  A transcendental entire function f is called geometrically finite if the
intersection of the set of singular values with the Fatou set is compact and
the intersection of the postsingular set with the Julia set is finite. (In
particular, this includes all entire functions with finite postsingular set.)
If f is geometrically finite, then the Fatou set of f is either empty or
consists of the basins of attraction of finitely many attracting or parabolic
cycles.
  Let z_0 be a repelling or parabolic periodic point of such a map f. We show
that, if f has finite order, then there exists an injective curve consisting of
escaping points of f that connects z_0 to infinity. (This curve is called a
dynamic ray.) In fact, the assumption of finite order can be weakened
considerably; for example, it is sufficient to assume that f can be written as
a finite composition of finite-order functions.
