Porosity of Collet-Eckmann Julia sets
Feliks Przytycki, Steffen Rohde
TL;DR
This paper proves that Julia sets of certain rational maps satisfying the Collet-Eckmann condition are mean porous, leading to the conclusion that their Minkowski dimension is less than 2, unless the set is the entire sphere.
Contribution
It establishes mean porosity and dimension bounds for Julia sets under Collet-Eckmann conditions, extending understanding of their geometric structure.
Findings
Julia sets are mean porous under specified conditions
Minkowski dimension of these Julia sets is less than 2
The results exclude the Julia set being the whole sphere
Abstract
We prove that the Julia set of a rational map of the Riemann sphere satisfying the Collet-Eckmann condition and having no parabolic periodic point is mean porous, if it is not the whole sphere. It follows that the Minkowski dimension of the Julia set is less than 2.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Analytic and geometric function theory · Meromorphic and Entire Functions