2.1 Existence of dominating surgeries

In Theorems 4 and 5 we start with any flow obtained from a suspension flow by finitely many surgeries and we exhibit orbits on which surgeries lead to ${\mathbb R}$ -covered or non- ${\mathbb R}$ -covered flows.

As promised at the end of the introduction, in this section we will state Theorems 6 and 7 which are more general versions of Theorems 4 and 5: given a finite set ${\mathcal E}$ of periodic orbits, there is one orbit $\gamma $ or two orbits $\gamma _{+}$ and $\gamma _{-}$ on which the surgeries dominate any surgery along the orbits in ${\mathcal E}$ . We furthermore notice in the addenda to Theorems 6 and 7 that most of these surgeries lead to hyperbolic $3$ -manifolds.

Interestingly enough, Theorem 6 implies that the surgeries along ${\mathcal E}$ seem negligible in comparison with the ones on $\gamma _{+}$ and $\gamma _{-}$ . This is also the case for Theorem 4, which also admits the following stronger version.

Thus Theorems 6 and 7 consider well-chosen sets of periodic orbits ${\mathcal Y}$ , on which surgeries dominate all surgeries on a given set ${\mathcal E}$ . We are still very far from understanding the general case.

In the next section we describe several settings where we can answer the previous question.

2.2 A geometric criterion on periodic orbits for surgery domination

As stated above, given an invariant finite set ${\mathcal E}$ , Theorems 6 and 7 assert the existence of periodic orbits ( $\gamma $ or $\gamma _{+}$ and $\gamma _{-}$ ) for which the surgeries dominate any surgery along ${\mathcal E}$ . We will see in this section that this phenomenon is due to the geometric properties of the relative position with respect to ${\mathcal E}$ of the announced periodic orbits ( $\gamma $ or $\gamma _{+}$ and $\gamma _{-}$ ). The aim of this section is to state Theorems 8 and 9 which provide criteria for surgery domination by using rectangles in the bifoliated plane.

Let us fix a hyperbolic matrix $A\in SL(2,{\mathbb Z})$ (not necessarily of positive trace), $X_{A}$ its associated suspension Anosov flow and two disjoint finite $f_{A}$ -invariant sets ${\mathcal X},{\mathcal Y}$ . Consider ${\mathcal X}$ (respectively, ${\mathcal Y}$ ) to be the union of the periodic orbits $ \lbrace x_{i}\rbrace _{i\in I}$ (respectively, $\lbrace y_{j}\rbrace _{j\in J}$ ). We denote by ${{\mathcal S}}urg(X_{A},{\mathcal X},{\mathcal Y})$ the set of Anosov flows obtained by performing surgeries along ${\mathcal X} \cup {\mathcal Y}$ , and by ${{\mathcal S}}urg(X_{A},{\mathcal X},{\mathcal Y},(m_{i})_{i\in I},\ast )$ the set of Anosov flows obtained by performing any kind of surgery along ${\mathcal Y}$ and surgeries with characteristic numbers $m_{i}$ along $x_{i}$ . We give an analogous meaning to the notation ${{\mathcal S}}urg(X_{A},{\mathcal X},{\mathcal Y},\ast , (n_{j})_{j\in J})$ . Similarly, ${{\mathcal S}}urg(X_{A},{\mathcal X},{\mathcal Y},(m_{i})_{i\in I}, (n_{j})_{j\in J})$ denotes the flow obtained by the surgeries with characteristic numbers $(m_{i})$ and $(n_{j})$ .

We consider the plane ${\mathbb R}^{2}$ (seen as the bifoliated plane associated to $X_{A}$ ) endowed with the lattice ${\mathbb Z}^{2}$ and the eigendirections $E^{s}_{A}, E^{u}_{A}$ . We fix an orientation for $E^{s}_{A}$ and $E^{u}_{A}$ . We denote by $F^{s}_{A}$ and $F^{u}_{A}$ the (trivial) foliations of ${\mathbb R}^{2}$ by affine lines parallel to the eigendirections. For any finite $f_{A}$ -invariant set ${\mathcal E}$ , we denote by $\tilde {{\mathcal E}}$ its lift on ${\mathbb R}^{2}$ . A rectangle is a topological disc $R\subset {\mathbb R}^{2}$ whose boundary consists of the union of two segments of leaves of $F^{s}_{A}$ and two segments of leaves of $F^{u}_{A}$ .

