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Hausdorff dimension versus smoothness

Authors
Publisher
Birkhäuser
Publication Date
Disciplines
  • Mathematics

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.

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