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A holomorphic motion with non-subharmonic Hausdorff dimension
Abstract
We present an example of a holomorphic motion of a Cantor set λ↦ Aλ for which neither the Hausdorff dimension dimH(Aλ) nor the lower Minkowski dimension dimMl(Aλ) is a subharmonic function of~λ. This answers negatively a question of Fuhrer--Ransford--Younsi, and distinguishes these dimensions from packing and upper Minkowski dimensions, for which subharmonicity holds. All four dimensions are computed explicitly.
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BibTeX
@misc{ransford2026holomorphic,
title = {{A holomorphic motion with non-subharmonic Hausdorff dimension}},
author = {Thomas Ransford and Malik Younsi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22644},
eprint = {2610.22644},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22644},
}
RIS
TY - UNPB
TI - A holomorphic motion with non-subharmonic Hausdorff dimension
AU - Thomas Ransford
AU - Malik Younsi
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22644
UR - https://arxiv.org/abs/2610.22644
AB - We present an example of a holomorphic motion of a Cantor set λ↦ A_λ for which neither the Hausdorff dimension dimH(A_λ) nor the lower Minkowski dimension dimMl(A_λ) is a subharmonic function of~λ. This answers negatively a question of Fuhrer--Ransford--Younsi, and distinguishes these dimensions from packing and upper Minkowski dimensions, for which subharmonicity holds. All four dimensions are computed explicitly.
ER -