A landing theorem for dynamic rays of geometrically finite entire functions

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(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(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.