Renormalization of Lorenz Maps

University dissertation from Stockholm : KTH Royal Institute of Technology

Abstract: This thesis is a study of the renormalization operator on Lorenz ?maps with a critical point. Lorenz maps arise naturally as first-return maps for three-dimensional geometric Lorenz flows. Renormalization is a tool for analyzing the microscopic geometry of dynamical systems undergoing a phase transition.In the first part we develop new tools to study the limit set of renormalization for Lorenz maps whose combinatorics satisfy a long return condition. This combinatorial condition leads to the construction of a relatively compact subset of Lorenz maps which is essentially invariant under renormalization. From here we can deduce topological properties of the limit set (e.g. existence of periodic points of renormalization) as well as measure theoretic properties of infinitely renormalizable maps (e.g. existence of uniquely ergodic Cantor attractors). After this, we show how Martens’ decompositions can be used to study the differentiable structure of the limit set of renormalization. We prove that each point in the limit set has a global two-dimensional unstable manifold which is a graph and that the intersection of an unstable manifold with the domain of renormalization is a Cantor set. All results in this part are stated for arbitrary real critical exponents  ?> 1.In the second part we give a computer assisted proof of the existence of a hyperbolic fixed point for the renormalization operator on Lorenz maps of the simplest possible nonunimodal combinatorial type. We then show how this can be used to deduce both universality and rigidity for maps with the same combinatorial type as the fixed point. The results in this part are only stated for critical exponenta ?= 2.