Partially Hyperbolic Sets with a Dynamically Minimal Invariant Lamination

TL;DR

This paper investigates the properties of partially hyperbolic sets with dynamically minimal laminations, showing they have empty interior and exploring their measure and spectral characteristics, with implications for C1-generic attractors.

Contribution

It establishes that such sets have empty interior and analyzes their measure and spectral properties, extending understanding of partially hyperbolic dynamics.

Findings

Partially hyperbolic sets with minimal laminations have empty interior.

Results apply to C1-generic robustly transitive attractors.

Analysis of Lebesgue measure and spectral decomposition of these sets.

Abstract

We study partially hyperbolic sets of C1-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 ap- plied to C1-generic/robustly transitive attractors with one-dimensional center bundle.