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The inverse problem for Julia limiting directions in the Eremenko--Lyubich class
Abstract
We study the inverse problem for Julia limiting directions in the Eremenko-Lyubich class ℬ, which is the class of transcendental entire functions with a bounded set of singular values, and its meromorphic generalization ℬmer. Every nonempty compact subset of the circle can be the set of Julia limiting directions of an entire f∈ℬ. For finite unions of nondegenerate closed arcs, the corresponding f∈ℬ, even with every finite order at or above an explicit bound determined by the arc lengths, can be attained. We also refine the series construction of the latter two authors to obtain f∈ ℬmer with any prescribed order and the set of Julia limiting directions.
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BibTeX
@misc{huang2026inverse,
title = {{The inverse problem for Julia limiting directions in the Eremenko--Lyubich class}},
author = {Jiaxing Huang and Jun Wang and Xiao Yao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39103},
eprint = {2610.39103},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39103},
}
RIS
TY - UNPB
TI - The inverse problem for Julia limiting directions in the Eremenko--Lyubich class
AU - Jiaxing Huang
AU - Jun Wang
AU - Xiao Yao
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39103
UR - https://arxiv.org/abs/2610.39103
AB - We study the inverse problem for Julia limiting directions in the Eremenko-Lyubich class ℬ, which is the class of transcendental entire functions with a bounded set of singular values, and its meromorphic generalization ℬ_mer. Every nonempty compact subset of the circle can be the set of Julia limiting directions of an entire f∈ℬ. For finite unions of nondegenerate closed arcs, the corresponding f∈ℬ, even with every finite order at or above an explicit bound determined by the arc lengths, can be attained. We also refine the series construction of the latter two authors to obtain f∈ ℬ_mer with any prescribed order and the set of Julia limiting directions.
ER -