Anosov flows on $3$-manifolds: the surgeries and the foliations

Abstract

To any Anosov flow X on a 3-manifold [Fe1] associated a bi-foliated plane (a plane endowed with two transverse foliations F s and F u ) which reflects the normal structure of the flow endowed with the center-stable and center unstable foliations. A flow is R-covered if F s (or equivalently F u ) is trivial.On the other hand, from one Anosov flow one can build infinitely many others by Dehn-Goodman-Fried surgeries. This paper investigates how these surgeries modify the bi-foliated plane.We first noticed that surgeries along some specific periodic orbits do not modify the bi-foliated plane: for instance,• surgeries on families of orbits corresponding to disjoint simple closed geodesics do not affect the bi-foliated plane associated to the geodesic flow of a hyperbolic surface (Theorem 1); • [Fe2] associates a (non-empty) finite family of periodic orbits, called pivots, to any non-R-covered Anosov flow. Surgeries on pivots do not affect the branching structure of the bi-foliated plane (Theorem 2) We consider the set Surg(A) of Anosov flows obtained by Dehn-Goodman-Fried surgery from the suspension flows of Anosov automorphisms A ∈ SL(2, Z) of the torus T 2 .Every such surgery is associated to a finite set of couples {(γ i , m i )} i , where the γ i are periodic orbits and the m i integers. When all the m i have the same sign, Fenley proved that the induced Anosov flow is R-covered and twisted according to the sign of the surgery. We analyse here the case where the surgeries are positive on a finite set of periodic points X and negative on another set Y . In particular we build non-R-covered Anosov flows on hyperbolic manifolds.Among other results, we show that given any flow X ∈ Surg(A) :• there exists > 0 such that for every ε-dense periodic orbit γ, every flow obtained from X by a non trivial surgery along γ is R-covered (Theorem 4). • there exist periodic orbits γ + , γ − such that every flow obtained from X by surgeries with distinct signs on γ + and γ − is non-R-covered (Theorem 5).

Goodman-Fried surgery, Birkhoff section, and R-covered Anosov flows

“…Remark that Bonatti and Iakovoglou [7] conjectured that for any periodic orbits sufficiently dense in the manifold we can obtain an R-covered Anosov flow by a Goodman-Fried surgery along the orbit.…”