Explore the fundamental concepts of dynamical systems in this 58-minute lecture on rotations of the circle and renormalization. Delve into the time evolution of discrete systems, focusing on maps of the circle. Examine key questions and main examples that illustrate the circle of concepts. Learn about the dichotomy in circle maps and understand the proof using the pigeonhole principle. Gain insights from speaker Corinna Ulcigrai of the University of Bristol, UK, as part of the Summer School in Dynamics (Introductory and Advanced) series organized by ICTP Mathematics.