Escaping Points and Symbolic Dynamics 4 3 Tails of Dynamic Rays 8 4 Dynamic Rays 14 5 Eventually Horizontal Escape 17 6 Classification of Escaping Points 21 7

We study the dynamics of iterated cosine maps E: z 7→ aez + be−z, with a, b ∈ C \ {0}. We show that the points which converge to ∞ under iteration are organized in the form of rays and, as in the exponential family, every escaping point is either on one of these rays or the landing point of a unique ray. Thus we get a complete classification of the escaping points of the cosine family, confirming a conjecture of Eremenko in this case. We also get a particularly strong version of the “dimension paradox”: the set of rays has Hausdorff dimension 1, while the set of points these rays land at has not only Hausdorff dimension 2 but infinite planar Lebesgue measure.