Questions tagged [julia-set]
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19 questions
7
votes
1
answer
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Does the airplane Julia set contain true circles?
The well known "airplane" Julia set looks like it contains a true circle. To be precise, let $c$ be the real root of $x^3+2x^2+x+1=0$. i.e., $c\approx -1.75$. The Julia set of $z^2+c$ is the ...
- 105
3
votes
1
answer
146
views
Hausdorff dimension of non-smooth, connected quadratic Julia sets
Q. Is is true that if $c$ is a parameter in the Mandelbrot set such that the corresponding Julia set $J_c$ of $z^2+c$ is not smooth, then the Hausdorff dimension of $J_c$ is greater than 1?
This seems ...
- 105
7
votes
3
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307
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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 ...
- 3,691
5
votes
1
answer
702
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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 $K_N$ ...
- 303
3
votes
0
answers
127
views
Transformation of Julia set sequence emerging from meromorphic function
I consider a sequence of meromorphic functions on the Riemann sphere $f_k:\hat{\mathbb{C}} \to \hat{\mathbb{C}}$ for $k\in\mathbb{N}$ of the form
$$f_k(z)=\sum_{j=1}^{n_k}\dfrac{1}{(z-p_j)^{c_j}}$$
...
- 377
3
votes
1
answer
164
views
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 (...
- 3,691
2
votes
1
answer
160
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 ...
- 613
5
votes
1
answer
180
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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. ...
- 3,691
-1
votes
1
answer
256
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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 ...
- 11
16
votes
3
answers
2k
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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 ...
- 2,270
4
votes
2
answers
557
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 ...
- 51
10
votes
1
answer
474
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 ...
- 16.1k
1
vote
1
answer
234
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 $...
- 3,691
5
votes
2
answers
1k
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 ...
- 597
5
votes
1
answer
232
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 ...
- 597
27
votes
5
answers
5k
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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 ...
- 381
13
votes
0
answers
334
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 ...
- 13.9k
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_\...
- 709
10
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1
answer
2k
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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. ...
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