Abstract
We study partially hyperbolic sets of \begin{document}$C^1$\end{document} -diffeomorphisms. For these sets there are defined the strong stable and strong unstable laminations.A lamination is called dynamically minimal when the orbit of each leaf intersects the set densely. We prove that partially hyperbolic sets having a dynamically minimal lamination have empty interior. We also study the Lebesgue measure and the spectral decomposition of these sets. These results can be applied to \begin{document}$C^1$\end{document} -generic/robustly transitive attractors with one-dimensional center bundle.
Highlights
Hyperbolicity of a proper set imposes quite specific properties of its “size” and “structure”, especially when the dynamics on it is transitive
We provide sufficient conditions guaranteing that these sets have empty interior or zero Lebesgue measure
In [17] we prove that there is a wide class of systems verifying this property: robustly/generically transitive attractors with one-dimensional center bundle
Summary
Hyperbolicity of a proper set imposes quite specific properties of its “size” and “structure”, especially when the dynamics on it is transitive. Transitivity, attractor and repeller, homoclinic classes, Lebesgue measure, partial hyperbolicity, spectral decomposition. Assuming that the central bundle is one-dimensional we prove that, C1-generically, s-minimal proper attractors have zero Lebesgue measure (see Theorem C) Another motivation of this paper concerns the spectral decomposition results for sets containing the relevant part of the dynamics (limit, non-wandering, chainrecurrent sets, etc.). A C1-generic robustly transitive partially hyperbolic proper attractor with one-dimensional center bundle has robustly empty interior. If Λf U is an s-minimal partially hyperbolic set with one-dimensional center bundle and U is a compatible neighborhood of f , for every hyperbolic periodic point p that x. Generically) transitive attractor that is robustly non-hyperbolic and partially hyperbolic with one-dimensional center bundle. If s-minimality is verified only in a p q residual p q subset we say of U, that a set
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