  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.
