06 → Thesis in Mathematics
CONTEXT:
Thesis in Partial Fulfillment of Honors in Degree in Mathematics from Williams College
Advisor: Thomas Garrity
Fall2019 - Spring 2020
SKILLS DEVELOPED:
Mathematical Research Skills, Dynamical Systems, Continued Fractions
ABSTRACT:
The relationship between traditional Continued Fractions and Interval Exchange Maps is known; my thesis maps a brief framework for variations on traditional Continued Fractions, as well as a framework for Interval Exchange Maps. I then link a process called Rauzy-Veech induction on Interval Exchange Maps to variations on the traditional Continued Fraction factoring algorithms in both two and three dimensions.
THE GAUSS + FAREY MAPS:
This decomposition of a fraction is actually utilizing the Gauss Map. Outlined below, the Gauss map codifies what we have done on the previous page with traditional mathematical notation. The Gauss Map is also closely related to something called the Farey Map also shown below. The Farey Map is simply a more step by step version of the Gauss Map and gives us a "additive" way of getting a continued faction: rather than taking all the "whole numbered parts" out at once, we take them out one at a time.
INTERVAL EXCHANGE MAPS:
If we start with a line segment and break it into sections, Interval Exchange Maps at their core are simply rearranging those sections of the line segment. There exists a process called Rauzy-Veech Induction (illustrated below), that takes this rearrangement and compares the final line segment section of the original with the last line segment of the permuted segment and cuts off the smaller of the two from the larger, resulting in a new map. This process is illustrated below.
INTERVAL EXCHANGE MAPS AND CONTINUED FRACTIONS:
The relationship between continued fractions and interval exchange maps is known. All we have to do is let our intervals λA and λB represent the fraction λA over λB, and we can perform the same operations on the line segments that we do the numerator and denominator of a fraction. In my thesis, I related variations on the traditional fraction algorithm obtained through the Gauss and Farey Maps, to interval Exchange Maps. In two dimensions, I found the Interval Exchange and Rauzy-Veech Induction steps for each of the 8 Farey and Gauss Variations.
3 DIMENSTIONAL CONTINUED FRACTIONS AND INTERVAL EXCHANGE MAPS:
Further developing this relationship, I studied how we represent continued fractions in 3 dimensions using Rauzy-Veech Induction and Interval Exchange Transformations. A 3 dimensional continued fraction simply takes two fractions and compares the two to one another rather than comparing the numerator and denominator of the fraction. In our Gauss and Farey Maps, instead of working on a 1 dimensional unit interval, we are working on a 2 dimensional triangle and partitioning the triangle in a similar manner. I developed a framework developing every Gauss case, and found all 21 Farey Case Variations.