Every Anosov flow on a 3-manifold is associated to a bifoliated 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 $\mathbb{R}$ -covered if $F^s$ (or equivalently $F^u$ ) is trivial. On the other hand, from any Anosov flow one can build infinitely many others by Dehn–Goodman–Fried surgeries. This paper investigates how these surgeries modify the bifoliated plane. We first observe that surgeries along orbits corresponding to disjoint simple closed geodesics do not affect the bifoliated plane of the geodesic flow of a hyperbolic surface (Theorem 1). Analogously, for any non- $\mathbb{R}$ -covered Anosov flow, surgeries along pivot periodic orbits do not affect the branching structure of its bifoliated plane (Theorem 2). Next, we consider the set $\mathcal{S}urg(A)$ of Anosov flows obtained by Dehn–Goodman–Fried surgeries from the suspension flow $X_A$ of any hyperbolic matrix $A \in SL(2,\mathbb{Z})$ . Fenley proved that performing only positive (or negative) surgeries on $X_A$ leads to $\mathbb{R}$ -covered Anosov flows. We study here Anosov flows obtained by a combination of positive and negative surgeries on $X_A$ . Among other results, we build non- $\mathbb{R}$ -covered Anosov flows on hyperbolic manifolds. Furthermore, we show that given any flow $X\in \mathcal{S}urg(A)$ there exists $\epsilon>0$ such that every flow obtained from $X$ by a non-trivial surgery along any $\epsilon$ -dense periodic orbit $\gamma$ is $\mathbb{R}$ -covered (Theorem 4). Analogously, for any flow $X \in \mathcal{S}urg(A)$ there exist periodic orbits $\gamma_+,\gamma_-$ such that every flow obtained from $X$ by surgeries with distinct signs on $\gamma_+$ and $\gamma_-$ is non- $\mathbb{R}$ -covered (Theorem 5).

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).

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We give an answer to a generalised form of Smale's problem in dimension 2 and 3 (Problem 6.6 in [S. Smale, Bull. Amer. Math. Soc. 73 (1967)]) concerning the realisability of Smale orders. More specifically, we are considering the question of whether a partial order on a finite set is realisable as the Smale order of a structurally stable diffeomorphism or flow acting on a closed manifold. We classify the orders that are realisable by 1) an Ω-stable diffeomorphism acting on a closed surface, 2) an Anosov flow on a closed 3-manifold, and 3) a stable diffeomorphism with trivial attractors and repellers acting on a closed surface.