On the Hyperbolicity of Lorenz Renormalization

TL;DR

This paper studies the structure of Lorenz maps under renormalization, proving the existence of periodic points, unstable manifolds, and properties of attractors for certain combinatorial types.

Contribution

It establishes the existence of periodic points and unstable manifolds for infinitely renormalizable Lorenz maps with specific combinatorial types and critical exponents.

Findings

Existence of periodic points of the renormalization operator.

Each map in the limit set has an associated unstable manifold.

Infinitely renormalizable maps have Cantor attractors of measure zero.

Abstract

We consider infinitely renormalizable Lorenz maps with real critical exponent $\alpha>1$ and combinatorial type which is monotone and satisfies a long return condition. For these combinatorial types we prove the existence 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.