Foundations for an iteration theory of entire quasiregular maps
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In this paper, the Julia set is defined as the set of all points such that the complement of the forward orbit of any neighbourhood has capacity zero, and it is shown that for maps which are not of polynomial type, the Julia sets is nonempty and has many properties of the classical Julia set of transcendental entire functions.Abstract:
The Fatou-Julia iteration theory of rational functions has been extended to uniformly quasiregular mappings in higher dimension by various authors, and recently some results of Fatou-Julia type have also been obtained for non-uniformly quasiregular maps. The purpose of this paper is to extend the iteration theory of transcendental entire functions to the quasiregular setting. As no examples of uniformly quasiregular maps of transcendental type are known, we work without the assumption of uniform quasiregularity. Here the Julia set is defined as the set of all points such that the complement of the forward orbit of any neighbourhood has capacity zero. It is shown that for maps which are not of polynomial type, the Julia set is non-empty and has many properties of the classical Julia set of transcendental entire functions.read more
Citations
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Journal ArticleDOI
The fast escaping set for quasiregular mappings
TL;DR: The fast escaping set of a transcendental entire function is defined as the set of all points which tend to infinity under iteration as fast as possible compatible with the growth of the function.
Journal ArticleDOI
Periodic domains of quasiregular maps
TL;DR: In this paper, the authors considered the problem of constructing a quasiregular map of transcendental type from R3 to R3 with a periodic domain in which all iterates tend locally uniformly to infinity.
Journal ArticleDOI
Hollow quasi-Fatou components of quasiregular maps
TL;DR: In this paper, the authors define a quasi-Fatou component of a quasiregular map as a connected component of the complement of the Julia set, and show that if U is bounded, then U has many properties in common with a multiply connected Fatou component, whereas U is completely invariant and has no unbounded boundary components.
Journal ArticleDOI
Slow escaping points of quasiregular mappings
TL;DR: In this paper, it was shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate, and an asymptotic rate of escape result was proved even for transcendental entire functions.
References
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TL;DR: In this article, the dynamics of iterated holomorphic mappings from a Riemann surface to itself are studied, focusing on the classical case of rational maps of the RiemANN sphere.
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