Clifford--Noor Orbits: A Generalization of the Mandelbrot Set via Three-Step Iterations in Geometric Algebra

In this paper, we introduce and study Clifford–Noor orbits by combining the geometric product of Clifford algebra with the three-step Noor iteration scheme, which include both the classical Mandelbrot set and its Clifford extension as special cases. As the parameters decrease toward zero, the iteration approaches the identity map and the fractal structures progressively smooth out, as illustrated through numerical experiments. Several applications of the results are discussed. This framework generates a three-parameter family of higher-dimensional fractals, opening new directions in fractal geometry, geometric algebra, data ascience and iterative dynamical systems.