42
votes
Did Gaston Julia ever get to see a computer-generated image of his eponymous set?
Not a definitive answer, but a close upper bound.
The same paper that had the first published computer-generated image of the
Mandelbrot set also includes an image of the Julia set (no idea if it's ...
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33
votes
Did Gaston Julia ever get to see a computer-generated image of his eponymous set?
It is very unlikely that Gaston Julia saw a computer-generated image of a julia set. The first images were obtained at the beginning of the eighties.
Note also that this would have been far less ...
- 18.5k
26
votes
Why are the Julia sets so simple? (quadratic family)
Julia sets are all very closely related to self-similar sets - each one can be thought of as the invariant set of something like an iterated function system. Specifically, the Julia set of $f(z)=z^2 +...
- 1,898
22
votes
Accepted
Poincaré metric on the Riemann sphere minus more than two points
Yes. The density of the Poincare metric with respect to the spherical metric is
a positive continuous function which tends to infinity at the punctures. Thus it
is bounded from below by some positive ...
- 88.8k
21
votes
Is the area of the Mandelbrot set known?
New Approximations for the Area of the Mandelbrot Set gives the state of the art from 2014, and a related paper from 2015 is On a numerical approximation of the boundary structure and of the area of ...
- 178k
17
votes
Accepted
On entire functions with polynomial Schwarzian derivative
The answer is this:
$$f(z)=\int_{z_0}^z e^{Q(\zeta)}d\zeta,$$
where $Q$ is a polynomial, and this is the general form of
an entire function whose Schwarzian is a polynomial. The crucial fact that $f$ ...
- 88.8k
16
votes
Did Gaston Julia ever get to see a computer-generated image of his eponymous set?
Quoting from SciAm's article Who Discovered the Mandelbrot Set?:
In 1976, he [John H. Hubbard of Cornell University] explains, he began using a computer to map out sets of complex numbers generated ...
- 416
15
votes
Why are the Julia sets so simple? (quadratic family)
To extend my comment and emphasize the self-similarity of Julia sets and the Dragon curve, here is an interpolation between the two.
Each frame is generated by two complex functions,
...
- 15.6k
14
votes
On complex dynamics in high dimensions
Main areas are dynamics of automorphisms (for example, Henon maps), dynamics of endomorphisms, dynamics of foliations, and local dynamics.
Eric Bedford, Tien-Cuong Dinh, John Fornaess, Misha Lyubich, ...
- 88.8k
13
votes
Exponential towers of $i$'s
The single-valued case of the exponential tower of $i$ was studied by Peter Lynch. The tower converges along three spirals to the fixed point $Q=i^Q$, given in terms of the Lambert W-function by
$$Q=\...
- 178k
13
votes
Accepted
Exponential towers of $i$'s
Let $f : \mathcal{P}(\mathbb{C}) \rightarrow \mathcal{P}(\mathbb{C})$ be the function on the powerset of the complex numbers defined by:
$$ f(A) = \left\{ \exp\left( \frac{\pi i}{2}(4n + 1)z \right) :...
- 12.2k
12
votes
Convergence of Newton's method
I don't think your initial assertion is accurate. Consider, for example, $f(z)=z^5-z-1$. If you iterate the Newton's method function $N(z) = z-f(z)/f'(z)$ from $z_0=0$, you'll quickly find an ...
- 1,898
11
votes
If I have zeros at the vertices of an icosahedron, where should the poles go?
MR1032073
Doyle, Peter; McMullen, Curt,
Solving the quintic by iteration.
Acta Math. 163 (1989), no. 3-4, 151–180.
- 88.8k
11
votes
Accepted
Tiling the plane with finitely many congruent pieces
Write $A_i=T_i(A)$ for $A=A_1$, where each $T_i$ is a rigid motion. For each $i$, we have $|A_i\cap B(r)|=|T_i(A\cap T_i^{-1}(B(r))|=|A\cap T_i^{-1}(B(r))|$. The symmetric difference between this set ...