A rectangle R has two diagonals. The orientations of $E^{s}_{A}$ and $E^{u}_{A}$ allow us to speak of the increasing and the decreasing diagonal. We endow the diagonals with the transverse orientation of $E^{s}_{A}$ , so that each diagonal has a first point (or else origin) and a last point.

If ${\mathcal E}\subset {\mathbb T}^{2}$ is a finite $f_{A}$ -invariant subset of the torus ${\mathbb T}^{2}= {\mathbb R}^{2}/{\mathbb Z}^{2}$ , we say that a rectangle R is a positive (respectively, negative) ${\mathcal E}$ -rectangle if the endpoints of its increasing (respectively, decreasing) diagonal belong to the lift $\tilde {\mathcal E}$ on ${\mathbb R}^{2}$ of ${\mathcal E}$ .

A positive or negative ${\mathcal E}$ -rectangle R is primitive if $R\cap \tilde {\mathcal E}$ consists of the endpoints of its increasing or decreasing diagonal.

We are ready to state our first geometric criterion, which is a geometric version of Theorem 4.

Obviously, the same statement holds:

• • by exchanging ${\mathcal X}$ with ${\mathcal Y}$ ;

• • by replacing ‘positive rectangle’ and ‘positively twisted’ by ‘negative rectangle’ and ‘negatively twisted’.

As we will see in the next observations, the geometric hypotheses in Theorem 8 for any ${\mathcal X}$ -rectangle are indeed conditions on a finite number of rectangles.

Indeed, for any finite $f_{A}$ -invariant set ${\mathcal E}$ , since A is orientation-preserving and the foliations $F^{s}_{A},F^{u}_{A}$ are invariant, one can make the following observation.

In the same way, the notion of primitive (respectively, positive, negative) ${\mathcal E}$ -rectangle is invariant under translations by elements of ${\mathbb Z}^{2}$ .

Therefore, the hypothesis in Theorem 8, namely the fact that every positive ${\mathcal X}$ -rectangle contains a point of $\tilde {{\mathcal Y}}$ , can be checked on a finite number of positive primitive ${\mathcal X}$ -rectangles.

Using Lemma 2.1, we get that many pairs $({\mathcal X},{\mathcal Y})$ satisfying the hypotheses of Theorem 8.

Theorem 8 states that if every positive ${\mathcal X}$ -rectangle contains a point of $\tilde {{\mathcal Y}}$ , then the surgeries on ${\mathcal Y}$ dominate the surgeries on ${\mathcal X}$ . It turns out that under the previous condition, it is not possible for the surgeries on ${\mathcal X}$ to also dominate the surgeries on ${\mathcal Y}$ . We thus get the following corollary to Theorem 8.

The following result can be seen as a geometric version of Theorem 5. It states our second geometric criterion which is based on the existence of ${\mathcal X}$ -rectangles disjoint from ${\mathcal Y}$ and vice versa, a hypothesis that complements the hypothesis of Theorem 8.

Once again the same statement holds by straightforward symmetries.

The hypothesis of Theorem 9 can be satisfied in a great variety of settings. Indeed, in Lemma 8.7 (see also Corollary 8.1) we will give a method for constructing for any $A \in SL(2,{\mathbb R})$ infinitely many pairs $({\mathcal X},{\mathcal Y})$ satisfying this condition.

3.1 Anosov flows: definitions, stability, orbital equivalence

Definition 3.1. A $C^{1}$ -vector field X on a closed manifold M is called an Anosov flow if the tangent bundle $TM$ admits a splitting

$$ \begin{align*} TM= E^{s}\oplus {\mathbb R} X\oplus E^{u}\end{align*} $$

satisfying the following properties.

