发表机构
School of Mathematical Sciences, East China Normal University（华东师范大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文用双曲面积与覆盖几何统一证明并推广了无游荡定理，覆盖亚纯映射、整函数及椭圆多项式斜积，并给出高维局部与全局无游荡准则。
AI 中文摘要
我们利用双曲面积和覆盖几何来研究游荡Fatou分量与奇异轨道。我们给出了有理函数无游荡定理以及Bergweiler问题9中已知无游荡结果的证明，且不依赖拟共形形变。该框架还排除了具有紧奇异集且其导出集位于Fatou集中的整函数的游荡现象。对于Bergweiler问题8，超越亚纯映射的每个游荡分量都存在一个迭代子序列，在局部一致收敛意义下趋于无穷。对于Bergweiler问题4，我们在删除任意有限个原始奇异值作为轨道生成元后，建立了无奇异Baker周期附近的双曲分离性。对于Bergweiler问题11，相应的有限删除边界估计在游荡尾部成立。在高维情形，一个相对面积定理排除了有界基上的多项式斜积的有界游荡分量，其中基上存在全支撑的绝对连续不变概率测度。将该定理应用于Siegel旋转域上的椭圆多项式斜积，得到局部无游荡定理，无需对纤维临界轨道或纤维Julia集的连续性施加条件。一个全局准则覆盖了$\u005cmathbb P^2$上的正则斜积，其有界基Fatou分量最终进入周期Siegel盘，包括$(\u006cambda z+z^d,q(z,w))$，其中$\u006cambda$为Brjuno数，$q$为全次数至多$d$且$w^d$系数非零的多项式。
英文摘要
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 $\mathbb P^2$ 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.
Comments58 pages, This manuscript develops the hyperbolic-area approach introduced in the earlier preprint arXiv:2609.23834v1 (https://arxiv.org/abs/2609.23834v1) and includes additional results
