MANDELBROT SETS and ITS THREE ...

MANDELBROT SETS and ITS THREE-DIMENSIONAL SLICES Pierre-Olivier Paris´ e Supervised by Prof. Dominic Rochon D´epartement de math´ematiques et d’informatique

Introduction

Results

Operations on tricomplex numbers

Experiments have been done in [1] for the Mandelbrot set of an integer order p ≥ 2 in the complex plane. These sets are generated by iterating the polynomial Qp,c(z) = zp + c from the starting point z = 0. Other authors generalized the classical Mandelbrot set to higher dimensions. In 2000, Dominic Rochon [4] used the algebra of bicomplex numbers to generate the classical Mandelbrot set in the three-dimensinal (3D) space. He obtained what he called the Tetrabrot. Then, in 2009, V. Garant-Pelletier, and Dominic Rochon [5] used the algebra of tricomplex numbers to generalized the Mandelbrot set in eight dimensions. This generalization allowed to do more 3D slices in the Mandelbrot set. They showed, up to conjugacy classes, that there are only eight principal classes.

Goals

The following figures show some examples of 3D slices.

Let ζ = w1 + w2i3, and η = s1 + s2i3 be two tricomplex numbers. Addition of two tricomplex numbers is defined as ζ + η := (w1 + s1) + (w2 + s2)i3. Multiplication of two tricomplex numbers is defined as ζ · η := (w1s1 − w2s2) + (w1s2 + w2s1)i3. The norm of a tricomplex number is defined as q p kηk3 := |z1|2 + |z2|2 + |z3|2 + |z4|2 = kw1k22 + kw2k22. With these operations, M(3) is an commutative unitary ring, but it is not a division ring. Moreover, it is also a Banach space with complex scalar multiplication.

Subsets of tricomplex numbers

• To establish a theory of 3D slices of the Mandelbrot sets. • To classify the 3D principal slices of the Mandelbrot sets.

The Tricomplex closed ball is defined as

That is, we choose three different units from the set of the eight units, and we make a linear combination with it. There is one last set that is useful for the classification of the principal 3D slices of the Mandelbrot sets :

η := x1 + x2i1 + x3i2 + x4i3 + x5i4 + x6j1 + x7j2 + x8j3 where xi ∈ R. Figure 1 shows the relations between each units.

· 1 i1 i2 i3 i4 j1 j2 j3

1 1 i1 i2 i3 i4 j1 j2 j3

i1 i1 −1 j1 j2 −j3 −i2 −i3 i4

i2 i2 j1 −1 j3 −j2 −i1 i4 −i3

i3 i3 j2 j3 −1 −j1 i4 −i1 −i2

i4 i4 −j3 −j2 −j1 −1 i3 i2 i1

j1 j1 −i2 −i1 i4 i3 1 −j3 −j2

j2 j2 −i3 i4 −i1 i2 −j3 1 −j1

j3 j3 i4 −i3 −i2 i1 −j2 −j1 1

Fig. 1: Relations between each units

The set of tricomplex numbers is denoted by M(3). It can also have the following form η = z1 + z2i2 + z3i3 + z4j3 where z1 := x1 + x2i1, z2 := x3 + x6i1, z3 := x4 + x7i1, and z4 := x8 + x5i1 are complex numbers M(1). Finally, there is another form η = w1 + w2i3 where w1 := z1 + z2i2, and w2 := z3 + z4i2 are bicomplex numbers M(2) (see [2]).

Fig. 4: T 8(i1, i2, j2)

Fig. 5: T 2(i1, i2, j2)

We say that two slices T1p(im, ik, il), and T2p(in, iq, is) are equivalent if there exists a bijective linear map φ from spanR {im, ik, il} to spanR {in, iq, is} that conjugates Qp,c1 with p p −1 Qp,c2 , that is φ ◦ Qp,c1 ◦ φ = Qp,c2 for any c1 ∈ T1 (im, ik, il), and c2 ∈ T2 (in, iq, is). This is an equivalence relation, and we have the following result.

1. T 3(1, i1, i2) called Tetrabric;

3. T 3(1, i1, j1) called Hourglassbric;

2. T 3(1, j1, j2) called Perplexbric;

4. T 3(i1, i2, i3) called Metabric.

B3(ζ, r) := {η ∈ M(3) : kη − ζk3 ≤ r}. T(im, ik, il) := {x1ik + x2il + x3im | x1, x2, x3 ∈ R}.

A tricomplex number η has the following form

Fig. 3: T 6(i1, i2, j1)

Th´ eor` eme 2. For p = 3, the Mandelbrot set M33 has only four principal slices:

Later, we will need a specific set to make 3D slices in Mandelbrot sets :

Tricomplexes numbers

Fig. 2: T 2(1, i1, i2)

The reduction of the number of 3D slices is caused by the particularity of the degree of the polynomial ζ 3 + c. Indeed, for any ζ ∈ spanR {im, ik, il}, ζ 3 ∈ M(im, ik, il). On the other hand, for p = 2, the square of a number ζ ∈ spanR {im, ik, il} may spread overall M(3). Thus, the degree p = 3 has a special feature.

M(ik, il, im) := {x1ik + x2il + x3im + x4ikilim : xi ∈ R, i = 1, . . . , 4}. In this case, we choose three distinct units, but we also consider, in the linear combination, the unit generated by the multiplication of the three others. Fig. 6: Tetrabric

Fig. 7: Perplexbric

Fig. 8: Hourglassbric

Fig. 9: Metabric

Tricomplex Mandelbrot sets The tricomplex Mandelbrot set of integer order p ≥ 2 is defined as the set of c ∈ M(3) for which the sequence {c, cp + c, (cp + c)p + c, . . .} is bounded, that is n o  ∞ p M3 := c ∈ M(3) : Qm p,c(0) m=1 is bounded

where Qp,c(ζ) := ζ p + c. Some authors also call these sets Multibrot sets. It has been showed in [3] that Mp3 ⊂ B3(0, 21/(p−1)), Mp3 is a compact and connected set. The following theorem helps to generate the Mandelbrot sets in 3D. p M3

if, and Th´ eor` eme 1. A number c ∈ 1/(p−1) (0)k ≤ 2 for any natural number m. kQm 3 p,c

only

Conclusion The nature of the integer p seems to influence the number of principal 3D slices. We have a conjecture, based on computer experiments, that for any odd integer p, there are four principal 3D slices and for any even integer p, there are eight principal 3D slices. One way to approach the problem would consist of using the Coxeter’s theory. Indeed, the bijective linear maps are signed permutation matrices. A second approach would use the idempotent representation, a very useful representation of tricomplex numbers.

if, R´ ef´ erences

A 3D principal slice of

Mp3

is defined as

o n  ∞ p m T (im, ik, il) := c ∈ T(im, ik, il) : Qp,c(0) m=1 is bounded .

1. Gujar, U. G. & Bhavsar, V. C. Fractals from z ← z α + c in the complex c-plane. Comput. & Graphics 15, 441–449 (1991). 2. Price, G. B. An Introduction To Multicomplex Spaces And Fonctions (Marcel Dekker, INC, Monographs, textbooks on pure et applied mathematics, 1991). 3. Paris´e, P.-O. & Rochon, D. A Study of Dynamics of the Tricomplex Polynomial η p + c. NonLinear Dynam. 82, 157–171 (2014). 4. Rochon, D. A Generalized Mandelbrot Set For Bicomplex Numbers. Fractals 8, 355–368 (2000). 5. Garant-Pelletier, V. & Rochon, D. On a Generalized Fatou-Julia Theorem in Multicomplex Spaces. Fractals 17, 241–255 (2008).

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