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On the Hausdorff dimension of the set of non-escaping points in the Julia set
Abstract
For an entire function f and a non-zero complex number λ, let fλ(z)=f(λz). We give conditions on f which imply that the Hausdorff dimension of the set of non-escaping points in the Julia set of fλ tends to $1$ as |λ|→ 0. In fact, we give an upper bound for this dimension in terms of λ. This generalizes earlier results concerned with the case that f(z)=exp z.
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BibTeX
@misc{bergweiler2026hausdorff,
title = {{On the Hausdorff dimension of the set of non-escaping points in the Julia set}},
author = {Walter Bergweiler and Weiwei Cui and Lingrui Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15412},
eprint = {2609.15412},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15412},
}
RIS
TY - UNPB
TI - On the Hausdorff dimension of the set of non-escaping points in the Julia set
AU - Walter Bergweiler
AU - Weiwei Cui
AU - Lingrui Wang
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15412
UR - https://arxiv.org/abs/2609.15412
AB - For an entire function f and a non-zero complex number λ, let f_λ(z)=f(λz). We give conditions on f which imply that the Hausdorff dimension of the set of non-escaping points in the Julia set of f_λ tends to $1$ as |λ|→ 0. In fact, we give an upper bound for this dimension in terms of λ. This generalizes earlier results concerned with the case that f(z)=exp z.
ER -