Questions tagged [julia-set]

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5
votes
2answers
433 views

Who proved that the Mandelbrot set's Julia sets are locally connected?

I'd be greatly interested in a reference to the respective article. Was it Douady? Julia? Hubbard? Fatou? Bonus question: Who gave the proof that can be found in the Orsay notes? EDIT: The question ...
5
votes
1answer
114 views

Is the function Point -> Julia set “injective”?

Consider the functions $f_c(z) := z^2 + c$ for $c \in \mathbb C$. For each such function, we may form the associated Julia set. My question: If $c, c' \in \mathbb C$ produce in this way the same Julia ...
24
votes
5answers
3k views

Why are the Julia sets so simple? (quadratic family)

I want to know why, when I look at the Julia sets of the quadratic family, I see only a finite number of repeating patterns, rather than a countable infinity of them. My question is specifically ...
12
votes
0answers
252 views

Diophantine approximation in the Julia set

Let $f : \mathbb{CP}^1 \to \mathbb{CP}^1$ be a rational map of degree $q > 1$; or just a quadratic binomial $z^2 + c$, if one prefers. The Julia set $J_f$ is the closure of the repelling periodic ...
8
votes
1answer
1k views

Area of filled Julia sets

The recent question Area of the boundary of the Mandelbrot set ? prompted me to ask this question. There has been some work on estimates for the area of the Mandelbrot set, e.g., a paper by John H. ...