Finite Dynamical Laminations

Abstract

We develop several combinatorial notions about laminations, some with clear implications for parameter space. We introduce a simplified class of laminations called finite dynamical laminations (FDL). In order to count FDL, we introduce sibling portraits, of which we provide a comprehensive counting theorem. We provide a characterization of which periodic polygons appear in invariant laminations. We introduce the pullback tree. The base of the pullback tree is a set of laminations, and we show that those laminations are proper and invariant, and all laminations in the base of the pullback tree correspond to a polynomial. We define the generational FDL graph, and it provides combinatorial information about polynomial parameter space.