Sectional-Anosov flows

Abstract

We define sectional-Anosov flow as a vector field on a manifold, inwardly transverse to the boundary, whose maximal invariant set is sectional-hyperbolic (Metzger and Morales in Ergodic Theory Dyn Syst 28:1587–1597, 2008). We obtain properties of sectional-Anosov flows without null-homotopic periodic orbits on compact irreducible 3-manifolds including: incompressibility of transverse torus, non-existence of genus 0 transverse surfaces nor hyperbolic attractors nor hyperbolic repellers and sufficient conditions for the existence of singularities non-isolated in the nonwandering set. These generalize some known facts about Anosov flows.