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

Authors

Michael J. Moorman, Harvard CollegeFollow
Gabriel B. Quijano, Auburn UniversityFollow
Matthew C. Williams Jr., Auburn UniversityFollow

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 of 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 p/q, 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.

Author Bio

This paper was written by three undergraduates, listed hereafter, who were supervised by Dr. Mayer - an emeritus professor in mathematics at the University of Alabama at Birmingham. The authors met Mayer through their teacher at Homewood High School and continued work on the paper into their separate undergraduate programs.

Matthew "Hugh" C. Williams Jr. is a Senior at Auburn University, majoring in Computer Science and Applied Mathematics with a minor in philosophy. He was born and raised in Birmingham, Alabama. Hugh has a personal interest in the philosophy of artificial intelligence with hopes of contributing to machine learning practically too.

Michael J. Moorman is a Senior at Harvard University, majoring in Computer Science with a minor in Mathematics and Spanish. He was born and raised in Birmingham, Alabama. Michael is interested in the intersection of pure mathematics and computer science, and hopes to work on cryptographic algorithms in a research or employment position in the future.

Gabriel B. Quijano is a Junior at Auburn University, majoring in Mathematics and minoring in Philosophy. From Birmingham Alabama, he has an interest in pure mathematics and applying its principles and ways of thinking to life and the natural sciences, and he hopes to be able to contribute to research in those areas.

Faculty Sponsor

John C. Mayer

Recommended Citation

Moorman, Michael J.; Quijano, Gabriel B.; and Williams, Matthew C. Jr. (2024) "Counting Rotational Sets for Laminations of the Unit Disk from First Principles," Rose-Hulman Undergraduate Mathematics Journal: Vol. 25: Iss. 2, Article 8.
Available at: https://scholar.rose-hulman.edu/rhumj/vol25/iss2/8