Björn Winckler: On the hyperbolicity of Lorenz renormalization

Abstract. I will be presenting an article I co-authored with Marco Martens entitled "On the hyperbolicity of Lorenz renormalization" (Comm.Math. Phys. 325, 2014).  In this article we study the dynamics of Lorenz maps using renormalization techniques.  Lorenz maps are interval maps whose dynamics are intimately related to three dimensional flows similar to the Lorenz butterfly.  As a result of this, they provide a link between one-dimensional maps (which we are able to study) and flows in three dimensions (which are not well understood and very difficult to analyze).  The idea of renormalization is to define an operator on the space of Lorenz maps and to then study it.  By choosing the operator carefully we get a new system whose dynamics has very strong implications on the dynamics of individual Lorenz maps.

In this talk I will introduce renormalization on Lorenz maps.  I will then outline our results on the renormalization operator and their consequences on the dynamics of Lorenz maps.