Quadratic Julia Sets with Positive Area
Xavier Buff, Arnaud Cheritat
TL;DR
This paper proves that certain quadratic polynomials can have Julia sets with positive area, specifically when they have Cremer points, Siegel disks, or undergo infinite renormalizations.
Contribution
It establishes the existence of quadratic Julia sets with positive measure in cases previously not confirmed, including Cremer points, Siegel disks, and infinite renormalizations.
Findings
Julia sets with positive Lebesgue measure exist for quadratic polynomials with Cremer points.
Julia sets with positive measure exist for quadratic polynomials with Siegel disks.
Julia sets with positive measure exist for quadratic polynomials with infinitely many satellite renormalizations.
Abstract
We prove the existence of quadratic polynomials having a Julia set with positive Lebesgue measure in three cases: the presence of a Cremer fixed point, the presence of a Siegel disk, the presence of infinitely many (satellite) renormalizations.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Meromorphic and Entire Functions · Advanced Differential Equations and Dynamical Systems