.e. the symmetric part of the polar decomposition of $A$. We consider the relation between two quasiregular mappings whose symmetric part of gradient are close. Our main result is the…

We consider the iteration of quasiregular maps of transcendental type from $\mathbb{R}^d$ to $\mathbb{R}^d$. We give a bound on the rate at which the iterates of such a map can escape to infinity in a periodic component of the quasi-Fatou…

We study one-parameter curves on the universal Teichm\"uller space $T$ and on the homogeneous space $M=\Diff S^1/\Rot S^1$ embedded into $T$. As a result, we deduce evolution equations for conformal maps that admit quasiconformal extensions…

Let $K\ge 1$ and $p\in(1,2]$. We obtain asymptotically sharp constant $c(K,p)$, when $K\to 1$ in the inequality $$\|\Im f\|_{p}\le c(K,p)\|\Re(f)\|_p$$ where $f\in \mathbf{h}^p$ is a $K-$quasiregular harmonic mapping in the unit disk…

Quasiconformal maps are homeomorphisms with useful local distortion inequalities; infinitesimally, they map balls to ellipsoids with bounded eccentricity. This leads to a number of useful regularity properties, including quantitative…

We study mappings with branching of a domain of Euclidean space. The H\"older and Lipschitz continuity are established for one class of spatial mappings whose characteristic satisfies the Dini type condition in a given domain. In addition,…

We prove several formulas for the Hilbert metric in the unit disk and apply these results to study quasiregular mappings of the unit disk $\mathbb{B}^2$ onto a bounded convex domain $D$. The main result deals with the H\"older continuity of…

In the present paper, questions about a local behavior of mappings $f:D\rightarrow \overline{{\Bbb R}^n},$ $n\ge 2,$ in $\overline{D}$ are studied. Under some conditions on a measurable function $Q(x),$ $Q:D\rightarrow [0, \infty],$ and…

In the field of dynamical systems, it is not rare to meet irregular functions, which are typically H{\"o}lder but not Lipschitz (e.g. the Weierstrass functions). Our goal is to scratch the surface of the following question: what happens if…

This paper deals with $(K_1, K_2)$-quasiregular mappings. It is shown, by Morrey's Lemma and isoperimetric inequality, that every $(K_1, K_2)$-quasiregular mapping satisfies a H\"older condition with exponent $\alpha$ on compact subsets of…

We construct a quasiregular analogue of the function $z\exp(z)$ in dimension 3, which gives the first explicit example of a quasiregular mapping of transcendental type that has exactly one zero. We then modify the construction to create a…

We define a new quasiregular mapping T in three dimensions that generalizes the tangent function on the complex plane and shares a number of its geometric properties. We investigate the dynamics of the family \lambda T for \lambda>0,…

We show that wandering domains can exist in the Fatou set of a polynomial type quasiregular mapping of the plane. We also give an example of a quasiregular mapping of the plane, with an essential singularity at infinity, which has a…

Since 1984, many authors have studied the dynamics of maps of the form $\mathcal{E}_a(z) = e^z - a$, with $a > 1$. It is now well-known that the Julia set of such a map has an intricate topological structure known as a Cantor bouquet, and…

We study quasiconformal mappings of the unit disk that have planar extension with controlled distortion. For these mappings we prove a bound for the modulus of continuity of the inverse map, which somewhat surprisingly is almost as good as…

The article is devoted to the study of mappings with finite distortion in metric spaces. Analogues of results relating to equicontinuity and normality of families of quasiregular mappings are obtained. It is proved that the indicated…

We introduce a higher dimensional quasiregular map analogous to the trigonometric functions and we use the dynamics of this map to define, for d>1, a partition of d-dimensional Euclidean space into curves tending to infinity such that two…

Given a uniformly quasiregular mapping, there is typically no reason to assume any relationship between linearizers at different repelling periodic points. However, in the current paper we prove that in the case where the uqr map arises as…

We prove that a harmonic quasiconformal mapping defined on a finitely connected domain in the plane, all of whose boundary components are either points or quasicircles, admits a quasiconformal extension to the whole plane if its Schwarzian…