It has been observed that two seemingly very different dynamical systems can bear striking resemblances when viewed at sufficiently small scales. Drawing motivations from physics, Feigenbaum (and independently, Collet and Tresser) introduced renormalization in the mid 1970’s as a conjectural explanation of this phenomenon. Since then, this idea has been successfully applied to a wide variety of fundamentally important examples of dynamical systems, leading to deep and rigorous mathematical theories that describe their long-term behaviors.
In this talk, I will outline the general structure of a fully developed renormalization theory. The aim will be to emphasize intuition over formal and technical details, and to avoid giving specifics that excessively narrow the scope of the discussion. At the end, I will conclude with some concrete results (both classical and new) that were obtained through applications of renormalization techniques.