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Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers. (English) Zbl 1071.37022

Let \(M\) be a boudaryless compact manifold, \({\mathcal X}^r(M)\) be the space of \(C\) vector fields on \(M\) endowed with the \(C^r\) topology, \(r\geq 1\). If \(X\in{\mathcal X}^r(M)\), then \(X_t\), \(t\in\mathbb{R}\), denotes the flow induced by \(X\). A compact invariant set \(\Lambda\) of \(X\) is called isolated if there exists an open set \(U\supset\Lambda\), called an isolating block, such that \(\Lambda= \bigcap_{t\in\mathbb{R}} X_t(U)\). If \(U\) can be chosen such that \(X_t(U)\subset U\) for \(t> 0\), we say that \(\Lambda\) is an attracting set. The set \(\Lambda\) is called transitive if it coincides with the \(\omega\)-limit set of an \(X\)-orbit. An attractor is a transitive attracting set, while a repeller is an attractor for the vector field \(-X\). An attractor (or repeller) which is not the whole manifold is called proper. An invariant set of \(X\) is nontrivial if it is neither a periodic point nor a singularity. Now, an isolated set \(\Lambda\) of a \(C\) vector field \(X\) is called robust transitive if it has an isolated block \(U\) such that, for any A robust transitive set containing singularities of a flow on a closed 3-manifold is either a proper attractor or a proper repeller.
As a corollary the authors prove: \(C^1\) vector fields on a closed 3-manifold with robust transitive nonwandering sets are Anosov. The authors next prove that every singularity \(\sigma\) of a robust attractor of \(X\) on a closed 3-manifold is Lorenz-like for \(X\), i.e., the eigenvalues \(\lambda_i\), \(1\leq i\leq 3\), of \(DX(\sigma)\) are real and satisfy \(\lambda_2< \lambda_3< 0< -\lambda_3< \lambda_1\).
The authors finally define the notion of partial (and singular) hyperbolicity for compact transitive sets of \(X\) and show a relation between the robust attractors and singular hyperbolicity.

MSC:

37D30 Partially hyperbolic systems and dominated splittings
37C10 Dynamics induced by flows and semiflows
37E99 Low-dimensional dynamical systems
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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