Renormalization and rigidity in one-dimensional dynamics

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Renormalization theory was first developed in physics in the context of quantum field theory and statistical mechanics. At the end of 70's in the pioneering work of M. Feigenbaum the renormalization ideas paved their way into the theory of dynamical systems. By now renormalization is one of the most powerful methods in the asymptotic analysis of dynamical systems. The main idea is to study long-term behaviour of dynamical systems in the rescaled coordinate system near places where trajectories return to the initial position on a smaller and smaller scales. It turns out that after such a rescaling the dynamical system exhibit universal behaviour which depends only on topological (or, combinatorial) type, and local structure of critical, or singular, points. Remarkably, in many examples such universal behaviour can be studied rigorously. Roughly speaking, it can be described in terms of fixed points of renormalization. Simple fixed points appear in the smooth case. Such fixed points are related to the theory of linearization which form an essential part of the celebrated Kolmogorov-Arnold-Moser theory. The case when dynamical system has critical, or singular, points is of special interest since it corresponds to highly nontrivial renormalization fixed points. In the case of maps with one critical point a beautiful renormalization theory was developed in 1990's by Sullivan, McMullen and Lyubich. The proposed research project aims to extend significantly our understanding of the renormalization behaviour. We intend to study systems with many singular, or critical points, and hope to reveal a rich structure of the renormalization attractor in this case. We also plan to study global universality in the case of general critical points, which is one of the central open problems in the theory of renormalization.