论文标题
余弦图的拓扑动力学
Topological dynamics of cosine maps
论文作者
论文摘要
在余弦地图的迭代中逃到无限的一组点，即$ c_ {a，b} \ colon z \ colon z \ mapsto ae^z+be^z+be^Z+be^{ - z} $ for $ a，b \ in \ in \ in \ mathbb {c}^\ ast $，由名为Dynerive dynaptimic rays组成。如果$ c_ {a，b} $的临界值将逃到无穷大，则其某些动态射线在关键点上对\ textit {split}重叠。我们考虑了一个巨大的余弦地图，其中包括临界值，包括地图$ z \ mapsto \ cosh（z）$。我们为他们在朱莉娅集合的动态提供了明确的拓扑模型。我们首先为任何余弦图附近的动力学提供模型，然后对其进行修改以反映我们研究的子类功能的射线的分裂。作为一个应用程序，我们对$ z \ mapsto \ cosh（z）$之间的重叠进行了明确的组合描述，并得出结论，其动态射线中没有两个。
The set of points that escape to infinity under iteration of a cosine map, that is, of the form $C_{a,b} \colon z \mapsto ae^z+be^{-z}$ for $a,b\in \mathbb{C}^\ast$, consists of a collection of injective curves, called dynamic rays. If a critical value of $C_{a,b}$ escapes to infinity, then some of its dynamic rays overlap pairwise and \textit{split} at critical points. We consider a large subclass of cosine maps with escaping critical values, including the map $z\mapsto \cosh(z)$. We provide an explicit topological model for their dynamics on their Julia sets. We do so by first providing a model for the dynamics near infinity of any cosine map, and then modifying it to reflect the splitting of rays for functions of the subclass we study. As an application, we give an explicit combinatorial description of the overlap occurring between the dynamic rays of $z\mapsto \cosh(z)$, and conclude that no two of its dynamic rays land together.
