The maximum modulus set of a quasiregular map

We study Hardy spaces H p \mathcal {H}^p , 0 > p > ∞ 0>p>\infty for quasiregular mappings on the unit ball B B in R n {\mathbb R}^n which satisfy appropriate growth and multiplicity conditions. Under these conditions we recover several classical results for analytic functions and quasiconformal mappings in H p \mathcal {H}^p . In particular, we characterize H p \mathcal {H}^p in terms of non-tangential limit functions and non-tangential maximal functions of quasiregular mappings. Among applications we show that every quasiregular map in our class belongs to H p \mathcal {H}^p for some p = p ( n , K ) p=p(n,K) . Moreover, we provide characterization of Carleson measures on B B via integral inequalities for quasiregular mappings on B B . We also discuss the Bergman spaces of quasiregular mappings and their relations to H p \mathcal {H}^p spaces and analyze correspondence between results for H p \mathcal {H}^p spaces and A \mathcal {A} -harmonic functions. A key difference between the previously known results for quasiconformal mappings in R n {\mathbb R}^n and our setting is the role of multiplicity conditions and the growth of mappings that need not be injective. Our paper extends results by Astala and Koskela, Jerison and Weitsman, Jones, Nolder, and Zinsmeister.

We define a quasi-Fatou component of a quasiregular map as a connected component of the complement of the Julia set. A domain in ℝd is called hollow if it has a bounded complementary component. We show that for each d ⩾ 2 there exists a quasiregular map of transcendental type f: ℝd → ℝd with a quasi-Fatou component which is hollow.Suppose that U is a hollow quasi-Fatou component of a quasiregular map of transcendental type. We show that if U is bounded, then U has many properties in common with a multiply connected Fatou component of a transcendental entire function. On the other hand, we show that if U is not bounded, then it is completely invariant and has no unbounded boundary components. We show that this situation occurs if J(f) has an isolated point, or if J(f) is not equal to the boundary of the fast escaping set. Finally, we deduce that if J(f) has a bounded component, then all components of J(f) are bounded.

The condition (1.1) means that for all D ~ G the coordinate functions o f f belong to the Sobolev space W,~(D) of functions in L"(D) whose distributional first order partial derivatives are also L" integrable in D. In the above definition f ' ( x ) is the formal derivative of f a t x defined by means of the partial derivatives, I f ' (x) l is the operator norm of f '(x), and Ji(x) is the Jacobian determinant of f a t x. These are uniquely defined almost everywhere by (1.1). It turns out that quasiregular mappings are differentiable almost everywhere and a nonconstant quasiregular map is discrete, open, and sense-preserving. For these and other basic properties of qr maps we refer to books [Re] by Reshetnyak, [Vu] by Vuorinen, and also to a forthcoming book [R6] by the second author. The definition extends immediately to the case f : M ~ N where M and N are connected oriented Riemannian n-manifolds. We say that a continuous map f: M ~ N is quasiregular if it has a local representation at each point of M which is quasiregular in the above sense, and if there exists K, 1 < K < oe, such that

The structure of a strange attractor

This article concerns the iteration of quasiregular mappings on Rd and entire functions on 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:Rd→Rd 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 fn 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 |fn(x)| is asymptotic to the iterated maximum modulus Mn(R,f).

In this paper, we investigate the boundary of the escaping set I(f) for quasiregular mappings on ℝn, both in the uniformly quasiregular case and in the polynomial type case. The aim is to show that ∂I(f) is the Julia set J(f) when the latter is defined, and shares properties with the Julia set when J(f) is not defined.

We show that a maximum modulus set K in a bounded strictly convex balanced domain Omega with C^{2} boundary can have a Lebesgue measure almost full on all slices. Additionally K has a so called slice continuity property.