2
votes
0answers
25 views

Blow up in holomorphic dynamics

Someone could explain the concept of blow up used in holomorphic dynamics? Especifaclly in the context of iteratation of holomorphics functions. This concept could be taken to some of the deformation ...
3
votes
0answers
59 views

Showing iterates of a complex function on the upper half plane converges uniformly on compact sets

The following is an irksome problem that my complex analysis class is having trouble solving: Let $*$ be an operator that takes a function $F:\mathcal{H}\to\mathcal{H}$ to a function ...
1
vote
1answer
105 views

Demonstrating that the Mandelbrot Set is connected

I know that demonstrating the Mandelbrot Set is connected requires a non-trivial proof, and that Mandelbrot himself was fooled at first. But can it be demonstrated visually that the set is connected? ...
7
votes
3answers
336 views

Why computing Fatou coordinate is so hard?

I'm trying to make images of Fatou coordinate for some polynomial maps. If I'm not wrong there is no explicit general formula/method for computing Fatou coordinate near parabolic fixed point. Is ...
5
votes
2answers
196 views

Dynamics of the repetitions of $f(z) = z^{2} +\frac{1}{4}$

This is probably a classic and most likely an easy question, but I was not able to find an answer. If we have the iteration $z_{n+1}=g(z_n)$, where $$g(z) = z + a z^{2},$$ with $a\ne 0$, then using a ...
2
votes
2answers
127 views

M-set interior point probability on the real axis

For the real axis, the Mandelbrot set consists of points from $[-2,0.25]$. Some of these points are in the interior of the m-set, and some are on the boundary. Those points in the interior are ...
1
vote
1answer
156 views

What is the shape of external rays landing on fixed points in case of quadratic discrete dynamical system?

In case of parabolic discrete dynamical system based on the complex quadratic polynomial fc(z) = z^2 + c some external rays land on alfa fixed point. Hera are 34 external rays landing on fixed ...
6
votes
2answers
209 views

How to figure out the starting point for this Mandelbrot?

My answer for another math stackexchange question, asked by Gottfried, involved observing Mandelbrot bifurcation for the iterated function in question, $f(z)\mapsto z-\log_b(z)$. In particular, for ...
5
votes
0answers
128 views

two fixed points, same fractional iteration

Suppose a function f(z) has two fixed points, one repelling, and the other attracting. Call the repelling fixed point f(-1)=-1, and the attracting fixed point f(+1)=+1. I'm interested in functions ...
2
votes
2answers
275 views

What is the shape of parabolic critical orbit?

The parabolic critical orbits of discrete dynamical system form n-th arm stars : which shapes are conjugated with "regular" n-th arm stars Here are 2 images of parabolic critical orbits for 2 ...
0
votes
2answers
64 views

Is $g(z)=\frac{1}{z}+\frac{1}{z^2+1}+\frac{1}{(z^2+1)^2 +1}+…$ analytic for $|z|>2$?

Let $z$ be a complex number. Let |.| denote be the absolute value. Let $n$ be a positive integer. Let $f_1(z)=z^2+1$. Let $f_n(z)=f_1(f_{n-1}(z)).$ Is ...
1
vote
1answer
72 views

Simplify the nonlinear system of dynamic equations

I am working with a set of nonlinear dynamic equations that Mathematica has problems with solving. It is of the form $$ f_1(x_{t+1},y_{t+1},x_{t},y_t) = g_1(x_{t+1},y_{t+1},x_{t},y_t),\\ ...
1
vote
0answers
66 views

About growth rate of the iterated exponential on the complex plane.

Let $n$ be a positive integer. Let $f(n,x) = exp(f(n-1,x))$ and $f(0,x)=x$. Let $Q(f(n,x)) =1$ if $Re(f(n,x))<2$. Let $S(n,x) = \Sigma_{1}^{n} Q(f(n,x))$. How to estimate $S(n,2+i)$ efficiently ? ...
2
votes
0answers
65 views

Is the half iterate of $2\sinh(x)$ - expanded from fixed point $0$ - analytic near the real line $\mathbb{R}$?

Is the half iterate of $2\sinh(x)$ - expanded from fixed point $0$ - analytic near the real line $\mathbb{R}$? I know this function has 2 other fixed points apart from $0$, so I'm not sure. Also ...
4
votes
2answers
154 views

Computing a Julia set

In Ermentrout's book he computes the Julia set for $z \mapsto z^2 + c$ by starting with a point inside the unit circle and then randomly choosing iterates $z\mapsto \pm\sqrt{z-c}$. Since the Julia set ...
3
votes
2answers
361 views

Classification of points in the Mandelbrot set

I am trying to understand the classification of points in the Mandelbrot set. There are an infinite number of baby Mandelbrots, each associated with a defined set of landing rays. There are the pre ...
0
votes
1answer
123 views

Question about iterations : $g(c,z) = a$ always has a solution where $z$ is strictly real?

Let $c$ be any positive real number. Let $z$ be a complex number. Let $g(c,z)$ be some locally analytic function that is the $c$ th iteration of the entire function $f(z)$ with $g(0,z)=f(z)$. Let ...
4
votes
1answer
192 views

Fixed Point of a complex dynamical spiral system

Last semester I finished my first class on complex variables and of course we had to show that $i^i$ was real. That got me wondering about quantities like $i^{i^i}$ and similar power towers. For my ...
0
votes
1answer
117 views

Boundary of $x^2+x$ Julia set

How do you calculate an infinite fixed point for $f=x^2+x$, so that $f^{o n}(x)$ never repeats, and doesn't go to infinity and doesn't go to zero? Can such a sequence of points densely cover the ...
9
votes
3answers
351 views

Quadratic Julia sets and periodic cycles

Consider the function $f_c(z) = z^2 + c$. Applying this function repeatedly, we get the familiar quadratic Julia sets that fractal enthusiasts burn compute cycles plotting. Infinity is always one ...
4
votes
1answer
308 views

Every basin of attraction contains a critical point?

Years and years ago, back when I first became interested in fractals [but didn't know much about anything], I vaguely remember coming across an interesting theorem. The gist of it was that "every ...