A landing theorem for entire functions with bounded post-singular sets

TL;DR

This paper extends the Douady-Hubbard landing theorem from polynomials to entire functions with bounded postsingular sets, introducing 'filaments' for infinite order functions and establishing their landing properties.

Contribution

It generalizes the landing theorem to entire functions, introducing 'filaments' as new structures for infinite order cases, and proves their landing behavior at repelling or parabolic points.

Findings

Periodic hairs land at repelling/parabolic points for finite order functions.

Periodic filaments land at repelling/parabolic points for infinite order functions.

Every point of a hyperbolic set is the landing point of a filament.

Abstract

The Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the study of polynomial dynamics. It states that, for a complex polynomial with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray. We prove an analogue of this theorem for an entire function with bounded postsingular set. If the function has finite order of growth, then it is known that the escaping set contains certain curves called "periodic hairs"; we show that every periodic hair lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic hair. For a postsingularly bounded entire function of infinite order, such hairs may not exist. Therefore we introduce certain dynamically natural connected sets, called "filaments". We show that every periodic filament lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic filament. More generally, we prove that every point of a hyperbolic set is the landing point of a filament.