Every complex polynomial with a locally connected Julia set generates a lamination of the unit disc — a closed set of non-crossing chords inside whose endpoints on are allowed to touch. This was a tool developed and explored by Thurston in order to study degree 2 complex polynomials, their Julia sets, and the parameter space of LC polynomials, the Mandelbrot set. The dimension of the corresponding `multi-brot’ sets increases in higher dimension so one usually restricts themselves to studying slices of the parameter space.
The restriction we make in this talk is to focus on symmetric polynomials, which we define as a degree complex polynomial whose locally connected Julia set — and therefore, whose lamination — has rotational symmetry. It turns out that this leads to behavior very similar to the degree 2 case.
Thurston utilized the Central Strip Lemma to help prove two main results in the degree 2 case — the No Wandering Triangles theorem (NWT) and the No Identity Return Triangles theorem (NIRT). The symmetric degree case has an analogous result, which we call the Critical Strip Lemma. In this talk we prove the Critical Strip Lemma, which puts restrictions on the placement of leaves in -symmetric laminations. We will then outline how it’s used to prove that in the -symmetric case, the NWT and NIRT theorems still hold.