Fractals

Table of Contents

Notation

Mandelbrot set

The Mandelbrot set is the set complex numbers for which the function

\begin{equation*} f_c(z) = z^2 + c \end{equation*}

does not diverge when iterated from $z = 0$, i.e. remains bounded in absolute value.

Formally, the Mandelbrot set is the set of values of $c$ in the complex plane for which the orbit of $0$ under iteration of the quadratic map

\begin{equation*} z_{n + 1} = z_n^2 + c \end{equation*}

remains bounded.

The Mandelbrot set $M$ is defined by a family of complex quadratic polynomials

\begin{equation*} P_c : \mathbb{C} \to \mathbb{C} \end{equation*}

given by

\begin{equation*} P_c : z \mapsto z^2 + c \end{equation*}

where $c$ is a complex parameter. For each $c$, one considers the behavior of the sequence

\begin{equation*} \Big( 0, P_c(0), (P_c \circ P_c)(0), \dots \Big) \end{equation*}

obtained by iteration $P_c(z)$ starting at critical point $z = 0$. This either escapes to infinity or stays within a disk of finite radius; the Mandelbrot set is then the set of all points $c$ such that the sequence stays within a disk of finite radius.

More formally, if $P_c^n(z)$ denotes the n-th iterate of $P_c(z)$ (i.e. $P_c(z)$ composed with itself $n$ times), the Mandelbrot set is the subset of the complex plane given by

\begin{equation*} M = \{ c \in \mathbb{C} : \exists s \in \mathbb{R}, \forall n \in \mathbb{N}, \ |P_c^n(0)| \le s \} \end{equation*}

As explained below, it is in fact possible to simplify this definition by taking $s = 2$.

Visualizations of Mandelbrot sets

  • Color all points $c$ that belong to the Mandelbrot set $M$ black and all other points white

Connection to Julia sets

An equivalent definition of the Mandelbrot set is as the subset of the complex plane consisting of those parameters $c$ for which the Julia set of $P_c$ is connected.

Julia set

Let $f(z)$ be a holomorphic function from the Riemann sphere to itself.

Such $f(z)$ are precisely the complex rational functions, that is

\begin{equation*} f(z) = \frac{p(z)}{q(z)} \end{equation*}

where $p(z)$ and $q(z)$ are complex polynomials.

Assume that $p$ and $q$ are non-constant, and that they have no common roots (i.e. $q$ does not divide $p$). Then there is a finite number of open sets $F_1, \dots, F_r$, that are left invariant by $f(z)$ and are such that:

  1. $\bigcup F_i$ is dense in the plane
  2. $f(z)$ behaves in a regular and equal way on each of the sets $F_i$

The last statement means that the termini of the sequences of iterations generated by the points of $F_i$ are either

  • the same set => finite cycle
  • $F_i$ are finite cycles of circular or annular shaped sets centered at the same point

These sets $F_i$ are the Fatou domains of $f(z)$, and their union is the Fatou set $F(f)$ of $f(z)$.

Further, each of the Fatou domains contains at least on critical point of $f(z)$, i.e. a (finite) point $z$ such that

\begin{equation*} f'(z) = 0 \quad \text{ or } \quad z = \infty \end{equation*}

if either of the following is the case:

  • degree of the numerator $p(z)$ is at least two larger than the degree of the denominator $q(z)$
  • $f(z) = 1 / g(z) + c$ for some $c$ and a rational function $g(z)$ satisfying this condition

The complement of the Fatou set $F(f)$ is the Julia set $J(f)$ of $f(z)$.

$J(f)$ is a nowhere dense set (it is without interior points) and an uncountable set (of the same cardinality as the real numbers).

Like $F(f)$, $J(f)$ is left invariant by $f(z)$, and on this set the iteration is repelling, meaning that

\begin{equation*} | f(z) - f(w) | > |z - w| \end{equation*}

for all $w$ in a neighborhood of $z$ (within $J(f)$).

This means that $f(z)$ behaves chaotically on the Julia set.

There are points in the Julia set whose sequence of iterations is finite, there are only a countable number of such points (and thus they only make up a infinitely small part of the Julia set). The sequences generated by points outside this set behave chaotically, a phenomenon called deterministic chaos.

Equivalent definitions of Julia sets are:

  • $J(f)$ is the smallest closed set containing at least three points which is completely invariant under $f$ (Möbius transformation?)
  • $J(f)$ is the closure of the set of repelling periodic points

  • For all but at most two points $z \in X$, the Julia set is the set of limit points of the full backwards orbit

    \begin{equation*} \bigcup_n f^{-n}(z) \end{equation*}

    i.e. limit points of the union of pre-images of all the iterations.

  • If $f$ is an entire function, then $J(f)$ is the boundary of the set of points which converge to infinity under iteration.
  • If $f$ is a polynomial, then $J(f)$ is the boundary of the filled Julia set: i.e. those points whose orbits under iterations of $f$ remain bounded.

Julia set of a Mandelbrot set

Mandelbrot sets are given by

\begin{equation*} f_c(z) = z^2 + c \end{equation*}

where $c$ is a complex parameter.

The Julia set of for this system is the subset of complex plane given by:

\begin{equation*} J(f_c) = \{ z \in \mathbb{C} : \forall n \in \mathbb{N}, \ |f_c^n(z)| \le 2 \} \end{equation*}

where $f_c^n(z)$ is the n-th iterator of $f_c(z)$, i.e. $f_c^n(z) = (f_c \circ \dots \circ f_c)(z)$, n times.