AI 中文总结
该研究将曼德博-朱利亚框架扩展至三变量复幂迭代，聚焦指数集，证明其不存在有限通用逃逸半径、存在逃逸时间函数不连续点及垂直有界性不对称等结构性质。
AI 中文摘要
由二次迭代 $z_{n+1}=z_n^2+c$ 生成的曼德博集与朱利亚集是复动力学中的基础对象。我们研究三变量主值复幂迭代 $z_{n+1}=z_n^x+c$，其中 $z_0,c,x\in\mathbb{C}$，所有产生完全良定义且有界轨道的三元组构成轨迹 $\mathcal{B}\subset\mathbb{C}^3$。固定两个坐标可得到三类自然坐标纤维族，记为 $M(z_0,x)$、$J(c,x)$ 和 $E(z_0,c)$。当 $x=2$ 时，$M(0,2)$ 是经典曼德博集，$J(c,2)$ 是经典填充朱利亚集，$\partial J(c,2)$ 是经典朱利亚集。我们聚焦于固定 $(z_0,c)$ 并改变复指数 $x$ 得到的指数集（简称 E-Set），证明三组结构结果：第一，对于显式参数族，包括纯幂实族、单位圆族及含非零实虚部的加性示例，不存在有限通用逃逸半径——对任意给定半径，均可选取一个指数，使其有界轨道超出该半径；第二，构造了一个边界点，在该点处，扩展值逃逸时间函数对某一严格阈值不连续；第三，证明了垂直有界性不对称性，其中主辐角约定明确介入。
英文摘要
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.
Comments34 pages, 3 figures; 6 numerical illustrations and 1 color legend
