Finite Dynamical Laminations

TL;DR

This paper introduces finite dynamical laminations (FDL), a simplified combinatorial framework for understanding polynomial parameter spaces, including counting methods and characterizations of invariant structures.

Contribution

It develops the concept of FDL, introduces sibling portraits and the pullback tree, and links these to polynomial parameter space analysis.

Findings

Count of FDL via sibling portraits

Characterization of periodic polygons in laminations

Construction of the pullback tree relating to polynomials

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.