CONTINUED FRACTION EXPANSIONS IN DYNAMICAL SYSTEMS: APPLICATION

Abstract

Continued fractions play a significant role in the study of rotation numbers and their connection with dynamical systems, especially in low-dimensional cases such as interval maps and circle maps. In this paper, we explore how continued fraction expansions can be used to analyze the behavior of orbits under simple dynamical rules. We consider examples of irrational rotation on the circle and examine the connection between the continued fraction of the rotation number and the system’s qualitative properties, such as periodicity, quasi-periodicity, and stability. Particular attention is given to the Gauss map and its dynamical interpretation as a generator of continued fraction digits. We also investigate how the depth and complexity of the continued fraction expansion influence the convergence rate of orbit approximations. Numerical simulations and visualizations are provided to support the theoretical analysis and to illustrate the fractal-like structures arising in related systems. This study serves as an introductory step for undergraduate students interested in number theory, dynamical systems, and mathematical modeling.