Topological dynamics of cosine maps

TL;DR

This paper studies the complex dynamics of cosine maps, especially those with escaping critical values, providing a topological model for their Julia sets and analyzing the behavior of dynamic rays, including their overlaps and landing properties.

Contribution

It introduces an explicit topological model for the dynamics of a broad class of cosine maps with escaping critical values, including the hyperbolic cosine, and analyzes the structure of their dynamic rays.

Findings

Dynamic rays form a collection of injective curves in the Julia set.

In certain cosine maps, some dynamic rays overlap and split at critical points.

For the map z→cosh(z), no two dynamic rays land together.

Abstract

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.