ON A GENERALIZED FATOU–JULIA THEOREM IN MULTICOMPLEX SPACES

In this article we introduce the hypercomplex 3D fractals generated from Multicomplex Dynamics. We generalize the well known Mandelbrot and filled-in Julia sets for the multicomplex numbers (i.e. bicomplex, tricomplex, etc.). In particular, we give a multicomplex version of the so-called Fatou-Julia theorem. More precisely, we present a complete topological characterization in ℝ2n of the multicomplex filled-in Julia set for a quadratic polynomial in multicomplex numbers of the form w2 + c. We also point out the symmetries between the principal 3D slices of the generalized Mandelbrot set for tricomplex numbers.

EXTENSION OF SINE–GORDON FIELD THEORY FROM GENERALIZED CLIFFORD ALGEBRAS

Linearization of homogeneous polynomials of degree n and k variables leads to generalized Clifford algebras. Multicomplex numbers are then introduced in analogy to complex numbers with respect to the usual Clifford algebra. In turn multi-complex extensions of trigonometric functions are constructed in terms of "compact" and "non-compact" variables. It gives rise to the natural extension of the d-dimensional sine–Gordon field theory in the n-dimensional multicomplex space. In two dimensions, the cases n = 1, 2, 3, 4 are identified as the quantum integrable Liouville, sine–Gordon and known deformed Toda models. The general case is discussed.