## Institute for Mathematical Sciences

## Preprint ims12-05

** Marco Martens, Björn Winckler**
*On the Hyperbolicity of Lorenz Renormalization*

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.

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