The paper is concerned with the dynamics of the entire functions \(E_\lambda(z)=\lambda \exp z= \exp(z+\kappa)\) where \(\lambda=\exp\kappa\). Let \(S\) be the set of integer sequences \(s=s_1s_2s_3\dots\) and let \(\sigma:S\to S\) be the shift map: \(\sigma(s_1s_2s_3\dots)=s_2 s_3 s_4\dots\). A dynamic ray (or external ray) is an injective curve \(g_s:(0,\infty)\to {\mathbf C}\) satisfying \(\lim_{t\to\infty} \operatorname{Re} g_s(t) =\infty\), \(E_\lambda(g_s(t))=g_{\sigma(s)}(e^t-1)\), and \(g_s(t)=t-\kappa+2\pi i s_1+r_s(t)\), where \(r_s(t)< 2 e^{-t}(| \kappa| +C)\) for some constant \(C\). Such dynamic rays are known to exist for bounded (and in fact more general) sequences \(s\). A dynamic ray is said to land if \(\lim_{t\to 0} g_s(t)\) exists. The limit is then called a landing point.
The paper gives a thorough study of dynamic rays and their landing properties. In particular it is shown that periodic rays always land if the orbit of \(0\) is bounded. On the other hand, it is shown that in many dynamically important cases every repelling or parabolic periodic point is a landing point of a periodic dynamic ray. It is also shown that if \(0\) is preperiodic, then \(0\) is the landing point of at least one preperiodic dynamic ray.
Note: Using results by D. Schleicher on the parameter space of exponentials, L. Rempe has recently shown that periodic dynamic rays always land.