The dynamics of quasiregular maps of punctured spaceTools Nicks, Daniel A. and Sixsmith, David J. (2017) The dynamics of quasiregular maps of punctured space. Indiana University Mathematics Journal . ISSN 00222518 (In Press)
AbstractThe FatouJulia iteration theory of rational and transcendental entire functions has recently been extended to quasiregular maps in more than two real dimensions. Our goal in this paper is similar; we extend the iteration theory of analytic selfmaps of the punctured plane to quasiregular selfmaps of punctured space. We define the Julia set as the set of points for which the complement of the forward orbit of any neighbourhood of the point is a finite set. We show that the Julia set is nonempty, and shares many properties with the classical Julia set of an analytic function. These properties are stronger than those known to hold for the Julia set of a general quasiregular map of space. We define the quasiFatou set as the complement of the Julia set, and generalise a result of Baker concerning the topological properties of the components of this set. A key tool in the proof of these results is a version of the fast escaping set. We generalise various results of MartiPete concerning this set, for example showing that the Julia set is equal to the boundary of the fast escaping set.
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