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I'm searching for a rigorous proof that $c$ is in the Mandelbrot set $M$ if and only if its Julia set $J_c$ is connected. I've read this question and answer , I understand the answer that is given however, first, I don't know how to prove the assertion he mentioned :
"the preimage of a curve wil...
In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets (Julia 'laces' and Fatou 'dusts') defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is 'regular', while on the Julia set its behavior is 'chaotic...
