Counting Rotational Sets for Laminations of the Unit Disk from First Principles

TL;DR

This paper provides an elementary geometric and combinatorial method to count rotational sets in laminations of the unit disk, offering insights into the structure of polynomial Julia sets and hyperbolic components.

Contribution

It derives a closed-form formula for counting rotational sets using simple geometric and combinatorial principles, making the results more accessible.

Findings

A closed-form formula for counting rotational sets is obtained.

The method offers an intuitive geometric and combinatorial explanation.

Insights into the structure of Julia sets and hyperbolic components are gained.

Abstract

By studying laminations of the unit disk, we can gain insight into the structure of Julia sets of polynomials and their dynamics in the complex plane. The polynomials of a given degree, $d$, have a parameter space. The hyperbolic components of such parameter spaces are in correspondence to rotational polygons, or classes of "rotational sets", which we study in this paper. By studying the count of such rotational sets, and therefore the underlying structure behind these rotational sets and polygons, we can gain insight into the interrelationship among hyperbolic components of the parameter space of these polynomials. These rotational sets are created by uniting rotational orbits, as we define in this paper. The number of such sets for a given degree $d$, rotation number $\frac pq$, and cardinality $k$ can be determined by analyzing the potential placements of pre-images of zero on the unit circle with respect to the rotational set under the $d$-tupling map. We obtain a closed-form formula for the count. Though this count is already known based upon some sophisticated results, our count is based upon elementary geometric and combinatorial principles, and provides an intuitive explanation.