The Generalized ``Filled-Julia" Set

It is now interesting to see what happens with the Julia set. First, we recall the definition of the ``filled Julia" set in the complex plane:

Definition 5   The ``filled-Julia" set is defined as follows: $(c\in\mathbb {C})$

$$\mathcal{K}_{c}=\{z\in\mathbb {C}: P_{c}^{\circ n}(z)\mbox{ is bounded }\forall n\in\mathbb {N}\}.$$

Moreover, the Julia set $\mathcal{J}_{c}:=\partial\mathcal{K}_{c}$.

We recall also the following beautiful relationship between the Mandelbrot set and the ``filled-Julia" set:

Theorem 4   $c\in\mathcal{M}\Leftrightarrow\mathcal{K}_{c}$ is connected.

It is possible to generalize the ``filled-Julia" set for bicomplex numbers:

Definition 6   The generalized ``filled-Julia" set for bicomplex numbers is defined as follows: $(c\in\mathbb {C}_{2})$

$$\mathcal{K}_{2,c}=\{w\in\mathbb {C}_{2}: P_{c}^{\circ n}(w)\mbox{ is bounded } \forall n\in\mathbb {N}\}.$$

The next lemma gives a charaterization of the ``filled-Julia" set for bicomplex numbers in terms of the ``filled-Julia" set for the complex plane. This lemma will be useful to give an analogue of Theorem 4 for the bicomplex numbers.

Lemma 2   $\mathcal{K}_{2,c}=\mathcal{K}_{2,(c_1-c_2i_1)e_1+(c_1+c_2i_1)e_2}= \mathcal{K}_{c_1-c_2i_1}\times_{e}\mathcal{K}_{c_1+c_2i_1}$.

Proof. The proof is along the same lines as the proof of the Lemma 1.$\Box$

Theorem 5   $c\in\mathcal{M}_{2}\Leftrightarrow\mathcal{K}_{2,c}$ is connected.

Proof. By Lemma 2, we know that $\mathcal{K}_{2,c}=\mathcal{K}_{c_1-c_2i_1}\times_{e}\mathcal{K}_{c_1+c_2i_1}$. Also, by the homeomorphism in the proof of Theorem 2, $\mathcal{K}_{c_1-c_2i_1}\times_{e}\mathcal{K}_{c_1+c_2i_1}$ is connected if and only if $\mathcal{K}_{c_1-c_2i_1}\times\mathcal{K}_{c_1+c_2i_1}$ is connected. Then, $\mathcal{K}_{c_1-c_2i_1}\times_{e}\mathcal{K}_{c_1+c_2i_1}$ is connected if and only if $\mathcal{K}_{c_1-c_2i_1}$ and $\mathcal{K}_{c_1+c_2i_1}$ are connected. Hence, by Theorem 4, $\mathcal{K}_{2,c}$ is connected if and only if $c_1-c_2i_1$, $c_1+c_2i_1\in\mathcal{M}_{1}$. Therefore, by Lemma 1, $\mathcal{K}_{2,c}$ is connected if and only if $c=(c_1-c_2i_1)e_{1}+(c_1+c_2i_1)e_{2}\in\mathcal{M}_{2}$.$\Box$