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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Dedicated to Arrigo Cellina and James Yorke
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Ferreira, F., Pinto, A.A., Rand, D.A. (2007). Hausdorff Dimension versus Smoothness. In: Staicu, V. (eds) Differential Equations, Chaos and Variational Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 75. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8482-1_15
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DOI: https://doi.org/10.1007/978-3-7643-8482-1_15
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