• • The splitting is invariant under the natural action of the derivative $DX^{t}$ of the flow on $TM$ :

  $$ \begin{align*} DX^{t}(E^{s}(x))=E^{s}(X^{t}(x)) \quad\mbox{and}\quad DX^{t}(E^{u}(x))=E^{u}(X^{t}(x)).\end{align*} $$

• • If $\|\cdot \|$ is a Riemannian metric on M, there exist $C>0$ and $0<\lambda <1$ such that, for any $x\in M$ , $t>0$ and any two vectors $u\in E^{s}(x)$ and $v\in E^{u}(x)$ ,

  $$ \begin{align*}\|DX^{t}(u)\|\leq C \lambda^{t}\|u\| \quad\mbox{and}\quad \|DX^{-t}(v)\|\leq C \lambda^{t}\|v\|.\end{align*} $$

An important property of Anosov flows is stated in the following theorem.

The homeomorphism h in the theorem can be chosen isotopic to the identity map.

We denote by ${\mathcal A}(M)$ the set of orbital equivalence classes of Anosov flows and by ${\mathcal A}_{0}(M)$ the set of equivalence classes of Anosov flows through orbital equivalence by homeomorphisms isotopic to the identity. Theorem 10 implies that the set ${\mathcal A}_{0}(M)$ is at most countable on any closed manifold M. The set ${\mathcal A}(M)$ , being a quotient of ${\mathcal A}_{0}(M)$ , is at most countable too.

There are simple examples of manifolds M for which ${\mathcal A}_{0}(M)$ is infinite (consider, for instance, the image of the geodesic flow of a hyperbolic surface by a vertical diffeomorphism of the unit tangent bundle). It remains unknown whether there are manifolds for which ${\mathcal A}(M)$ is infinite. There are $3$ -manifolds for which ${\mathcal A}(M)$ has a cardinal greater than any given number, see [Reference Béguin, Bonatti and YuBeBoYu]. An example of manifold M for which ${\mathcal A}(M)$ is infinite has recently been proposed in [Reference Clay and PinskyClPi].

3.2 Foliations

Another important property of Anosov flows is that the stable, centre stable, unstable and centre unstable fibre bundles $E^{s},E^{cs}=E^{s}\oplus {\mathbb R} X, E^{u}, E^{cu}=E^{u}\oplus {\mathbb R} X$ are uniquely integrable.

The foliations ${\mathcal F}^{s},{\mathcal F}^{cs}, {\mathcal F}^{u}$ and ${\mathcal F}^{cu}$ are respectively called stable, centre stable, unstable and centre unstable.

In dimension $3$ the centre stable and centre unstable foliations provide the main known obstructions for a $3$ -manifold M to carry an Anosov flow.

As a direct corollary of the above, the manifold M carries foliations ( ${\mathcal F}^{cs}$ and ${\mathcal F}^{cu}$ ) with no compact leaves and thus with no Reeb component. Under these hypotheses, a consequence of Novikov’s theorem implies that M admits ${\mathbb R}^{3}$ as a universal cover.

A simple argument also allows us to check that the leaves have exponential growth. As a consequence of this, the fundamental group of M has exponential growth (see [Reference Plante and ThurstonPlaTh]).

3.3 The bifoliated plane associated to an Anosov flow on a $3$ -manifold, ${\mathbb R}$ -covered and non- ${\mathbb R}$ -covered Anosov flows

Before beginning this section, the reader can refer to §1.2 for the definitions of:

• • the bifoliated plane $({\mathcal P}_{X}, F^{s}_{X},F^{u}_{X})$ associated to an Anosov flow X on a $3$ -manifold;

• • a non- ${\mathbb R}$ -covered Anosov flow;

• • an ${\mathbb R}$ -covered Anosov flow;

• • a twisted ${\mathbb R}$ -covered Anosov flow;

• • a positively and negatively twisted ${\mathbb R}$ -covered Anosov flow (these notions are only defined on oriented manifolds and depend on a choice of the orientation of the manifold).

