Hausdorff dimension for non-hyperbolic repellers II: DA diffeomorphisms
Vanderlei Horita Marcelo Viana
Discrete & Continuous Dynamical Systems - A 2005, 13(5): 1125-1152 doi: 10.3934/dcds.2005.13.1125
We study non-hyperbolic repellers of diffeomorphisms derived from transitive Anosov diffeomorphisms with unstable dimension 2 through a Hopf bifurcation. Using some recent abstract results about non-uniformly expanding maps with holes, by ourselves and by Dysman, we show that the Hausdorff dimension and the limit capacity (box dimension) of the repeller are strictly less than the dimension of the ambient manifold.
keywords: Repeller. Dimension theory Non-uniform hyperbolicity
Basin problem for Hénon-like attractors in arbitrary dimensions
Vanderlei Horita Nivaldo Muniz
Discrete & Continuous Dynamical Systems - A 2006, 15(2): 481-504 doi: 10.3934/dcds.2006.15.481
We prove that Hénon-like strange attractors of diffeomorphisms in any dimensions, such as considered in [2], [7], and [9] support a unique Sinai-Ruelle-Bowen (SRB) measure and have the no-hole property: Lebesgue almost every point in the basin of attraction is generic for the SRB measure. This extends two-dimensional results of Benedicks-Young [4] and Benedicks-Viana [3], respectively.
keywords: Hénon-like attractor non-uniformly hyperbolic system SRB measure.

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