14 - Escaping points of the cosine family

Summary

Abstract. We study the dynamics of iterated cosine maps E: z ↦ ae^z + 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 and sometimes full planar Lebesgue measure.

INTRODUCTION

The dynamics of iterated polynomials has been investigated quite successfully, particularly in the past two decades. The study begins with a description of the escaping points: those points which converge to ∞ under iteration. It is well known that the set of escaping points is an open neighborhood of ∞ which can be parametrized by dynamic rays. The Julia set can then be studied in terms of landing properties of dynamic rays.

For entire transcendental functions, the point ∞ is an essential singularity (rather than a superattracting fixed point as for polynomials). This makes the investigation of the dynamics much more difficult. In particular, there is no obvious structure of the set of escaping points.