It has been observed in various settings—such as dripping faucets, insect populations, thermal convection in fluids, and electronic circuits—that the transition from regular dynamics to chaos occurs through a characteristic sequence of period-doubling bifurcations. Remarkably, the quantitative characteristics of this phenomenon appear to be universal, meaning they do not depend on the specific details of the systems under consideration. These observations have been rigorously established for 1D unimodal maps by Sullivan, McMullen, and Lyubich. In their work, they pioneered a new technique, known as renormalization, which has since become fundamental to the field. Together with S. Crovisier, M. Lyubich, and E. Pujals, we have recently extended this theory to the far more intricate 2D setting of Hénon-like maps.
In my talk, I will provide an intuitive explanation of the renormalization method, both in the classical 1D setting and in the new 2D context. I will then discuss how the tools we developed for the renormalization of Hénon-like maps contribute to the broader study of 2D dynamical systems.
