Hausdorff Dimension versus Smoothness

Abstract

There is a one-to-one correspondence between C^{1+H} Cantor exchange systems that are C^{1+H} fixed points of renormalization and C^{1+H} diffeomorphisms f on surfaces with a codimension 1 hyperbolic attractor Λ that admit an invariant measure absolutely continuous with respect to the Hausdorff measure on Λ. However, there is no such C^{1+α} Cantor exchange system with bounded geometry that is a C^{1+α} fixed point of renormalization with regularity α greater than the Hausdorff dimension of its invariant Cantor set. The proof of the last result uses that the stable holonomies of a codimension 1 hyperbolic attractor Λ are not C^{1+θ} for θ greater than the Hausdorff dimension of the stable leaves of f intersected with Λ.