Annular itineraries for entire functions

TL;DR

This paper introduces the concept of annular itineraries to analyze the growth behavior of iterates of transcendental entire functions, demonstrating the existence of points with various prescribed itineraries using new covering results.

Contribution

It introduces annular itineraries for entire functions and proves the existence of points with prescribed itineraries, supported by novel annuli covering theorems.

Findings

Different types of annular itineraries can occur for any transcendental entire function.

Points with various prescribed annular itineraries can be found.

New annuli covering results are established that have broader applications.

Abstract

In order to analyse the way in which the size of the iterates $(f^n(z))$ of a transcendental entire function $f$ can behave, we introduce the concept of the {\it annular itinerary} of a point $z$. This is the sequence of non-negative integers $s_0s_1...$ defined by \[ f^n(z)\in A_{s_n}(R),\;\;\text{for}n\ge 0, \] where $A_0(R)=\{z:|z|<R\}$ and \[ A_n(R)=\{z:M^{n-1}(R)\le |z|<M^n(R)\},\;\;n\ge 1. \] Here $M(r)$ is the maximum modulus of $f$ and $R>0$ is so large that $M(r)>r$, for $r\ge R$. We consider the different types of annular itineraries that can occur for any transcendental entire function $f$ and show that it is always possible to find points with various types of prescribed annular itineraries. The proofs use two new annuli covering results that are of wider interest.