Abstract
Abstract We prove that given any closed 3-manifold M 3, there is an A-flow f t on M 3 such that the non-wandering set NW (f t ) consists of 2-dimensional non-orientable expanding attractor and trivial basic sets.
Highlights
A-flows were introduced by Smale [15]
Recall that a Morse-Smale flow has a nonwandering set consisting of finitely many hyperbolic periodic trajectories and hyperbolic singularities, while any Anosov flow has hyperbolic structure on the whole supporting manifold
A-flows have hyperbolic nonwandering sets that are the topological closure of periodic trajectories
Summary
A-flows were introduced by Smale [15] (see basic definitions bellow). This class of flows contains structurally stable flows including Morse-Smale flows and Anosov flows. Recall that a Morse-Smale flow has a nonwandering set consisting of finitely many hyperbolic periodic trajectories and hyperbolic singularities, while any Anosov flow has hyperbolic structure on the whole supporting manifold. A nontrivial basic set Ω is called expanding if its topological dimension coincides with the dimension of unstable manifold at each point of Ω. Given any closed 3-manifold M3, there is an A-flow f t on M3 such that the non-wandering set NW ( f t) consists of a two-dimensional non-orientable expanding attractor and trivial basic sets. This result contrasts with the case for 3-dimensional A-diffeomorphisms. In the end of the paper, we discuss the result and formulate some conjectures
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