A landing theorem for dynamic rays of geometrically finite entire functions

Abstract

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.

19 pages, 4 figures. results have been extended to a larger class of functions

Cited by in corpus (8)

• Brushing the hairs of transcendental entire functions
• Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds
• A landing theorem for entire functions with bounded post-singular sets
• Geometrically finite transcendental entire functions
• Dynamic rays of bounded-type entire functions
• Orbifold expansion and entire functions with bounded Fatou components
• The escaping set in transcendental dynamics
• A Converse Landing Theorem in Parameter Spaces