Any (non-singular) foliation on the plane is orientable and transversally orientable. We will sometimes use a choice of orientation of the foliations $F^{s}_{X}$ , $F^{u}_{X}$ . By convention, in that case the orientation chosen on ${\mathcal P}_{X}$ is the orientation on $F^{s}_{X}$ followed by the orientation on $F^{u}_{X}$ .

As we consider flows X on oriented manifolds M, the normal bundle of X is naturally oriented as follows: the orientation of X followed by the orientation of its normal bundle is the orientation of M. The orientation on ${\mathcal P}_{X}$ can be seen as the orientation of the normal bundle of X.

As M is oriented, the strong stable foliation ${\mathcal F}^{s}_{X}$ is oriented if and only if ${\mathcal F}^{u}_{X}$ is oriented. In that case, by convention, the orientation on ${\mathcal F}^{s}_{X}$ followed by the orientation of ${\mathcal F}^{u}_{X}$ is the normal orientation of the normal bundle of X.

If ${\mathcal F}^{s}_{X}$ and ${\mathcal F}^{u}_{X}$ are not oriented, then their local orientations define a $2$ -fold cover on M and the lift of X on this cover is an Anosov flow with oriented strong stable and strong unstable foliations.

The fundamental group $\pi _{1}(M)$ acts by the deck transformation group on the universal cover $\tilde M\simeq {\mathbb R}^{3}$ of M. This action preserves the lift $\tilde X$ of X on $\tilde M$ and also preserves the lifts $\tilde {\mathcal F}^{cs}_{X},\tilde {\mathcal F}^{cu}_{X}$ of the centre stable and centre unstable foliations. Therefore, the action passes to the quotient by the equivalence relation ‘belonging in the same orbit’. The space obtained by this quotient is ${\mathcal P}_{X}$ and we will denote the projection map by $\pi : \tilde {M}\rightarrow {\mathcal P}_{X}$ . We thus obtain an action of $\pi _{1}(M)$ on ${\mathcal P}_{X}$ , which preserves the foliations $F^{s}_{X}$ and $F^{u}_{X}$ . This action is called the natural action of $\pi _{1}(M)$ on the bifoliated plane $({\mathcal P}_{X}, F^{s}_{X}, F^{u}_{X})$ and we denote it by

$$ \begin{align*}\theta_{X}\colon \pi_{1}(M)\to \mathrm{Homeo}({\mathcal P}_{X}, F^{s}_{X}, F^{u}_{X}),\end{align*} $$

where $\mathrm {Homeo}({\mathcal P}_{X}, F^{s}_{X}, F^{u}_{X})$ is the group of homeomorphisms of the plane ${\mathcal P}_{X}$ preserving the foliations $F^{s}_{X}$ and $F^{u}_{X}$ .

As we consider only Anosov flows on oriented manifolds, the action $\theta _{X}$ takes values in $\mathrm {Homeo}_{+}({\mathcal P}_{X}, F^{s}_{X}, F^{u}_{X})$ and thus preserves the orientation on ${\mathcal P}_{X}$ .

However, $\theta _{X}$ may not preserve the orientations of the foliations $F^{s}_{X}$ and $F^{u}_{X}$ .

If X is a transitive Anosov flow then the action on ${\mathcal P}_{X}$ admits dense orbits. Furthermore, the orbit of any half leaf of $F^{s}_{X}$ and of $F^{u}_{X}$ is dense in ${\mathcal P}_{X}$ .

Let $x_{0}$ be the base point of $\pi _{1}(M)$ in M and $\tilde {x_{0}}$ a lift of $x_{0}$ on $\tilde {M}$ . Consider an element $\tilde \gamma \in {\mathcal P}_{X}$ corresponding to a periodic orbit $\gamma \subset M$ , and $\tilde {\Gamma }$ the lift of $\gamma $ on $\tilde {M}$ corresponding to $\tilde {\gamma }$ . Then one has a well-defined element $[\tilde \gamma ]\in \pi _{1}(M)$ which is the homotopy class of a closed path obtained by the concatenation $\sigma \gamma \sigma ^{-1}$ , where $\sigma $ is the projection on M of a path in $\tilde {M}$ joining $\tilde {x_{0}}$ to a point of the orbit $\tilde \Gamma $ .

