Anosov Foliations Research Articles

Contact structures, deformations and taut foliations

Using deformations of foliations to contact structures as well as rigidity properties of Anosov foliations we provide infinite families of examples which show that the space of taut foliations in a given homotopy class of plane fields is in general not path connected. Similar methods also show that the space of representations of the fundamental group of a hyperbolic surface to the group of smooth diffeomorphisms of the circle with fixed Euler class is in general not path connected. As an important step along the way we resolve the question of which universally tight contact structures on Seifert fibered spaces are deformations of taut or Reebless foliations when the genus of the base is positive or the twisting number of the contact structure in the sense of Giroux is non-negative.

Helicity in Differential Topology and Incompressible Fluids on Foliated 3-Manifolds

We review the interpretation of the helicity of the velocity fields of incompressible fluids on closed 3-manifolds as the asymptotic linking pairing of vorticity fields and further develop this point of view. For codimension 1 foliated manifolds, this idea has a strong relation with the 1st foliated cohomology and the secondary invariants, such as the Godbillon-Vey invariant and the Reeb class. The main purpose of the present article is, based on these frameworks, to give a description of the space of velocity fields of incompressible fluids which are holonomically constrained to the leaves of a foliated 3-manifold. In particular for algebraic Anosov foliations we see how these ideas work effectively to understand the space of incompressible foliated flows.

Branching in the universal cover of taut foliations

We study the topology of codimension one taut foliations of closed orientable 3-manifolds which are smooth along the leaves. In particular, we focus on the lifts of these foliations to the universal cover, specifically when any set of leaves corresponding to nonseparable points in the leaf space can be totally ordered. We use the structure of branching in the lifted foliation to find conditions that ensure two nonseparable leaves are left invariant under the same covering translation. We also determine when the set of leaves nonseparable from a given leaf is finite up to the action of covering translations. The hypotheses for the results are satisfied by all Anosov foliations.

Continuous extension of Anosov foliations in 3-manifolds with negatively curved fundamental group

We study Anosov flows in 3-manifolds whose stable and unstable foliations in the universal cover have Hausdorff leaf space. We show that the intrinsic ideal boundaries of distinct stable leaves can be canonically identified and similarly for the unstable foliation. This is then applied to the case when the 3manifold has negatively curved fundamental group and leaves of the above foliations extend continuously to the ideal boundaries. We prove that the continuous extension restricted to the ideal boundaries respects the identifications of intrinsic ideal points mentioned above. We also analyse the non injectivity of the extension to the boundaries and show that there are uncountably many almost periodic, non periodic orbits of the flow which lift to flow lines with same ideal point in both directions. Finally we prove that the image of any open set in the domain ideal boundary, contains open sets in the range ideal boundary.

On sided branching in Anosov foliations

We study topologically transitive Anosov flows in 3-manifolds. We show that if one of the stable or unstable foliations in the universal cover does not have Hausdorff leaf space (branching occurs) then both have branching in the positive and negative directions.

The invariant connection of a 1/2-pinched Anosov diffeomorphism and rigidity

Let / be a C°° Anosov diffeomorphism of a compact manifold M, preserving a smooth measure. If / satisfies the |pinching assumption defined below, it must preserve a continuous affine connection for which the leaves of the Anosov foliations are totally geodesic, geodesically complete, and flat (its tangential curvature is defined along individual leaves). If this connection, which is the unique /-invariant affine connection on M, is Cr-differentiable, r > 2, then / is conjugate via a Cr+2-affine diffeomorphism to a hyperbolic automorphism of a geodesically complete flat manifold. If / preserves a smooth symplectic form, has C3 Anosov foliations, and satisfies the 2 : 1-nonresonance condition (an assumption that is weaker than pinching), then / is C°° conjugate to a hyperbolic automorphism of a complete flat manifold. (In the symplectic case, the invariant connection is the one previously defined by Kanai in the context of geodesic flows.) If the foliations are C2 and the holonomy pseudo-groups satisfy a certain growth conditipn, the same conclusion holds.

Brownian motion on Anosov foliations and manifolds of negative curvature

Brownian motion on Anosov foliations and manifolds of negative curvature

$PL$ representations of Anosov foliations

En choisissant une certaine section de Birkhoff pour le flot géodésique d’une surface compacte à courbure négative, E. Ghys a montré que le feuilletage instable du flot géodésique admet une structure transversalement affine par morceaux. Nous explicitons l’holonomie globale induite par cette structure transversalement affine par morceaux et calculons son invariant de Godbillon-Vey discret.

Erratum to ‘Anosov Foliations and Cohomology’

We regret to say that there is an error in the proof of theorem 1 of [F]. The algebras E*(U), E*(S) used there are not preserved by d, as claimed, save in the restricted case when U and S commute. A counterexample is the nilmanifold automorphism of [AA], Appendix 23. Thus the results of §1 and §2 are dubious, although the third section is unaffected.

Anosov foliations and cohomology

AbstractThe cohomology action of an Anosov diffeomorphism on a nilmanifold resembles that of a Cartesian product map. Corresponding results hold for infranilmanifolds, giving an invariant bigrading of the cohomology and a fourfold symmetry that extends Poincaré duality. Holonomy invariant cocycles are applied to the action on first cohomology.

Anosov Foliations are Hyperfinite

Anosov Foliations are Hyperfinite