Visualization of Mandelbrot and Julia Sets of Möbius Transformations

Abstract

This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function η(z)=z2+λ. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented.

The Proof of a Conjecture Relating Catalan Numbers to an Averaged Mandelbrot-Möbius Iterated Function

“…Remark 1. We remark that Mork and Ulness [16] posed a slightly different conjecture. In fact, we can express their question by defining h…”

“…Very recently, Mork and Ulness [16] continued the previous line of research by dealing with the so-called j-averaged Mandelbrot set which is a set generated by iterating a function obtained by composing the function η λ and the Möbius transformation µ A (z) = az+b cz+d , where A = (a, b, c, d) ∈ C 4 . Thus,…”

“…The name "j-averaged" is used here since the points of the resulting fractal are colored according to the total number of members of the following sequence of iterations (H n ) 0≤n≤j , that escaped from the circle with radius 2 (the concrete algorithm for coloring of points of this fractal you can find in Appendix 1 of [16]), see Figure 1,…”

“…2, 3, 4 (the first row from the left to the right) and for and j = 5, 7, 10, 100 (the second row from the left to the right). We used functions in the software Mathematica ® (see [17]) that are defined in Appendix 1 of [16].…”

“…Mork and Ulness ( [16] Theorem 1) proved that the j-averaged Mandelbrot set for the Möbius transformation µ A with A = (0, 1, 1, 0) has threefold rotational symmetry and dihedral mirror symmetry. Additionally, they raised a conjecture (see [16], Conjecture 2)) concerning the coefficients of these iterations. Before stating their conjecture, we introduce some basic notations.…”