A transcendental entire function f is called geometrically finite if the intersection of the set S(f) of singular values with the Fatou set ?(f) is compact and the intersection of the postsingular set P(f) with the Julia set J(f) is finite. (In particular, this includes all entire functions with finite postsingular set.) If f is geometrically finite, then ?(f) is either empty or consists of the basins of attraction of finitely many attracting or parabolic cycles. Let z0 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 z0 to ∞. (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.
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