In recent years, there has been significant progress in the understanding of the dynamics of transcendental entire functions with bounded postsingular set. In particular, for certain classes of such functions, a complete description of their topological dynamics in terms of a simpler model has been given inspired by methods from polynomial dynamics. In this paper, and for the 1st time, we give analogous results in cases when the postsingular set is unbounded. More specifically, we show that if $f$ is of finite order, has bounded criticality on its Julia set $J(f)$, and its singular set consists of finitely many critical values that escape to infinity and satisfy a certain separation condition, then $J(f)$ is a collection of dynamic rays or hairs, which split at critical points, together with their corresponding landing points. In fact, our result holds for a much larger class of functions with bounded singular set. Moreover, this result is a consequence of a significantly more general one: we provide a topological model for the action of $f$ on its Julia set.

中文翻译：

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具有超验全功能的分毛

近年来，在有界后奇异集的超越全函数动力学的理解方面取得了重大进展。特别是，对于某些类别的此类函数，在多项式动力学方法的启发下，已经根据更简单的模型给出了对其拓扑动力学的完整描述。在本文中，我们第一次在后奇异集无界的情况下给出了类似的结果。更具体地说，我们证明如果 $f$ 是有限阶的，在其 Julia 集 $J(f)$ 上具有有限临界，并且其奇异集由有限的许多临界值组成，这些临界值逃逸到无穷大并满足特定的分离条件，那么 $J(f)$ 是动态射线或毛发的集合，它们在临界点以及它们相应的着陆点处分裂。实际上，我们的结果适用于更大的一类有界奇异集的函数。此外，这个结果是一个明显更普遍的结果的结果：我们为 $f$ 在其 Julia 集上的动作提供了一个拓扑模型。