A Computational and Conformal Analysis of Multibrot Sets
Abstract
For the unicritical family fc,j(z)=zj+c, the Multibrot sets Mj converge geometrically to the closed unit disk as j→∞, but this limit does not quantify their finite-degree area behavior. We estimate Area(Mj) 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∈{25,100,500,1000} and ρ∈{128,256,512,1024}, the final grid refinement changes the reported areas by at most 1.1×10−3. At K=1000 and ρ=1024, the estimates are well described by C(j)≈π−2.49216j−0.57103. Independently, Laurent coefficients of the exterior conformal map and Gronwall's area formula yield finite-truncation upper bounds U1000(j)≈π−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.
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BibTeX
@misc{senapati2026computational,
title = {{A Computational and Conformal Analysis of Multibrot Sets}},
author = {Rohan Senapati},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26837},
eprint = {2609.26837},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26837},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Computational and Conformal Analysis of Multibrot Sets
AU - Rohan Senapati
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.26837
UR - https://arxiv.org/abs/2609.26837
AB - 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→∞, 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∈{25,100,500,1000} and ρ∈{128,256,512,1024}, the final grid refinement changes the reported areas by at most 1.1×10^−3. At K=1000 and ρ=1024, the estimates are well described by C(j)≈π−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)≈π−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.
ER -