All Questions
Tagged with complex-dynamics julia-set
13 questions
7
votes
2
answers
186
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Non-locally connected polynomial Julia sets
What are some examples of complex polynomials whose Julia sets are connected, but not locally?
In the book Complex Dynamics by Carleson and Gamelin, I found:
They seem to reference:
But what is a ...
- 6,571
5
votes
1
answer
550
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Connected set in a filled Julia set
Let $K$ be the filled Julia set of a complex polynomial of degree at least 2. Suppose that $K$ is connected. Let $p_1, \dots, p_N \in K$ be some points. Does there exist a connected set $K_N$ ...
- 63.9k
2
votes
1
answer
118
views
Uniformization of Julia sets and lacunary series
If the complement of a Julia set of quadratic polynomial z^2+c is locally connected and simply connected, it is uniformized by the complement of the unit disk. Consider the uniformization map and its ...
CommunityBot
- 1
2
votes
1
answer
115
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Dense orbits for a rational map
Given a complex rational function $f$ and $z\in \mathbb C$, let $O^+(z)=\{f^n(z):n\geq 1\}$. Let $$D(f)=\big\{z\in \mathbb C:\overline{O^+(z)}=J(f)\big\}.$$
So $D(f)$ is the set of points whose (...
- 11.8k
5
votes
1
answer
120
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Jordan curve boundaries of Fatou components
Let $f:\mathbb C\to \mathbb C$ be a rational map and let $J(f)$ and $F(f)$ denote the Julia and Fatou sets of $f$, respectively.
Let $\mathcal S$ be the set of all boundaries of Fatou components. ...
- 28.1k
16
votes
3
answers
1k
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 ...
- 2,370
9
votes
2
answers
1k
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Is this a Julia set (and if so, for which function family is it the Julia set)?
Consider the function family given by $f_\lambda(z) = z - p_\lambda(z)/p_\lambda'(z)$ where $p_\lambda(z) = (z^2 - 1)(z - \lambda)$. Every attracting cycle and every rational neutral cycle of $f_\...
- 4,713
4
votes
2
answers
432
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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 ...
- 6,548
10
votes
1
answer
419
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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 ...
- 3,709
1
vote
1
answer
165
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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 $...
CommunityBot
- 1
5
votes
2
answers
1k
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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 ...
- 91.8k
27
votes
5
answers
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 ...
- 106
13
votes
0
answers
311
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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 ...
- 13.8k