Definition:Mandelbrot Set

Definition

Definition 1

The Mandelbrot set $M$ is the subset of the complex plane defined as follows:

Let $c \in \C$ be a complex number.

Let $T_c: \C \to \C$ be the complex function defined as:

$\forall z \in \C: \map {T_c} z = z^2 + c$

Then $c \in M$ if and only if the sequence:

$\tuple {0, \map {T_c} 0, \map { {T_c}^2} 0, \ldots}$

is bounded.

Definition 2

The Mandelbrot set $M$ is the subset of the complex plane defined as follows:

Let $c \in \C$ be a complex number.

Let $T_c: \C \to \C$ be the complex function defined as:

$\forall z \in \C: \map {T_c} z = z^2 + c$

Then $M$ is the set of points $c$ for which the Julia set of $T_c$ is connected.

Also see

• Results about the Mandelbrot set can be found here.

Source of Name

This entry was named for Benoît B. Mandelbrot.