Diophantine Approximation and Dynamical Systems

Abstract: The subject of diophantine approximation is a classical mathematic problem, as old as it is well studied. There are many different texts describing its connection to more modern areas of study, but few which do so with the aim of exploring the connections themselves. This paper aims to serve as an introduction to diophantine approximation, and to expose some properties common between two dynamical systems where it occurs. This is done in the style of a booklet, starting from the basics in each of the areas of diophantine appproximation, continued fractions, symbolic sequences, and hyperbolic geometry. Focus on each of the chapters following the first is on how to they connect back to diophantine equation. The chapters are then capped off with additional notes which explore things related to their respective subjects, for example the modern advancements made in the subject, or other interesting trivia for the interested reader. For complete comprehension of the text, the reader is assumed to have basic knowledge of the relation between rational and real numbers, analysis, matrices, number theory, and function theory. The text largely succeeds in its goals as an educatory text and is thought to be a somewhat novel contribution to the body of literature on the subject. Further work could expand on this by incorporating further areas in mathematics where diophantine approximation appears. Another avenue of exploration is to explore the underlying reasons for the similarities exposed here.