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Rigidity and non local connectivity of J...
Levin, Genadi...

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Rigidity and non local connectivity of Julia sets of some quadratic polynomials by Levin, Genadi ( Author )
Lemonjar Open Access
N.A
07-08-2023
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.
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Article
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English
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MYR 0.01
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http://arxiv.org/abs/0710.4406
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