Questions tagged [julia-set]

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How does a connected Julia set imply a member of the Mandelbrot Set?

I'm doing an introductory online course in complex analysis. In one of the lectures its stated that a complex number $c$ belongs to the Mandelbrot Set iff the Julia set $J(z^2 + c)$ is connected. I ...
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2answers
973 views

If I have zeros at the vertices of an icosahedron, where should the poles go?

I've been tinkering with Newton's method applied to polynomials. E.g., Newton's method for $z^5 - 1 = 0$ gives: There aren't a lot of symmetric patterns of finite sets of points in the plane, so I ...
4
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2answers
231 views

Hausdorff dimension of Julia set

Can anyone show me the proof "Hausdorff dimension of Julia set is strictly positive"? For purpose to prove this we might have to prove the green function of basin of attraction to infinity ...
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0answers
81 views

Can anyone prove that an IFS cubic Julia Mandelbrot set is not a "real" Mandelbrot set?

This question is the "null question": my proof that these Julia sets are three attraction set Mandelbrot sets is two fold: The Point at Infinity sets of cubic Julia Mandelbrot sets are ...
10
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1answer
304 views

Convex Julia sets

Consider the classical Julia set $J_f$ associated with $f(z)=z^2+c$. Since $J_c$ is completely invariant, we know that $f^{-1}(J_f) \subseteq J_f$. Now, let $H_f$ be the convex hull of $J_f$. Is it ...
1
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1answer
97 views

Is the set of non-escaping points in a Julia set always totally disconnected?

I am looking for examples of transcendental entire functions $f:\mathbb C\to \mathbb C$ such that the set of non-escaping points in the Julia set of $f$ is not totally disconnected. I denote this set $...
5
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2answers
729 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
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1answer
145 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 ...
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5answers
4k 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 ...
13
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0answers
290 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
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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. ...