The Lorenz Renormalization Conjecture

The renormalization paradigm for low-dimensional dynamical systems is that of hyperbolic horseshoe dynamics. Does this paradigm survive a transition to more physically relevant systems in higher dimensions? This article addresses this question in the context of Lorenz dynamics which originates in homoclinic bifurcations of flows in three dimensions and higher. A conjecture classifying the dynamics of the Lorenz renormalization operator is stated and supported with numerical evidence.