Estimation of dilatations for mappings more general than quasiregular mappings

We consider the so-called ring Q-mappings, which are natural generalizations of quasiregular mappings in a sense of V?is?l?’s geometric definition of moduli. It is shown that, under the condition of nondegeneracy of these mappings, their inner dilatation is majorized by a function Q(x) to within a constant depending solely on the dimension of the space.     

Slow escaping points of quasiregular mappings

This article concerns the iteration of quasiregular mappings on \(\mathbb {R}^d\) and entire functions on \(\mathbb {C}\). It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate. Moreover, an asymptotic rate of escape result is proved that is new even for transcendental entire functions. Let \(f:\mathbb {R}^d\rightarrow \mathbb {R}^d\) be quasiregular of transcendental type. Using novel methods of proof, we generalise results of Rippon and Stallard in complex dynamics to show that the Julia set of f contains points at which the iterates \(f^n\) tend to infinity arbitrarily slowly. We also prove that, for any large R, there is a point x with modulus approximately R such that the growth of \(|f^n(x)|\) is asymptotic to the iterated maximum modulus \(M^{n}(R,f)\).     

Fixed points and normal families of quasiregular mappings

LetF be a family of mappingsK-quasiregular in some domainG. We show that if for eachf∈F, there existsk>1 such that thek-th iteratef ^k off has no fixed point, thenF is normal. Moreover, we examine to what extent this result holds if we consider only repelling fixed points, rather than fixed points in general. We also prove thatF is quasinormal, ifF contains only quasiregular mappings that do not have periodic points of some period greater than one inG. This implies that a quasiregular mappingf:ℝ ⁿ with an essential singularity in ∞ has infinitely many periodic points of any period greater than one. These results generalize results of M. Essén, S. Wu, D. Bargmann and W. Bergweiler for holomorphic functions.     

On quasiregular mappings between smooth Jordan domains

It is proved that every proper quasiregular C² mapping w between two plane Jordan domains Ω∈C1,α and G∈C2,α, 0<α?1, satisfying the differential inequality |Δw|?M²|∇w|+N is Lipschitz continuous. This extends the main result of the author and M. Mateljevi? (Kalaj and Mateljevi?, 2006 [7]).     

On Dyakonov type theorems for harmonic quasiregular mappings

We prove two Dyakonov type theorems which relate the modulus of continuity of a function on the unit disc with the modulus of continuity of its absolute value. The methods we use are quite elementary, they cover the case of functions which are quasiregular and harmonic, briefly hqr, in the unit disc.     

On branch points of three-dimensional mappings with unbounded characteristic of quasiconformality

For open discrete mappings f:D\{ b } ? \mathbbR³ f:D\backslash \left\{ b \right\} \to {\mathbb{R}^3} of a domain D ì \mathbbR³ D \subset {\mathbb{R}^3} satisfying relatively general geometric conditions in D \ {b} and having an essential singularity at a point b ? \mathbbR³ b \in {\mathbb{R}^3} , we prove the following statement: Let a point y ₀ belong to [`(\mathbbR³)] \f( D\{ b } ) \overline {{\mathbb{R}^3}} \backslash f\left( {D\backslash \left\{ b \right\}} \right) and let the inner dilatation K _I (x, f) and outer dilatation K _O (x, f) of the mapping f at the point x satisfy certain conditions. Let B _f denote the set of branch points of the mapping f. Then, for an arbitrary neighborhood V of the point y ₀, the set V ∩ f(B _f ) cannot be contained in a set A such that g(A) = I, where I = { t ? \mathbbR:| t | < 1 } I = \left\{ {t \in \mathbb{R}:\left| t \right| < 1} \right\} and g:U ? \mathbbRⁿ g:U \to {\mathbb{R}^n} is a quasiconformal mapping of a domain U ì \mathbbRⁿ U \subset {\mathbb{R}^n} such that A ⊂ U.     

