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.

Figures (9)

• Figure 1: The bijection considered in the proof of \ref{['firstcount']} in the case of a degree three round gap with a triangle in its image.
• Figure 2: The bijection considered in the proof of \ref{['count2']} where the domain is the case of a degree three round gap with a triangle in its image.
• Figure 3: The pullback tree starting with the polygon with vertices {$\_001$,$\_010$,$\_100$} up to level 5, also known as the rabbit pullback tree.
• Figure 4: The canonical rabbit pullback lamination.
• Figure 5: \ref{['ex56']}

Theorems & Definitions (68)

• Definition 1.1
• Definition 1.2
• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Lemma 2.8