CONTINUED FRACTION EXPANSIONS IN DYNAMICAL SYSTEMS: EXAMPLES

Abstract

Continued fractions and dynamical systems are very connected to each other, and in this article, we show how simple iterative maps can generate continued fraction expansions of real numbers. We focus on several important systems — including the Gauss Map, Farey Map, Slow Continued Fraction Map, Rényi Map, β-Transformation, and a simplified billiard model — and investigate how they uncover number-theoretic properties such as rational approximations, Diophantine analysis, and chaotic behavior. Building on the theoretical groundwork from Continued Fractions by Wieb Bosma and Cor Kraaikamp, we offer clear definitions and interpret each system from a dynamical perspective. All models are implemented in MATLAB to simulate orbits, extract continued fraction digits, and visualize invariant measures. These simulations reveal the surprisingly structured yet chaotic nature of these systems and demonstrate their value in studying the real number line. This work is for both learning and further research in number theory and dynamical systems.