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

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.