# Dynamical generation of parameter laminations

## TL;DR

This paper explores the combinatorial structure of quadratic Julia sets and the Mandelbrot set using geodesic laminations, revealing how non-renormalizable laminations can be generated through limits of pullbacks of minors.

## Contribution

It introduces a new method to generate quadratic invariant laminations via pullbacks of minors, linking Julia sets and the Mandelbrot set in a combinatorial framework.

## Key findings

- Non-renormalizable laminations can be obtained by limits of pullbacks.
- The Quadratic Minor Lamination (QML) models the Mandelbrot set locally.
- A new combinatorial construction for quadratic laminations is proposed.

## Abstract

Local similarity between the Mandelbrot set and quadratic Julia sets manifests itself in a variety of ways. We discuss a combinatorial one, in the language of geodesic laminations. More precisely, we compare quadratic invariant laminations representing Julia sets with the so-called Quadratic Minor Lamination (QML) representing a locally connected model of the Mandelbrot set. Similarly to the construction of an invariant lamination by pullbacks of certain leaves, we describe how QML can be generated by properly understood pullbacks of certain minors. In particular, we show that the minors of all non-renormalizable quadratic laminations can be obtained by taking limits of "pullbacks" of minors from the main cardioid. This is the second, amended version of the paper, to appear in Contemporary Mathematics