MONOTONE TWIST MAPS: EXAMPLES AND THEIR MINIMIZING ORBITS

This thesis explores the theory of monotone twist maps, minimal and periodic orbits, and rotation numbers. Monotone twist maps, a fundamental concept in dynamical systems, serve as the primary focus, with an in-depth analysis of their properties and behaviors. The study delves into the existence and characterization of minimal and periodic orbits within these systems, alongside a thorough investigation of rotation numbers, which play a crucial role in understanding the dynamics of twist maps. Furthermore, the thesis highlights various applications of these theoretical concepts, particularly emphasizing the example of billiard maps applied to convex billiards. The convex billiard field is used to illustrate the practical implications and utility of the discussed mathematical frameworks, providing a concrete case study that bridges abstract theory with real-world applications. By integrating theoretical insights with practical examples, this work contributes to the broader understanding of dynamical systems and their intricate behaviors, offering valuable perspectives on both the mathematical underpinnings and the applied aspects of monotone twist maps and their associated phenomena.