Articles of complex dynamics

Why does the mandelbrot set and its different variants follow similar patterns to epi/hypo trochodis and circular multiplication tables?

So the $z^2 + c$ variant has a cardioid shape at the center. This shape is made by an epitrochoid with a ratio of the radi being one, or from the two times table when we display it in a circle (as seen in this video The next variation $z^3 +c$ has a nephroid […]

Mandelbrot set: periodicity of secondary and subsequent bulbs as multiples of their parent bulbs

In the Mandelbrot set, all points of the main carodioid are asymptotic (that is, the iterations of c^2 + c approach a constant). In contrast, it seems that all bulbs have a periodicity greater than 1, that is, the iterations settle into a cycle with a certain period. There are several questions to be asked […]

The density — or otherwise — of $\{\{2^N\,\alpha\}:N\in\mathbb{N}\}$ for ALL irrational $\alpha$.

Problem Is there an irrational $\alpha\in\mathbb{R}\backslash\mathbb{Q}$ such that the set $S= \{\,\{2^N\alpha\} :N\,\in\mathbb{N}\}$ is not dense in $[0,1]$. Here $\{x\}=x-\lfloor x\rfloor$ is the fractional part of $x$. Questions very similar to this have been asked in the past. For example Multiples of an irrational number forming a dense subset However right now I can’t adjust […]

About fractional iterations and improper integrals

Let $g(x,0) = x$ and $g(x,t+1) = g(x,t) – \dfrac{1}{g(x,t)}$ for every real $t$. From the fact \begin{align} \int_{-\infty}^{\infty}f\left(x-\frac{1}{x}\right)dx&=\int_{0}^{\infty}f\left(x-\frac{1}{x}\right)dx+\int_{-\infty}^{0}f\left(x-\frac{1}{x}\right)dx=\\ &=\int_{-\infty}^{\infty}f(2\sinh T)\,e^{T}dT + \int_{-\infty}^{\infty}f(2\sinh T)\,e^{-T}dT=\\ (collecting\space terms ) &=\int_{-\infty}^{\infty}f(2\sinh T)\,2\cosh T\,d T=\\ &=\int_{-\infty}^{\infty}f(x)\,dx. \end{align} as discussed here : Why is this integral $\int_{-\infty}^{+\infty} F(f(x)) – F(x) dx = 0$? I am tempted to conclude $\int_{-\infty}^{\infty}f(g(x,t))=\int_{-\infty}^{\infty}f(x)\,dx.$ for […]

Are mini-Mandelbrots known to be found in any fractals other than the Mandelbrot set itself?

This is a generalization of the question Are there mini-mandelbrots inside the julia set? @Hagen raises an issue I was afraid of, which is that even the mini-Mandelbrots in the Mandelbrot set are not exact copies. I don’t know enough to give a rigorous answer to this question, but I’m thinking of whatever (implicit) standard […]

Why Study Critical Polynomials?

In dynamical systems, I often read about the post-critical orbits. As in take a moduli space of functions $f$ which are self maps. Find general critical points, and see where they orbit. They would then be polynomials in some variables if we allow $f$ to be parameterised. Those are called critical polynomials. It could go […]

Help locating mini mandelbrots

I would like to be able to list the coordinates of all the first level minibrots. Here is a picture of the mandelbrot set generated by fraqtive: zooming in to the circled area we see a slightly distorted copy of the original fractal the julia set associated to the point right in the middle of […]

Do there exist periodic fractals $A_f$ of this type?

Let $z$ be a complex number. Meromorphic here means meromorphic on all of the complex plane $C$. Lets define a fractal $A_f$ on the complex plane as the result of iterating a meromorphic function $f(z)$ an infinite amount of times and then coloring according to the value. Thus $A_f(z)=f(f(f(…f(z)…)))$ and the color is decided upon […]

Is $\frac{1}{\exp(z)} – \frac{1}{\exp(\exp(z))} + \frac{1}{\exp(\exp(\exp(z)))} -\ldots$ entire?

Let $z$ be a complex number. Is the alternating infinite series $$ f(z) = \frac{1}{\exp(z)} – \frac{1}{\exp(\exp(z))} + \frac{1}{\exp(\exp(\exp(z)))} -\dotsb $$ an entire function ? Does it even converge everywhere ? Additional questions (added dec 16) Consider the similar case for $z$ being real or having a small imaginary part: $$ g(z) = \frac{1}{z} + […]

Properties of the Mandelbrot set, accessible without knowledge of topology?

Are there any properties of the Mandelbrot set that can be analysed without a knowledge of complicated topology? Considering the fact that the set is based on a quadratic function, are there any interesting properties of the set that can be proved using algebra or relatively simple geometry? If there are, then what are those […]