A Generalized Mandelbrot Set for Bicomplex Numbers¹

Dominic Rochon

June, 2000

Résumé

We use a commutative generalization of complex numbers called bicomplex numbers to introduce bicomplex dynamics. In particular, we give a generalization of the Mandelbrot set and of the ``filled-Julia" sets in dimensions three and four. Also, we establish that our version of the Mandelbrot set with quadratic polynomial in bicomplex numbers of the form $w^{2}+c$ is identically the set of points where the associated generalized ``filled-Julia" set is connected. Moreover, we prove that our generalized Mandelbrot set of dimension four is connected.

Keywords: Bicomplex numbers, Mandelbrot set, Julia set.
AMS subject classification: 00A69, 58F13, 30D05.