A characterization of singular-hyperbolicity. (English) Zbl 1380.37062
Let \(\Lambda\) be a compact invariant set of a three-dimensional flow, such that all singularities \(\sigma\) are Lorenz like and satisfies \(W^{\mathrm{ss}} (\sigma) \cap \Lambda = \sigma\). The author proves that \(\Lambda\) is singular hyperbolic if and only if \(\Lambda^*\) has a dominated splitting of index 1 with respect to the linear Poincaré flow and every ergodic measure on \(\Lambda\) is of hyperbolic saddle type.
Reviewer: Tomas Persson (Lund)
MSC:
| 37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |
| 37D30 | Partially hyperbolic systems and dominated splittings |
| 37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |
| 37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |
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