The Dynamical Fine Structure of Iterated Cosine Maps and a Dimension Paradox

Abstract

We discuss in detail the dynamics of maps $z\mapsto ae^z+be^{-z}$ for which both critical orbits are strictly preperiodic. The points which converge to $\infty$ under iteration contain a set $R$ consisting of uncountably many curves called ``rays'', each connecting $\infty$ to a well-defined ``landing point'' in $\C$, so that every point in $\C$ is either on a unique ray or the landing point of finitely many rays. The key features of this paper are the following two: (1) this is the first example of a transcendental dynamical system where the Julia set is all of $\C$ and the dynamics is described in detail using symbolic dynamics; and (2) we get the strongest possible version (in the plane) of the ``dimension paradox'': the set $R$ of rays has Hausdorff dimension 1, and each point in $\C\sm R$ is connected to $\infty$ by one or more disjoint rays in $R$; as a complement of a 1-dimensional set, $\C\sm R$ has of course Hausdorff dimension 2 and full Lebesgue measure.