The Exponent Set: A Third Natural Extension of the Mandelbrot-Julia Framework

Abstract

The Mandelbrot and Julia sets, generated by the quadratic iteration $z_{n+1}=z_n^2+c$, are foundational objects in complex dynamics. We study the three-variable principal-value complex-power iteration $z_{n+1}=z_n^x+c$, where $z_0,c,x\in \mathbb{C}$. The triples producing all-time well-defined and bounded orbits form a locus $\mathcal{B}\subset {\mathbb{C}}^3$. Fixing two coordinates yields three natural families of coordinate fibers, denoted $M(z_0,x)$, $J(c,x)$, and $E(z_0,c)$. For $x=2$, $M(0,2)$ is the classical Mandelbrot set, $J(c,2)$ is the classical filled Julia set, and $\partial J(c,2)$ is the classical Julia set. We focus on the Exponent Set, or E-Set, obtained by fixing $(z_0,c)$ and varying the complex exponent $x$. We prove three groups of structural results. First, for explicit parameter families, including pure-power real and unit-circle cases and an additive example with nonzero real and imaginary parts, no finite universal escape radius exists: for every prescribed radius, one can choose an exponent whose bounded orbit makes a finite excursion beyond that radius. Second, we construct a boundary point at which an extended-valued escape-time function is discontinuous for one strict threshold. Third, we prove a vertical boundedness asymmetry in which the principal-argument convention enters explicitly.

34 pages, 3 figures; 6 numerical illustrations and 1 color legend