We discuss in detail the dynamics of maps $z\mapsto ae^z+be^{-z}$ for which both critical orbits are strictly preperiodic. The points that 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 $\mathbb {C}$ , so that every point in $\mathbb {C}$ is either on a unique ray or the landing point of several rays.
¶ The key features of this article are the following:
¶ (1) this is the first example of a transcendental dynamical system, where the Julia set is all of $\mathbb {C}$ and the dynamics is described in detail for every point using symbolic dynamics;
¶ (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 $\mathbb {C}{\textbackslash}R$ is connected to $\infty$ by one or more disjoint rays in $R$ .
¶ As the complement of a $1$ -dimensional set, $\mathbb {C}{\textbackslash}R$ of course has Hausdorff dimension $2$ and full Lebesgue measure