On the Lipschitz continuity of certain quasiregular mappings between smooth Jordan domains

We first investigate the Lipschitz continuity of (K,K’)-quasiregular C ² mappings between two Jordan domains with smooth boundaries, satisfying certain partial differential inequalities concerning Laplacian. Then two applications of the obtained result are given: As a direct consequence, we get the Lipschitz continuity of ρ-harmonic (K,K’)-quasiregular mappings, and as the other application, we study the Lipschitz continuity of (K,K’)- quasiconformal self-mappings of the unit disk, which are the solutions of the Poisson equation Δw = g. These results generalize and extend several recently obtained results by Kalaj, Mateljevi? and Pavlovi?.     

On finite matroids with one more hyperplane than points

In this paper a complete classification of finite matroids with one more hyperplane than points is obtained.Dedicated to Professor Maria Scafati Tallini on the occasion of her 65th birthday     

Subharmonicity of the modulus of quasiregular harmonic mappings

In this note we determine all numbers q∈R such that q|u| is a subharmonic function, provided that u is a K-quasiregular harmonic mappings in an open subset Ω of the Euclidean space Rn.     

On finite matroids with two more hyperplanes than points

One of the most interesting results about finite matroids of finite rank and generalized projective spaces is the result of Basterfield, Kelly and Green (1968/1970) (J.G. Basterfield, L.M. Kelly, A characterization of sets of n points which determine n hyperplanes, in: Proceedings of the Cambridge Philosophical Society, vol. 64, 1968, pp. 585-588; C. Greene, A rank inequality for finite geometric lattices, J. Combin Theory 9 (1970) 357-364) affirming that any matroid contains at least as many hyperplanes as points, with equality in the case of generalized projective spaces. Consequently, the goal is to characterize and classify all matroids containing more hyperplanes than points. In 1996, I obtained the classification of all finite matroids containing one more hyperplane than points. In this paper a complete classification of finite matroids with two more hyperplanes than points is obtained. Moreover, a partial contribution to the classification of those matroids containing a certain number of hyperplanes more than points is presented.     

Existence of fixed points for mappings of finite sets

We show that the existence theorem for zeros of a vector field (fixed points of a mapping) holds in the case of a “convex” finite set X and a “continuous” vector field (a self-mapping) directed inwards into the convex hull co X of X. The main goal is to give correct definitions of the notions of “continuity” and “convexity”. We formalize both these notions using a reflexive and symmetric binary relation on X, i.e., using a proximity relation. Continuity (we shall say smoothness) is formulated with respect to any proximity relation, and an additional requirement on the proximity (we shall call it the acyclicity condition) transforms X into a “convex” set. If these two requirements are satisfied, then the vector field has a zero (i.e., a fixed point).     

Approximate fixed points of nonexpansive mappings in unbounded sets

It follows from Banach’s fixed point theorem that every nonexpansive self-mapping of a bounded, closed and convex set in a Banach space has approximate fixed points. This is no longer true, in general, if the set is unbounded. Nevertheless, as we show in the present paper, there exists an open and everywhere dense set in the space of all nonexpansive self-mappings of any closed and convex (not necessarily bounded) set in a Banach space (endowed with the natural metric of uniform convergence on bounded subsets) such that all its elements have approximate fixed points.     

Remarks on the local index of quasiregular mappings

Journal d'Analyse Mathématique -     

The oscillation of harmonic and quasiregular mappings

Abstract. If u(z) is harmonic in with and we set A result is obtained which shows, in particular that if and then a bound for can be obtained in terms of for a suitable constant , so that the logarithm of the oscillation has an approximate convexity property. The proof uses classical inequalities of Hadamard and Borel–Carathéodory and this suggests a generalization to quasiregular mappings in . Such results are obtained, though necessarily in a less precise form because of the lack of good explicit estimates for -harmonic measures in spherical ring domains. Received: 9 November 2000 / Published online: 18 January 2002     

On Liouville’s theorem for locally quasiregular mappings inR n

Letf:R ⁿ→Rⁿ be locally quasiregular in the sense that the restriction off to any ball |x|<r has finite inner dilatationK ₁(r). Suppose that the growth condition ∫^∞r^-1K₁(r)^1/(1-n) holds. Then Liouville’s theorem is valid:If f is bounded, f is a constant. An example shows that this growth condition is relatively sharp.