Rigidity and non local connectivity of Julia sets of some quadratic polynomials

TL;DR

This paper investigates the local connectivity of Julia and Mandelbrot sets for certain quadratic polynomials, establishing conditions under which these sets are connected or disconnected, thus advancing understanding of complex dynamics.

Contribution

It provides new criteria linking renormalization periods and rotation numbers to the local connectivity properties of Julia and Mandelbrot sets, refining previous theoretical conditions.

Findings

Mandelbrot set is locally connected at certain parameters under specific conditions.

Julia set is not locally connected when certain rotation number conditions are met.

The results extend and refine earlier work by Douady, Hubbard, and Milnor.

Abstract

For an infinitely renormalizable quadratic map $f_c: z\mapsto z^2+c$ with the sequence of renormalization periods ${k_m}$ and rotation numbers ${t_m=p_m/q_m}, we prove that if $\limsup k_m^{-1}\log |p_m|>0$, then the Mandelbrot set is locally connected at $c$. We prove also that if $\limsup |t_{m+1}|^{1/q_m}<1$ and $q_m\to \infty$, then the Julia set of $f_c$ is not locally connected and the Mandelbrot set is locally connected at $c$ provided that all the renormalizations are non-primitive (satellite). This quantifies a construction of A. Douady and J. Hubbard, and weakens a condition proposed by J. Milnor. Abstract of the Addendum: We improve one of the main results of the above paper.