ON THE HYPERBOLICITY OF LORENZ RENORMALIZATION

Abstract. We consider infinitely renormalizable Lorenz maps with real crit-ical exponent α> 1 and combinatorial type which is monotone and satisfies a long return condition. For these combinatorial types we prove the exis-tence of periodic points of the renormalization operator, and that each map in the limit set of renormalization has an associated unstable manifold. An unstable manifold defines a family of Lorenz maps and we prove that each infinitely renormalizable combinatorial type (satisfying the above conditions) has a unique representative within such a family. We also prove that each infinitely renormalizable map has no wandering intervals and that the closure of the forward orbits of its critical values is a Cantor attractor of measure zero. Stony Brook IMS Preprint #2012/