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 ...
- 91.8k
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 ...
- 188k
18
votes
Accepted
Can the topologist's sine curve be realized as a Julia set?
The answer is negative. Since every neighborhood of a point
on the Julia set is mapped onto the whole Julia set by some iterate of the rational function, it follows that if a small piece of the Julia ...
- 91.8k
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 $f$ ...
- 91.8k
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, ...
- 91.8k
13
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 ...
- 2,083
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
$$Q=\...
- 188k
13
votes
Accepted
Exponential towers of $i$'s
Let $f : \mathcal{P}(\mathbb{C}) \rightarrow \mathcal{P}(\mathbb{C})$ be the function on the powerset of the complex numbers defined by:
$$ f(A) = \left\{ \exp\left( \frac{\pi i}{2}(4n + 1)z \right) :...
- 12.4k
12
votes
Connected set in a filled Julia set
Not if the Julia set is an interval. Consider the Chebyshev polynomial $f(z)=z^2-2$ for example whose Julia set/filled Julia set is given by $[-2,2]$. The only connected subset containing Julia points ...
- 3,599
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.
- 91.8k
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 ...
- 28.2k
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 ...
- 6,548
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-...
- 3,274
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.
...
- 91.8k
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 ...
- 3,709
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 ...
- 91.8k
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, $z$ ...
- 23.2k
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-...
- 6,548
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 ...
- 188k
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 ...
- 2,705
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. ...
- 91.8k
8
votes
Accepted
What is the logical complexity of the Mandelbrot Local Connectivity conjecture? (Is it equivalent to a statement of arithmetic?)
Here is an argument that MLC is equivalent to an arithmetic statement. Let $M$ denote the Mandelbrot set, and let $\mathbb{C}_{\mathbb{Q}}$ denote the set of rational complex numbers. First note that ...
- 12.4k
7
votes
Accepted
Newton method and Siegel disks
Implementing Alexandre's suggestion, I found that for $\lambda\approx 0.483096 + 1.00504\ i$, we have
$$
\begin{align}
P_{\lambda}(z) &= (z - 1)\ (z + 1/2 - \lambda)\ (z + 1/2 + \lambda) \\
&...
- 2,083
7
votes
Convergence of Newton's method
Your statement that iterates of the Newton method converge to a cycle almost everywhere is equivalent to the statement that for every polynomial $f$
the Julia set of the rational function $z-f(z)/f'(z)...
- 91.8k
7
votes
Accepted
Smooth Julia set for quadratic polynomials
The answer to a) is yes, and this was proved by Fatou in 1919.
Sur les équations fonctionnelles
Bulletin de la S. M. F., tome 48 (1920), p. 208-314.
There are many generalizations of this fact. For ...
- 91.8k
7
votes
Who proved that the Mandelbrot set's Julia sets are locally connected?
If you mean
are the Julia sets that correspond to parameter values of the Mandelbrot set connected, then I do believe that both Julia and Fatou proved this (in 1918 and 1916 respectively).
are the ...
- 11.8k
7
votes
Non-locally connected polynomial Julia sets
There is a necessary and sufficient condition due to Yoccoz in terms of continued fraction expansion. The condition is that $\lambda=e^{2\pi i \theta}$ should be such that $\theta$ is not a Brjuno ...
- 2,154
6
votes
Accepted
Linearizing a power series by conjugation
You may find useful information in a recent article by D. Sauzin and al. "Explicit linearization of one-dimensional germs through tree-expansions" here, where they use "mould calculus" (introduced by ...
- 5,428
Only top scored, non community-wiki answers of a minimum length are eligible