A Reduction Theorem for Clifford–Mandelbrot Sets with Real Exponent

Source abstract

Let V be an n-dimensional real inner product space, $\vec{n}\in V$ a unit vector and p > 1 a real exponent. We study the Clifford--Mandelbrot set $M_{p,\vec{n}}$, formed by the parameters $\vec{c}\in V$ for which the orbit of the origin under $\vec{x}\mapsto$ ($\vec{x}$ ⋄ $\vec{n}$)p ⋄ $\vec{n}$ + $\vec{c}$ is defined and bounded, where ⋄ is the geometric product. We prove that every such orbit stays in the plane spanned by $\vec{n}$ and $\vec{c}$, and that on this plane the map coincides with the principal branch of $z\mapsto z^{p}$. Consequently, $M_{p,\vec{n}}$ is the solid of revolution of the planar principal multibrot set about the $\vec{n}$-axis, so the construction does not produce genuinely higher-dimensional dynamics. The cause is structural: every map obtained by applying a function to $\vec{x}$ ⋄ $\vec{n}$ and multiplying by $\vec{n}$ behaves in the same way. From the reduction we derive the identity $\lVert F_{p}(\vec{x})\rVert=\lVert\vec{x}\rVert^{p}$, the escape radius $2^{1/(p-1)}$ (attained for even integers $p$), the exact intersection of $M_{p,\vec{n}}$ with the $\vec{n}$-axis, an inner ball of radius $\kappa_{p}=(1-1/p)\,p^{-1/(p-1)}$, the limiting behaviour as $p\to1^{+}$ and $p\to\infty$, the Hausdorff dimension of the boundary, and the discontinuity of the map across its branch locus. All statements are checked numerically.