Equivalence of Definitions of Mandelbrot Set

Theorem

The following definitions of the concept of Mandelbrot Set are equivalent:

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 $\overline \C$.

Proof

By definition of the Julia set:

The Julia set of $f$ is the boundary of the set of those points in $\overline \C$ whose orbits under $f$ are bounded.

Definition $(1)$ implies Definition $(2)$

Let $M$ be the Mandelbrot set by definition $1$.

Let $c \in M$.

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Thus $M$ is the Mandelbrot set by definition $2$.

$\Box$

Definition $(2)$ implies Definition $(1)$

Let $M$ be the Mandelbrot set by definition $2$.

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Thus $M$ is the Mandelbrot set by definition $1$.

$\blacksquare$