Hyperbolic Area Methods in Meromorphic Dynamics and Elliptic Polynomial Skew Products
Abstract
We use hyperbolic area and covering geometry to study wandering Fatou components and singular orbits. We give proofs without quasiconformal deformation of the rational no-wandering theorem and the known no-wandering results in Bergweiler's Question~9. The framework also excludes wandering for entire functions with compact singular sets whose derived sets lie in the Fatou set. For Bergweiler's Question~8, every wandering component of a transcendental meromorphic map has a subsequence of iterates converging locally uniformly to infinity. For Bergweiler's Question~4, we establish hyperbolic separation near singular-free Baker cycles after deleting any finite set of original singular values as orbit generators. For Bergweiler's Question~11, corresponding finite-deletion boundary estimates hold along wandering tails. In higher dimension, a relative-area theorem excludes bounded wandering components of polynomial skew products over bounded bases carrying an absolutely continuous invariant probability of full support. Applied to elliptic polynomial skew products over Siegel rotation domains, it yields a local no-wandering theorem without conditions on the fiber critical orbits or continuity of fiber Julia sets. A global criterion covers regular skew products on ℙ2 whose bounded base Fatou components eventually enter periodic Siegel disks, including (λz+zd,q(z,w)) for Brjuno λ and polynomial q of total degree at most d with a nonzero wd coefficient.
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BibTeX
@misc{ye2026hyperbolic,
title = {{Hyperbolic Area Methods in Meromorphic Dynamics and Elliptic Polynomial Skew Products}},
author = {Zihao Ye},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30067},
eprint = {2609.30067},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30067},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hyperbolic Area Methods in Meromorphic Dynamics and Elliptic Polynomial Skew Products
AU - Zihao Ye
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30067
UR - https://arxiv.org/abs/2609.30067
AB - We use hyperbolic area and covering geometry to study wandering Fatou components and singular orbits. We give proofs without quasiconformal deformation of the rational no-wandering theorem and the known no-wandering results in Bergweiler's Question~9. The framework also excludes wandering for entire functions with compact singular sets whose derived sets lie in the Fatou set. For Bergweiler's Question~8, every wandering component of a transcendental meromorphic map has a subsequence of iterates converging locally uniformly to infinity. For Bergweiler's Question~4, we establish hyperbolic separation near singular-free Baker cycles after deleting any finite set of original singular values as orbit generators. For Bergweiler's Question~11, corresponding finite-deletion boundary estimates hold along wandering tails. In higher dimension, a relative-area theorem excludes bounded wandering components of polynomial skew products over bounded bases carrying an absolutely continuous invariant probability of full support. Applied to elliptic polynomial skew products over Siegel rotation domains, it yields a local no-wandering theorem without conditions on the fiber critical orbits or continuity of fiber Julia sets. A global criterion covers regular skew products on ℙ^2 whose bounded base Fatou components eventually enter periodic Siegel disks, including (λz+z^d,q(z,w)) for Brjuno λ and polynomial q of total degree at most d with a nonzero w^d coefficient.
ER -