Please use this identifier to cite or link to this item: https://hdl.handle.net/10216/92969
Author(s): Flávio Ferreira
Alberto A. Pinto
David A. Rand
Title: Hausdorff Dimension versus Smoothness
Issue Date: 2008
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.
Description: 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.
Subject: Matemática
Mathematics
Scientific areas: Ciências exactas e naturais::Matemática
Natural sciences::Mathematics
URI: https://repositorio-aberto.up.pt/handle/10216/92969
Source: Differential Equations, Chaos and Variational Problems - Progress in Nonlinear Differential Equations and Their Applications, Volume 75, Springer.
Document Type: Capítulo ou Parte de Livro
Rights: restrictedAccess
Appears in Collections:FCUP - Capítulo ou Parte de Livro

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