- 27.4k
10
votes
Is the area of the Mandelbrot provably computable?
Your statement that the area of the union of bounded hyperbolic components is lower-computable is very plausible.
So if, as conjectured, hyperbolic parameters have full measure, then the area is ...
- 6,455
10
votes
Did Gaston Julia ever get to see a computer-generated image of his eponymous set?
I have a recollection of having seen a computer picture due to Cherry, an Autralian mathematician, who died in 1966.
You would think I would have noted exactly where and when, but I didn't. Does ...
- 101
10
votes
Why are the Julia sets so simple? (quadratic family)
As @GNiklasch points out, you seem to be zooming into two places which are both preimages of the same repelling periodic point. So the images of the Julia set are locally related by a conformal map, ...
- 6,455
10
votes
Accepted
An entire function all whose forward orbits are bounded
Given an entire function $f\colon\mathbb{C}\to\mathbb{C}$, the escaping set, $I(f)$, is the set of $z\in\mathbb{C}$ such that $f^n(z)\to\infty$. Per the Wikipedia article, the escaping set of a non-...
- 3,204
10
votes
Who proved that the Mandelbrot set's Julia sets are locally connected?
Nobody. This is the principal unsolved problem in the area, which is called MLC
(That the Mandelbrot set is locally connected). Two Fields medals were awarded for
partial progress in this problem.
...
- 88.8k
10
votes
Convex Julia sets
Edited: The previously found sufficient condition is indeed necessary, but even better, it is satisfied by all polynomials of degree at least two. Thus the conjecture is true:
Theorem: Let $p$ be a ...
- 3,659
10
votes
Accepted
Does the Mandelbrot set have dense interior?
The answer is positive and this is not difficult (a normal families argument). The boundary
of the Mandelbrot set is the set of $J$-instability. Every point $c_0$ of this set is a limit of $c_n$ such ...
- 88.8k
9
votes
Accepted
Symmetries for Julia sets of perturbations of polynomial maps
Yes. Let $\zeta=e^{2\pi i/(k+n)}$. Then $f(\zeta^j z)=\zeta^{nj}f(z)$. Iterating this, the orbit of $\zeta^j z$ agrees with the orbit of $z$ up to multiplying by powers of $\zeta$. In particular, $z$ ...
- 22.5k
9
votes
Cutting a Julia set into infinitely many pieces at finitely many points
For a polynomial, this is equivalent to asking whether there can be infinitely many external rays landing at a single point. This could happen only if the function has a Cremer point (i.e., a non-...
- 6,455
9
votes
Renormalization in physics vs. dynamical systems
The renormalization approach to dynamical systems pioneered by Chen, Goldenfeld and Oono [1] applies the Gell-Mann and Low renormalization group from quantum physics [2] to extract the global behavior ...
- 178k
8
votes
Accepted
A question about Julia set for quadratic family
I expect that you will find the answer (and a lot more) in the following paper, since it's easy to figure out which quadratic polynomials (if any) commute with one another:
Commuting polynomials and ...
- 45.7k
8
votes
Accepted
Why are the Julia sets so simple? (quadratic family)
Given that I have started making pictures, I thought it might be worthwhile adding another, shorter, direct answer to your questions, in addition to my longer, more detailed one.
Question 1. Are the ...
- 6,455
8
votes
Dynamics of Riemann zeta function
One can observe that $s=-0.295905\ldots$ is an attracting fixed point.
One can do computations which give pictures of the Mandelbrot set of this iteration. (points are those which have after 20 ...
- 2,725
8
votes
Accepted
Ahlfors' proof of Bloch's theorem
He explains his choice in lines 8-9 on p. 364 of the paper:
"This metric has the curvature $-4$ for it is obtained from the hyperbolic metric by the transformation $w'=w^{1/2}$ ."
Remarks. ...
- 88.8k
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