A generalized Mandelbrot set for bicomplex numbers (Q2709705)

The author considers the set \(\mathbb C_2\) of so-called bicomplex numbers \(a+i_1b+i_2c+jd\), where \(a, b, c, d \in \mathbb R\), \(i_1^2=i_2^2=-1\), \(j^2=1\), \(i_2j=ji_2=-i_1\), \(i_1j=ji_1=-i_2\), \(i_2i_1=i_1i_2=j\). Such a number can be written as \((a+i_1b)+i_2(c+i_1d)=z_1+i_2z_2\), where \(z_1\), \(z_2 \in \mathbb C_1 := \{ x+i_1y : i_1^2=-1 \}\) so that \(\mathbb C_2\) can be viewed as the complexification of the usual complex numbers \(\mathbb C\). Then \(\mathbb C_2\) is a commutative unitary ring. Another representation of a bicomplex number is given by \(z_1+i_2z_2=(z_1-i_1z_2)e_1+(z_1+i_1z_2)e_2\), where \(e_1=(1+j)/2\) and \(e_2=(1-j)/2\). Then a set \(X \subset \mathbb C_2\) is called a \(\mathbb C_2\)-Cartesian product of the sets \(X_1\), \(X_2 \subset \mathbb C_1\) if \(X = X_1 \times_e X_2 := \{ w_1e_1+w_2e_2 : (w_1,w_2) \in X_1 \times X_2 \}\). NEWLINENEWLINENEWLINEFor \(c \in \mathbb C_2\) consider the quadratic polynomial \(P_c(w)=w^2+c\), \(w \in \mathbb C_2\), and denote by \(P_c^n\) the \(n\)-th iterate of \(P_c\). Then the author defines the generalized Mandelbrot set \(\mathcal M_2\) as the set of all \(c \in \mathbb C_2\) such that the sequence \((P_c^n(0))_{n=1}^\infty\) is bounded. The first result states that \(\mathcal M_2\) is connected. This is an immediate consequence of the fact that \(\mathcal M_2 = \mathcal M \times_e \mathcal M\) (where \(\mathcal M\) is the usual Mandelbrot set) and the well-known but deep result that \(\mathcal M\) is connected. Also, the author introduces the generalized filled Julia set \(\mathcal K_{2,c}\) of \(P_c\) as the set of all \(w \in \mathbb C_2\) such that the sequence \((P_c^n(w))_{n=1}^\infty\) is bounded, and he proves that \(c \in \mathcal M_2\) if and only if \(\mathcal K_{2,c}\) is connected. NEWLINENEWLINENEWLINEFurthermore, the author considers the so-called Tetrabrot \(\mathcal T\) which is a certain 3-dimensional slice of the \(4\)-dimensional Mandelbrot set \(\mathcal M_2\), and he gives a method to produce \(3\)-dimensional computer graphics of \(\mathcal T\). At least one third of the article is covered with such nice pictures. Finally, the author conjectures that \(\mathcal T\) is disconnected, and he gives some hints why this could be true.