Definition:Multibrot Set

Definition

Let $n \in \Z$ be an integer such that $n > 2$

The multibrot $n$ set $M_n$ 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:

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

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

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

is bounded.

Graphics

Multibrot $4$

The following is a depiction of the complete multibrot $4$ set.

The multibrot $4$ set itself is white.

The region of the complex plane depicted is the rectangle $\closedint {-2.2} {2.2} \times \closedint {-1.65 i} {1.65 i}$.

The iterated mapping here is $z \mapsto z^4 + c$

Also see

• Definition:Mandelbrot Set, the set of which this is a generalization

• Results about multibrot sets can be found here.

Source of Name

This entry was named for Benoît B. Mandelbrot.

Online Fractal Generator

Fractals presented on this page were generated using this online fractal generator.

Sources

There are no source works cited for this page. Source citations are highly desirable, and mandatory for all definition pages. Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted. To discuss this page in more detail, feel free to use the talk page.