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

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

2007-10-24

arxiv.org/abs/0710.4406v2

Mathematics

Dynamical Systems

Final version. To appear in Commun. Math. Phys

Scientific paper

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.

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