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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 ( 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  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 ∞. (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.

A landing theorem for dynamic rays of geometrically finite entire functions