pseudo-Anosov Flow Research Articles

Foliations with good geometry

The goal of this article is to show that there is a large class of closed hyperbolic 3-manifolds admitting codimension one foliations with good large scale geometric properties. We obtain results in two directions. First there is an internal result: A possibly singular foliation in a manifold is quasi-isometric if, when lifted to the universal cover, distance along leaves is efficient up to a bounded multiplicative distortion in measuring distance in the universal cover. This means that leaves reflect very well the geometry in the large of the universal cover and are geometrically tight—this is the best geometric behavior. We previously proved that nonsingular codimension one foliations in closed hyperbolic 3-manifolds can never be quasi-isometric. In this article we produce a large class of singular quasi-isometric, codimension one foliations in closed hyperbolic 3-manifolds. The foliations are stable and unstable foliations of pseudo-Anosov flows. Our second result is an external result, relating (nonsingular) foliations in hyperbolic 3-manifolds with their limit sets in the universal cover, that is, showing that leaves in the universal cover have good asymptotic behavior. Let G \mathcal G be a Reebless, finite depth foliation in a closed hyperbolic 3-manifold. Then G \mathcal G is not quasi-isometric, but suppose that G \mathcal G is transverse to a quasigeodesic pseudo-Anosov flow with quasi-isometric stable and unstable foliations—which are given by the internal result. We then show that the lifts of leaves of G \mathcal G to the universal cover extend continuously to the sphere at infinity and we also produce infinitely many examples satisfying the hypothesis. The main tools used to prove these results are first a link between geometric properties of stable/unstable foliations of pseudo-Anosov flows and the topology of these foliations in the universal cover, and second a topological theory of the joint structure of the pseudo-Anosov foliation in the universal cover.

Incompressible surfaces and pseudo-Anosov flows

As the main result of this paper, we show that any closed surface immersed transverse to a suitable pseudo-Anosov flow on a closed, hyperbolic three-manifold is geometrically infinite if and only if the surface lifts to be an embedded fiber in a finite cover in which the flow lifts to be the suspension flow. This is an extension of results known for surfaces immersed in bundles (Cooper et al., 1994).

Geodesic flows on hyperbolic orbifolds, and universal orbifolds

The authors discuss a class of flows on 3-manifolds closely related to Anosov flows, which they call Anosov flows. These are flows which are Anosov outside of a finite number of periodic singular orbits'', such that each orbit has a Poincare section on which the first return map has an singularity'' for some n≥1, n≠2. If only 1-pronged singularities occur the flow is called V-Anosov; the authors observe, for example, that the geodesic flow of a compact, hyperbolic 2-orbifold is V-Anosov. The main theorem is that every closed 3-manifold has a Anosov flow. The theorem is proved by constructing a certain link L in the 3-sphere such that L is a universal branching link, so every closed 3-manifold M is a branched cover of the 3-sphere branched over L, and L is the set of orbits of some V-Anosov flow on S3, so the lifted flow is a Anosov flow on M. In the literature, a Anosov flow whose n-pronged singularities always satisfy n≥3 is called pseudo-Anosov. The main theorem should be contrasted with the fact that an Anosov or pseudo-Anosov flow can only occur on an aspherical 3-manifold—an irreducible 3-manifold with infinite fundamental group. The literature contains many constructions of Anosov and pseudo-Anosov flows, but it remains unknown exactly which aspherical 3-manifolds support such flows

Surfaces and branched surfaces transverse to pseudo-Anosov flows on 3-manifolds

Surfaces and branched surfaces transverse to pseudo-Anosov flows on 3-manifolds

Correction to ‘Equivariant spectral decomposition for flows with a ℤ-action’

The statement and proof of the ℤ-spectral decomposition theorem for a pseudo-Anosov flow φ on a 3-manifold M are in error (see [1]). There are counter-examples which show that the theorem as stated is false. We remark that the results of § 9 of [1], concerning analogues of the ℤ-spectral decomposition theorem for basic sets of Axiom A flows, are unaffected by this error.