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Renormalization, equipotential annuli and the Hausdorff measure
Abstract
For a complex single variable polynomial f of degree d, let K be its filled Julia set, i.e., the union of all bounded orbits. Assume that K has an invariant component K* on which f acts as a degree d*<d map. This is a simplest instance of holomorphic polynomial-like renormalization (Douady-Hubbard). One can associate a certain Cantor-like subset G' of the circle with K*; it is defined as the set of arguments of all smooth or broken rays to K*. We will describe a role the Hausdorff dimension of G' and the respective Hausdorff measure play in geometry of K*. In particular, we give upper and lower bounds on the modulus of renormalization in terms of the Hausdorff measure of K*.
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BibTeX
@misc{blokh2026renormalization,
title = {{Renormalization, equipotential annuli and the Hausdorff measure}},
author = {Alexander Blokh and Lex Oversteegen and Vladlen Timorin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.20188},
eprint = {2606.20188},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.20188},
}
RIS
TY - UNPB
TI - Renormalization, equipotential annuli and the Hausdorff measure
AU - Alexander Blokh
AU - Lex Oversteegen
AU - Vladlen Timorin
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2606.20188
UR - https://arxiv.org/abs/2606.20188
AB - For a complex single variable polynomial f of degree d, let K be its filled Julia set, i.e., the union of all bounded orbits. Assume that K has an invariant component K^* on which f acts as a degree d_*<d map. This is a simplest instance of holomorphic polynomial-like renormalization (Douady-Hubbard). One can associate a certain Cantor-like subset G' of the circle with K^*; it is defined as the set of arguments of all smooth or broken rays to K^*. We will describe a role the Hausdorff dimension of G' and the respective Hausdorff measure play in geometry of K^*. In particular, we give upper and lower bounds on the modulus of renormalization in terms of the Hausdorff measure of K^*.
ER -