On McMullen-like mappings
Antonio Garijo, S\'ebastien Godillon

TL;DR
This paper generalizes McMullen's family of rational maps by defining McMullen-like mappings associated with hyperbolic postcritically finite polynomials and pole data, characterizing their Julia sets through an arithmetic condition.
Contribution
It introduces a broad class of McMullen-like mappings, providing necessary and sufficient conditions for their dynamics using Thurston obstructions and quasiconformal surgery.
Findings
Characterization of Julia sets as Cantor sets of circles under certain conditions.
Development of an arithmetic criterion for McMullen-like mappings.
Application of Thurston's theory and quasiconformal surgery to establish results.
Abstract
We introduce a generalization of the McMullen family . In 1988, C. McMullen showed that the Julia set of is a Cantor set of circles if and only if and the simple critical values of belong to the trap door. We generalize this behavior defining a McMullen-like mapping as a rational map associated to a hyperbolic postcritically finite polynomial and a pole data where we encode, basically, the location of every pole of and the local degree at each pole. In the McMullen family, the polynomial is and the pole data is the pole located at the origin that maps to infinity with local degree . As in the McMullen family , we can characterize a McMullen-like mapping using an arithmetic condition depending only on the polynomial and the pole data…
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Taxonomy
TopicsMathematical Dynamics and Fractals · History and Theory of Mathematics · Analytic and geometric function theory
