Critical classes of non-degenerate laminations.
by
Doug Childers
University of Alabama at Birmingham (UAB)
Coauthors: Alexander Blokh Lex Oversteegen John C. Mayer (UAB)

Let S¹ = R/Z denote the complex unit circle and define s: S¹ → S¹ by s(t) = 2t mod 1. W.P. Thurston defined an equivalence relation ~ on the complex unit disk [`D] whose induced quotient space is conjecturally homeomorphic to the Mandelbrot set M (the set of all values c in the complex plane, such that z²+c has connected Julia set). Each ~ -class represents an equivalence relation ≈ on S¹ such that (s, ≈ ) induces a map F: S¹/ ≈  → S¹/ ≈ ; such an equivalence relation ≈ is called a lamination. Often times there exists a quadratic polynomial P(z) = z²+c (c ∈ M) with Julia set J such that P|_J is semi-conjugate to F. However there are obstructions to this being true in general. One of these obstructions is that S¹/ ≈ could reduce to a point. In this case we call ≈ the degenerate lamination.

A critical class C ⊂ S¹ of a lamination ≈ is a ≈ -class such that s|_C is not injective. Clearly the degenerate lamination has a critical class. However, a given closed subset of S¹ must satisfy certain dynamical properties for it to be the critical class of a non-degenerate lamination. We give these properties, which are necessary and sufficient conditions for this to occur.

Date received: February 28, 2006