Renormalization in dynamics

The simplest form of dynamical systems are hyperbolic systems. Their properties are preserved after perturbation, they are structurally stable. Hyperbolic dynamical systems are very well understood. This is not the case for non-hyperbolic systems. There is definitely not a general theory to describe non-hyperbolic dynamics.

This research project aims to contribute  to the understanding of non-hyperbolic systems, their topological and geometrical properties. Do they also have some form of stability? The main tool is renormalization. Given a dynamical system one can consider the first return map to a carefully chosen subdomain.

This first return map is again a dynamical system. Renormalization describes the dynamics of the original system on a smaller scale. It acts like a microscope.

The asymptotic behavior of renormalization gives information on the topological and geometrical properties of the system. Renormalization has been the main force behind the development of the theory for one-dimensional dynamics. Moreover beyond one-dimensional systems, a general theory is still completely out of reach.

The aim of the project is to explore the renormalization techniques, the topological and geometrical properties of non-hyperbolic higher dimensional systems. They are, in fact, a better description of phenomena in nature.

We will in particular focus on two-dimensional Lorenz dynamics and Hénon dynamics, looking at the one-dimensional case as a reference and inspiration case.