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

To any Anosov flow X on a 3-manifold Fe1 associated a bi-foliated plane (a plane endowed with two transverse foliations Fs and Fu) which reflects the normal structure of the flow endowed with the center-stable and center unstable foliations. A flow is R-covered if Fs (or Fu) is trivial. 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 in SL(2,Z) of the torus T2. Every such surgery is associated to a finite set of couples (C,m(C)), where the C are periodic orbits and the m(C) integers. When all the m(C) 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 X and negative on another set Y. Among other results, we show that given any flow X in Surg(A) : - there exists e>0 such that for every e-dense periodic orbit C, every flow obtained from X by a non trivial surgery along C is R-covered (Theorem 4). - there exist periodic orbits C+,C- such that every flow obtained from X by surgeries with distinct signs on C+ and C- is non-R-covered (Theorem 5).