Definition:Mandelbrot Set/Definition 1

Definition

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.

Also see

• Equivalence of Definitions of Mandelbrot Set

• Results about the Mandelbrot set can be found here.

Source of Name

This entry was named for Benoît B. Mandelbrot.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Mandelbrot set (B. B. Mandelbrot, 1980)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Mandelbrot set (B.B. Mandelbrot, 1980)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Mandelbrot set