The ``Filled-Julia" Set for the Tetrabrot

The same process as for the Tetrabrot yields a version of the ``filled-Julia" set in $\mathbb {R}^{3}$. We define the ``filled-Julia" set for the Tetrabrot.

Definition 7   The associated ``filled-Julia" set for the Tetrabrot is defined as follows: ( $c\in\mathbb {C}_{2}$)

$$\mathcal{L}_{2,c}=\{w=w_1+w_2i_2\in\mathbb {C}_{2}:Im(w_2)=0... ..._{c}^{\circ n}(w) \mbox{ is bounded }\forall n\in\mathbb {N}\}.$$

Figure 15 is an illustration of the ``filled-Julia" set for the Tetrabrot at the same point $c=-1.754878$ as the ``filled-Julia" set B of Fig. 1. Hence, Fig. 15 is a kind of generalization of the ``filled-Julia" set $\mathcal{K}_{c}$ in the complex plane. In the same manner, Figure 16 is the generalization of Fig. 1.C, and Fig. 17 the generalization of Fig. 1.D.

In [1], L. Carleson and T. W. Gamelin have remarked this interesting fact: ``One striking feature of $\mathcal{M}$ is that shapes of certain of the Julia sets $\mathcal{J}_{c}$ in dynamic space (z-space) are reflected in the shape (c-shape).". For the Tetrabrot, we seem to have something similar. For example, Fig. 18 is Fig. 17 with the same kind of cut as for the Tetrabrot in Fig. 7. Hence we see that inside Fig. 17 we have the same shape as inside the Tetrabrot near the point $c=0,25$. It is also possible to see the same phenomenon with the ``filled-Julia" set of Fig. 16. This phenomenon has been illustrated in Fig. 19 where we have put together the border of the Tetrabrot and the associated ``filled-Julia" set at the point $c=-1.16-0.25i_1$ on the border. We see clearly that this ``filled-Julia" set imitates the border of the Tetrabrot.

Finaly, in Figs. 20, 21, 22 and 23 we show the ``filled-Julia" set at $c=i_1$ for different infinite divergence layers. We remark that Fig. 23 is a good approximation of this set.