ABSTRACT

Consider a meromorphic function f with a countable compact set of essential singularities and whose Fatou set consists only of finitely many cycles of Leau domains and finitely many families of invariant Baker domains and their preimages. We provide sufficient conditions on a sequence of meromorphic functions $\{f_n\}$ with only Leau domains and uniformly convergent on compact sets to f, so that the Julia sets of $f_n$ converge in the Hausdorff metric to the Julia set of f. In particular, the Leau domains of $f_n$ approximate in the sense of Carathèodory, either the Leau or the Baker domains of f.

中文翻译：

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Leau 和 Baker 域的 Carathèodory 收敛

摘要

考虑亚纯函数f具有可数紧致的基本奇点集，其 Fatou 集仅由有限多个 Leau 域循环和有限多个不变 Baker 域家族及其原像组成。我们提供了一系列亚纯函数的充分条件$\{F_n\}$只有 Leau 域并且在紧集上一致收敛到f，因此 Julia 集$F_n$在 Hausdorff 度量中收敛到f的 Julia 集。尤其是 Leau 域$F_n$在 Carathèodory 的意义上近似，f的 Leau 域或 Baker 域。