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Holomorphic motions, Assouad dimension and quasiconformal mappings

  • Katheryn Menssen
  • Malik Younsi
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Abstract

We study the variation of the quasi-Assouad dimension of a set moving under a holomorphic motion. We show that the reciprocal of the quasi-Assouad dimension is inf-harmonic in the sense of Fuhrer--Ransford--Younsi. As a consequence, we obtain quasiconformal distortion bounds for quasi-Assouad dimension as well as an improved version of Smirnov's celebrated theorem on the dimension of quasicircles. Our approach is elementary in that it does not require optimal Sobolev regularity for quasiconformal mappings.

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Cite

BibTeXRIS
BibTeX
@misc{menssen2026holomorphic,
  title = {{Holomorphic motions, Assouad dimension and quasiconformal mappings}},
  author = {Katheryn Menssen and Malik Younsi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.19522},
  eprint = {2609.19522},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.19522},
}
RIS
TY  - UNPB
TI  - Holomorphic motions, Assouad dimension and quasiconformal mappings
AU  - Katheryn Menssen
AU  - Malik Younsi
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.19522
UR  - https://arxiv.org/abs/2609.19522
AB  - We study the variation of the quasi-Assouad dimension of a set moving under a holomorphic motion. We show that the reciprocal of the quasi-Assouad dimension is inf-harmonic in the sense of Fuhrer--Ransford--Younsi. As a consequence, we obtain quasiconformal distortion bounds for quasi-Assouad dimension as well as an improved version of Smirnov's celebrated theorem on the dimension of quasicircles. Our approach is elementary in that it does not require optimal Sobolev regularity for quasiconformal mappings.
ER  -