The following lemma is a classical result in the theory (see, for instance, [Reference BarbotBa]).

Using the previous notation, we say that a curve $L^{s}\subset W^{s}(\gamma )$ is a complete (stable) transversal if it is transverse to X, cuts all the orbits in $W^{s}(\gamma )$ and also is such that the first return map of X induces a homeomorphism $P_{\gamma }\colon L^{s}\to L^{s}$ (which is a contraction). One defines in the same way a complete (unstable) transversal $L^{u} \subset W^{u}(\gamma )$ and the first return map $P_{\gamma }\colon L^{u}\to L^{u}$ (which is a dilation).

Take complete stable and unstable transversals $L^{s}$ and $L^{u}$ that contain a point $x\in \gamma $ . Now using the previous notation, take any lift $\tilde {x}$ of x in $\tilde {\Gamma }$ . $L^{s}$ and $L^{u}$ admit canonical lifts $L^{s}_{\tilde x}$ and $L^{u}_{\tilde x}$ on $\tilde M$ through $\tilde {x}$ . Let us denote the lift maps by $\pi ^{s}_{\tilde {x}}: L^{s} \rightarrow L^{s}_{\tilde x}$ and $\pi ^{u}_{\tilde {x}}: L^{u} \rightarrow L^{u}_{\tilde x}$ . $L^{s}_{\tilde x}$ and $L^{u}_{\tilde x}$ project on ${\mathcal P}_{X}$ injectively. We can therefore define two bijections $h^{s}:=\pi \circ \pi ^{s}_{\tilde {x}}$ and $h^{u}:=\pi \circ \pi ^{u}_{\tilde {x}}$ from $L^{s}$ and $L^{u}$ to $F_{X}^{s}(\gamma )$ and $F_{X}^{u}(\gamma )$ . We denote by $P_{\tilde \gamma }\colon F^{s}_{X}(\tilde \gamma )\to F^{s}_{X}(\tilde \gamma )$ and $P_{\tilde \gamma }\colon F^{u}_{X}(\tilde \gamma )\to F^{u}_{X}(\tilde \gamma )$ the homeomorphisms $h^{s} P_{\gamma } (h^{s})^{-1}$ and $h^{u} P_{\gamma } (h^{u})^{-1}$ . We can easily convince ourselves that the following lemma holds.

Lemma 3.2 is a consequence of the following lemma.

A proof of Lemma 3.3 can be found in [Reference BarbotBa].

3.4 A characterization of ${\mathbb R}$ -covered Anosov flows by complete and incomplete quadrants

In this section, each time we consider an Anosov flow X, we will assume that the bifoliated plane ${\mathcal P}_{X}$ is endowed with a choice of orientation of the foliations $F^{s}_{X},F^{u}_{X}$ .

Let $({\mathcal F},{\mathcal G})$ be two oriented transverse foliations on the plane ${\mathcal P}={\mathbb R}^{2}$ . This defines four quadrants at each point x: for any $\omega =(\omega _{1},\omega _{2})\in \{-,+\}^{2}$ , the (closed) quadrant $C_{\omega }(x)$ is the closure of the connected component of ${\mathcal P}\setminus ({\mathcal F}(x)\cup {\mathcal G}(x))$ bounded by the half leaves ${\mathcal F}_{\omega _{1}}(x),{\mathcal G}_{\omega _{2}}(x)$ .

Lemma 3.5. A transitive Anosov flow X is a suspension if and only if there exist $x\in {\mathcal P}_{X}$ and $\omega \in \{+,-\}^{2}$ such that the quadrants $C_{\omega }(x)$ and $C_{-\omega }(x)$ are trivially foliated.

