Modeling Julia Sets with Laminations: An Alternative Definition
by
Debra Mimbs
University of Alabama at Birmingham
Coauthors: Lex Oversteegen

Given a topological space Z and a function, f: Z → Z, one may examine the sequence of iterates of f, i.e. {fⁿ(z)}_{n ∈ N} for z ∈ Z, called the orbit of z. Then one may classify the points of Z based upon their behavior under iterates of f. Specifically, let P:C → C, where C denotes the complex plane, be a polynomial of degree at least two. We denote by F(P) the Fatou set, which is the maximal open set on which the iterates of P form a normal family in the sense of Montel. Further, we let J(P) = C \F(P) denote the Julia set, the set on which the dynamics is chaotic. It is well known that J(P) is a nonempty, perfect, compact set, which is either connected or has uncountably many components. We consider the case where J(P) is connected. As J(P) is where complex, chaotic behavior occurs, and the dynamics on F(P) is well understood, we are interested in studying the behavior of J(P).

Typically, Julia sets exhibit very complex behavior. Thus, we desire to simplify our study of Julia sets by using a less complicated model. One method of modeling Julia sets is to use a lamination, which is a closed collection of chords in the unit disc, D, any two of which intersect at most at an endpoint on the boundary of D. By requiring that the lamination be d-invariant, one achieves a space whose dynamics are easier to study than a Julia set, while the dynamics on the two spaces are related.

However, the original definition of d-invariant given by Thurston requires gap-invariance. In many situations, gap-invariance is cumbersome to prove. In general, gaps are often difficult to study. Hence, we present an alternative definition of d-invariance, which avoids the notion of gap-invariance, making the definition easier to use in many situations. We prove that the two definitions are equivalent.

Date received: February 25, 2010