It has been observed in various settings (such as a population of rabbits, thermal convection in a fluid, etc) that the transition from regular dynamics to chaos happens through a sequence of period-doubling bifurcations. Moreover, the quantitative characteristics of this phenomenon appear to be universal, meaning they do not depend on the specific details of the systems under consideration.
For 1D unimodal maps, these observations have been rigorously justified through the work of Sullivan, McMullen, and Lyubich using renormalization techniques. Together with S. Crovisier, M. Lyubich, and E. Pujals, we extended this theory to the 2D setting of Hénon-like maps. In this talk, I will provide an overview of our key results and discuss how the tools we developed contribute to the broader study of 2D dynamical systems.
