Real Laminations of Cubic Polynomials on Boundaries of Hyperbolic Components

TL;DR

This paper characterizes the real laminations of cubic polynomials on the boundaries of hyperbolic components of types (A), (B), and (C), revealing their structure and implications for combinatorial rigidity.

Contribution

It provides a detailed description of the real laminations on tame boundaries of certain hyperbolic components and establishes non-rigidity results for most cubic polynomials.

Findings

Real laminations are minimal containing the lamination of the hyperbolic map and a characteristic equivalence class.

Most hyperbolic cubic polynomials (except type D) are not combinatorially rigid.

The characterization applies to boundary maps of types (A), (B), and (C).

Abstract

Milnor divides all bounded hyperbolic components of cubic polynomials into 4 types (A), (B), (C) and (D). In this article, we characterize the real laminations of cubic polynomials on the tame boundary of all bounded hyperbolic components of type (A), (B), or (C). For such maps, we prove that the real lamination is the smallest lamination which contains the real lamination of maps in the hyperbolic component and an equivalence relation generated by one characteristic equivalence class. As an application, we show that every hyperbolic cubic polynomial except type (D) is not combinatorially rigid.