Cantor exchange systems and renormalization |
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Authors: | A.A. Pinto D.A. Rand F. Ferreira |
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Affiliation: | aDepartamento de Matemática, Universidade do Minho, 4710-057 Braga, Portugal;bMathematics Institute, University of Warwick, Coventry CV4 7AL, UK;cESEIG, Instituto Politécnico do Porto, R.D. Sancho I, 981, 4480-876 Vila do Conde, Portugal |
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Abstract: | We prove a one-to-one correspondence between (i) C1+ conjugacy classes of C1+H Cantor exchange systems that are C1+H fixed points of renormalization and (ii) C1+ conjugacy classes of C1+H diffeomorphisms f with a codimension 1 hyperbolic attractor Λ that admit an invariant measure absolutely continuous with respect to the Hausdorff measure on Λ. However, we prove that there is no C1+α Cantor exchange system, with bounded geometry, that is a C1+α fixed point of renormalization with regularity α greater than the Hausdorff dimension of its invariant Cantor set. |
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