Figure 4

In this figure the white circle should be considered as a point at infinity.

By Remark 3.1 and Lemma 3.4, we get the following criteria for deciding whether an Anosov flow is ${\mathbb R}$ -covered or not.

3.5 Stable and unstable holonomies and the completeness of the quadrants

Fix $x\in {\mathcal P}_{X}$ and consider $y\in F^{u}_{X}(x)$ . We call the map $h^{u}_{X,x,y}$ from $F^{s}_{X}(x)$ to $F^{s}_{X}(y)$ defined by

$$ \begin{align*}\{h^{u}_{X,x,y}(z)\}= F^{u}_{X}(z)\cap F^{s}_{X}(y),\quad\mbox{for } z\in F^{s}_{X}(x), \quad\mbox{if } F^{u}_{X}(z)\cap F^{s}_{X}(y)\neq \emptyset\end{align*} $$

an unstable holonomy from x to y. This definition is consistent as the intersection of a stable and an unstable leaves is at most one point.

The domain ${\mathcal D}(h^{u}_{X,x,y})$ is an interval of $F^{s}_{X}(x)$ and the image is an interval of $F^{s}_{X}(y)$ .

One defines in a analogous way the stable holonomy $h^{s}_{X,x,z}$ for $z\in F_{X}^{s}(x)$ .

Remark 3.3. For $x\in {\mathcal P}_{X}$ the quadrant $C_{+,+}(x)$ is complete if for any $y\in F^{u}_{+}(x)$ one has

$$ \begin{align*}F^{s}_{+}(x)\subset {\mathcal D}(h^{u}_{X,x,y}).\end{align*} $$

The quadrant $C_{+,+}(x)$ is undertwisted if there is $y\in F^{u}_{+}(x)$ such that ${\mathcal D}(h^{u}_{X,x,y})$ is a relatively compact interval in $F^{s}_{+}(x)$ .

Analogous statements hold in every quadrant and by exchanging the unstable holonomy for the stable holonomy.

3.6 Dehn–Goodman–Fried surgery

As explained in the introduction, it has recently been proven that the topological flow built by Fried’s surgery is orbitally equivalent to the Anosov flow obtained by Goodman’s surgery. Thanks to its explicitness, throughout the next pages we will make use of the action of Fried’s surgery on the bifoliated plane rather than its general definition which we quickly recall now.

Let X be an Anosov flow on a oriented $3$ -manifold M and let $\gamma $ be a periodic orbit with positive eigenvalues.

Consider the blow-up $\pi _{\gamma }\colon M_{\gamma }\to M$ of M along $\gamma $ , that is:

• • $M_{\gamma }$ is a manifold with boundary and $\partial M_{\gamma }$ is a torus $T_{\gamma } \simeq {\mathbb T}^{2}$ ;

• • $\pi _{\gamma }$ induces a diffeomorphism from the interior of $M_{\gamma }$ to $M\setminus \gamma $ ;

• • for every $x\in \gamma $ the fibre $\pi _{\gamma }^{-1}$ is a circle which is canonically identified with the unit normal bundle $N^{1}(x)$ of $\gamma $ in M at the point x.

In other words, consider two segments $\sigma _{1},\sigma _{2}$ in $M_{\gamma }$ transverse to the boundary $\partial M_{\gamma }$ at $\sigma _{i}(0)$ . Then $\sigma _{1}(0)=\sigma _{2}(0)$ if and only if $\pi _{\gamma }(\sigma _{1}(0))=\pi _{\gamma }(\sigma _{2}(0))=c$ and the segments $c_{1}=\pi _{\gamma }\circ \sigma _{1}$ and $c_{2}=\pi _{\gamma }\circ \sigma _{2}$ have the following property: the vector $({\partial c_{2}}/{\partial t})(0)$ belongs to the half plane of $T_{c(0)}M$ containing $\pm X(c(0))$ and $({\partial c_{1}}/{\partial t})(0)$ .

