Nonsingular expansive flows on 3-manifolds and foliations with circle prong singularities

Geodesic Flows Modelled by Expansive Flows

“…Bowen [6] and Bowen and Walters [8] pointed out that expansive systems with a local product structure are C 0 -topologically stable (see also [20,44] in the case of flows); Lewowicz [31] and Hiraide [25] showed that expansive homeomorphisms of compact surfaces have an 'almost' local product structure. Paternain [37] and Inaba and Matsumoto [26] extended Lewowicz's work to show that expansive geodesic flows of compact surfaces have a local product structure and hence that they are C 0 -topologically stable. Ruggiero [42] showed that expansive geodesic flows of compact manifolds without conjugate points have a local product structure, which also implies topological stability.…”

Rescaled expansivity and separating flows

“…Take ρ > 0 such that BCρ/(1 − ρB) < δ. We will show that (12) if dist(p, q) ≤ ρ X(p) then dist r (p, q) ≤ δ.…”

Research Announcement: Partially Hyperbolic Diffeomorphisms Homotopic to the Identity on 3-Manifolds

“…As mentioned earlier in Appendix D, thanks to the work of Paternain [Pat93] and Inaba and Matsumoto [IM90], the definition of topological Anosov flow can be replaced by asking for the flow to be expansive and to preserve two (non singular, i.e., without prongs) foliations. Note also that just condition (i) is not enough for a flow to be topological Anosov as condition (i) does not imply condition (ii).…”