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Quasiregular mappings of polynomial type in \(\mathbb{R}^{2}\). (English) Zbl 1233.30013
From the introduction: More recently, the iteration of quasiregular mappings in \(\mathbb R^n\) has been studied, motivated by the fact that several key tools in complex dynamics have analogues for quasiregular mappings, for example, Rickman’s theorem generalizing Picard’s theorem and Montel’s theorem. In fact, direct analogues of the Fatou and Julia sets can be defined for a special class of quasiregular mappings, all of whose iterates have distortion bounded by some fixed number. These are the so-called uniformly quasiregular mappings, introduced by T.  Iwaniec and G.  Martin.
For general quasiregular mappings, it is no longer possible to define the Fatou set for the simple reason that the iterates will have no common bound on the distortion.
It is, however, possible to define the escaping set \[ I(f):=\{z\in\mathbb C:f^n(z)\to\infty\} \] which is a key object in complex dynamics. It is well known that for an analytic function, the boundary of \(I(f)\) coincides with the Julia set. It is therefore natural to consider \(\partial I(f)\) for quasiregular mappings and see to what extent it can be considered an analogue of the Julia set.
The authors analyze the boundary of the escaping set for the simplest quasiregular mappings with non-trivial dynamics; namely the composition of quadratic polynomials and quasiconformal mappings with constant complex dilatation.
A canonical form for compositions of quadratic polynomials and affine stretches is derived, and connectedness properties of \(\partial I(f)\) are considered.
A generalization of the Mandelbrot set is introduced and some of its properties are studied.

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
Full Text: DOI arXiv
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