The vector field $\pi _{\gamma }^{-1}(X)$ is well defined on the interior of $M_{\gamma }$ , and extends by continuity on the boundary $T_{\gamma }$ by the natural action of the derivative $DX^{t}$ on the normal bundle over $\gamma $ . We denote by $X_{\gamma }$ this (smooth) vector field on $M_{\gamma }$ .

The flow on $T_{\gamma }$ is a Morse–Smale flow with four periodic orbits, which correspond to the normal vectors to $\gamma $ tangent to the stable and unstable manifolds of $\gamma $ . These four periodic orbits are freely homotopic to one another and are non-trivial in $\pi _{1}(T_{\gamma })$ . The homotopy (or homology) class $b\in {\mathbb Z}^{2}=\pi _{1}(T_{\gamma })$ of these periodic orbits is called the parallel.

On the other hand, the fibres of $\pi _{\gamma }\colon T_{\gamma }\to \gamma $ inherit an orientation from the orientation of M, and the corresponding homotopy class $a\in {\mathbb Z}^{2}=\pi _{1}(T_{\gamma })$ is called the meridian.

Given any integer $n\in {\mathbb Z}$ , one easily checks the existence of foliations ${\mathcal G}_{n}$ on $T_{\gamma }$ , transverse to the flow $X_{\gamma }$ , whose leaves are simple closed curves of homotopy class $a+nb$ . By reparametrizing the flow $X_{\gamma }$ , one gets a new smooth vector field $Y_{\gamma }$ on $M_{\gamma }$ that leaves the foliation ${\mathcal G}_{n}$ invariant.

Let $M_{\gamma ,n}$ be the manifold obtained from $M_{\gamma }$ by collapsing the leaves of ${\mathcal G}_{n}$ . The flow $Y_{\gamma }$ passes to the quotient and becomes a topological Anosov flow $X_{\gamma ,n}$ on $M_{\gamma ,n}$ .

It is easy to do this construction in a way such that $X_{\gamma ,n}$ is a Lipschitz vector field, but it is not clear at all that it can be smooth and Anosov. It is not even clear that the orbital equivalence class of the construction does not depend on the choice of the foliation ${\mathcal G}_{n}$ . Shannon proved that it is orbitally equivalent (by a homeomorphism isotopic to the identity) to an Anosov flow (the one built by Goodman), proving at the same time that the orbital equivalence class of this construction (in fact, the element of ${\mathcal A}_{0}(M_{\gamma ,n})$ ) is well defined. This element of ${\mathcal A}_{0}(M_{\gamma ,n})$ is called the Anosov flow $X_{\gamma ,n}$ obtained from Xby a surgery along $\gamma $ with characteristic number n.

4.1 The key tool: a common cover

We will denote by $\tilde \Gamma _{X}\subset \tilde M$ and $\tilde \Gamma _{Y}\subset \tilde N$ the lifts of $\Gamma $ to the universal covers $\tilde M$ and $\tilde N$ .

By a convenient abuse of language, we will also denote by $\tilde \Gamma _{X}$ and $\tilde \Gamma _{Y}$ the corresponding (discrete) sets in ${\mathcal P}_{X}$ and ${\mathcal P}_{Y}$ .

Let $V=M\setminus \Gamma =N\setminus \Gamma $ and Z be the restriction of X to V (or equivalently of Y to V).

The space of orbits in $\tilde V$ is a bifoliated plane, denoted by $({\mathcal P}_{\Gamma }, F^{s}_{\Gamma }, F^{u}_{\Gamma })$ . This bifoliated plane is the universal cover of $({\mathcal P}_{X},F^{s}_{X},F^{u}_{X})\setminus \tilde \Gamma _{X}$ and of $({\mathcal P}_{Y},F^{s}_{Y},F^{u}_{Y})\setminus \tilde \Gamma _{Y}$ . We denote by $\Pi _{X}$ and $\Pi _{Y}$ the natural projections of $\tilde V$ onto ${\mathcal P}_{X}$ and ${\mathcal P}_{Y}$ :

$$ \begin{align*} \begin{array}{rcccl} & &{\mathcal P}_{\Gamma} & & \\ &\overset{\Pi_{X}\;\;\;}\swarrow& &\overset{\;\;\;\Pi_{Y}}\searrow& \\ {\mathcal P}_{X}\setminus \tilde \Gamma_{X}& & & &{\mathcal P}_{Y}\setminus\tilde{\Gamma}_{Y} \end{array} \end{align*} $$

This simple fact has an important (straightforward) consequence.

