Multibrot 集的计算与共形分析
A Computational and Conformal Analysis of Multibrot Sets
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中文总结 AI 辅助
本文通过像素计数和共形映射上界估计 Multibrot 集的面积，发现其亏缺随次数呈幂律衰减，且网格估计低于共形上界。
中文摘要 AI 辅助
对于单临界族 $f_{c,j}(z)=z^j+c$，Multibrot 集 $M_j$ 在 $j\to\infty$ 时几何上收敛于闭单位圆盘，但这一极限并未量化其有限次数的面积行为。我们使用逃逸时间像素计数来估计 Area($M_j$)，同时分离迭代阈值 $K$ 和空间采样密度 $ρ$ 的影响。有限迭代集构成嵌套递减序列，而网格细化则通过经验方式处理。在 $K\in\{25,100,500,1000\}$ 和 $ρ\in\{128,256,512,1024\}$ 范围内，最终的网格细化使报告的面积变化至多为 $1.1\times10^{-3}$。在 $K=1000$ 和 $ρ=1024$ 时，估计值很好地由 $C(j)\approxπ-2.49216j^{-0.57103}$ 描述。独立地，外部共形映射的 Laurent 系数和 Gronwall 面积公式给出了有限截断上界 $U_{1000}(j)\approxπ-2.07405j^{-0.643427}$。在所有测试的次数中，网格估计值均低于相应的共形上界，且两个序列都表现出从 $π$ 的面积亏缺的相似幂律衰减。
英文摘要
For the unicritical family $f_{c,j}(z)=z^j+c$, the Multibrot sets $M_j$ converge geometrically to the closed unit disk as $j\to\infty$, but this limit does not quantify their finite-degree area behavior. We estimate Area($M_j$) using escape-time pixel counting while separating the effects of the iteration threshold $K$ and spatial sampling density $ρ$. The finite-iteration sets form a nested decreasing sequence, while grid refinement is treated empirically. Across $K\in\{25,100,500,1000\}$ and $ρ\in\{128,256,512,1024\}$, the final grid refinement changes the reported areas by at most $1.1\times10^{-3}$. At $K=1000$ and $ρ=1024$, the estimates are well described by $C(j)\approxπ-2.49216j^{-0.57103}$. Independently, Laurent coefficients of the exterior conformal map and Gronwall's area formula yield finite-truncation upper bounds $U_{1000}(j)\approxπ-2.07405j^{-0.643427}$. Across all tested degrees, the grid estimates remain below the corresponding conformal bounds, and both sequences exhibit similar power-law decay of the area deficit from $π$.
