6 - Residual Julia sets of rational and transcendental functions

Summary

Abstract. The residual Julia set, denoted by J_r(f), is defined to be the subset of those points of the Julia set which do not belong to the boundary of any component of the Fatou set. The points of J_r(f) are called buried points of J(f) and a component of J(f) which is contained in J_r(f) is called a buried component. In this paper we survey the most important results related to the residual Julia set for several classes of functions. We also give a new criterion to deduce the existence of buried points and, in some cases, of unbounded curves in the residual Julia set (the so-called Devaney hairs). Some examples are the sine family, certain meromorphic maps constructed by surgery and the exponential family.

INTRODUCTION

Given a map f : X → X, where X is a topological space, the sequence formed by its iterates will be denoted by f⁰ ≔ Id, fⁿ ≔ f ∘ fⁿ⁻¹, n ∈ ℕ. When f is a holomorphic map and X is a Riemann surface the study makes sense and is non-trivial when X is either the Riemann sphere, the complex plane ℂ or the complex plane minus one point ℂ \ {0}. All other interesting cases can be reduced to one of these three. All other interesting cases can be reduced to one of these three.

In this paper we deal with the following classes of maps (partially following [12]).