Category:Definitions/Mandelbrot Set

This category contains definitions related to Mandelbrot Set.
Related results can be found in Category:Mandelbrot Set.

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 $c \in \C$ for which the Julia set of $T_c$ is connected in the extended complex plane $\overline \C$.