The parameter space of cubic laminations with a fixed critical leaf

TL;DR

This paper extends Thurston’s lamination parameterization from quadratic to cubic polynomials by analyzing slices of cubic invariant laminations with a fixed critical leaf, using advanced combinatorial techniques.

Contribution

It introduces a new parameterization method for cubic invariant laminations with a fixed critical leaf, generalizing Thurston’s quadratic lamination framework.

Findings

Parameterization of cubic laminations with fixed critical leaves.

Extension of quadratic lamination techniques to cubic case.

Foundation for further cubic polynomial combinatorial models.

Abstract

Thurston parameterized quadratic invariant laminations with a non-invariant lamination, the quotient of which yields a combinatorial model for the Mandelbrot set. As a step toward generalizing this construction to cubic polynomials, we consider slices of the family of cubic invariant laminations defined by a fixed critical leaf with non-periodic endpoints. We parameterize each slice by a lamination just as in the quadratic case, relying on the techniques of smart criticality previously developed by the authors.