One can use the following generalization of this argument.

4.2 Two ways to associate a path in ${\mathcal P}_{X}$ to a path in ${\mathcal P}_{Y}$

Let $\sigma _{X}\colon {\mathbb R} \to {\mathcal P}_{X}$ be a locally injective continuous path, obtained by the concatenation of locally finitely many stable/unstable leaf segments. One can define a transverse orientation as follows: the transverse orientation followed by the orientation of $\sigma _{X}$ is the orientation of ${\mathcal P}_{X}$ .

Assume that $p_{X}= \sigma _{X}(0) \notin \tilde \Gamma _{X}$ and $p_{Y}\in \Pi _{Y}(\Pi _{X}^{-1}(p_{X}))$ .

Let $\sigma _{X,t}\colon {\mathbb R} \to {\mathcal P}_{X}$ , $t\in [-1,1)$ be a continuous family of paths such that:

• • $\sigma _{X,0}=\sigma _{X}$ ;

• • $\sigma _{X,t}(0)=p_{X}$ ;

• • $\sigma _{X,t}$ is disjoint from $\tilde {\Gamma }_{X}$ ;

• • $\sigma _{X,t}$ tends to $\sigma _{X,0}$ from the positive side as $t\to 0$ and $t>0$ and from the negative side as $t\to 0$ and $t<0$ .

Corollary 4.1. Using the above notation, there are uniquely defined paths $\sigma _{Y,t}$ for $t>0$ (respectively, $t<0$ ) such that

• • $\sigma _{Y,t}(0)=p_{Y}$ , and

• • $\sigma _{Y,t}(s)\in \Pi _{Y}(\Pi _{X}^{-1}(\sigma _{X,t}(s)))$ , $s\in {\mathbb R}$ ,

and the limits

$$ \begin{align*}\lim_{t\to 0_{+}}\sigma_{Y,t}(s)=\sigma_{Y,+}(s) \quad\mbox{and}\quad\lim_{t\to 0_{-}}\sigma_{Y,t}(s)=\sigma_{Y,-}(s)\end{align*} $$

are well-defined continuous paths which are a concatenation of locally finitely many stable/unstable segments. Furthermore, $\sigma _{Y,+}$ and $\sigma _{Y,-}$ only depend on the choice of $p_{Y}$ and not on the choice of the homotopies $\sigma _{X,t}$ .

It is fairly easy to see that, in general, $\sigma _{Y,+}$ and $\sigma _{Y,-}$ do not coincide. An example of such a case is given in Figure 5. In this example, we consider a black path and a family of green and blue paths all with the same endpoints in ${\mathcal P}_{X}$ . Because of the surgery performed on p, by applying $\Pi _{Y} \circ \Pi _{X}^{-1}$ to a blue and a green path, we obtain two paths in ${\mathcal P}_{Y}$ that do not share the same endpoints. This is made more precise in Proposition 4.2.

Figure 5

In this picture the black points represent points in $\tilde {\Gamma }_{X,Y}$ on which we have performed non-trivial surgeries. Colour available online.

4.3 Comparison of holonomies: the main tool

Recall that, given an Anosov flow X, the orientations of the manifold M, of the bifoliated plane ${\mathcal P}_{X}$ and the foliations $F^{s}_{X}$ and $F^{u}_{X}$ are related by the convention introduced in §3.3.

Our main tool for comparing the holonomies of the foliations associated to X and Y is the next proposition.

In Proposition 4.2 one considers an orbit segment joining the points $b,c \in F^{s}_{+}(\tilde \gamma ) $ by turning around $\tilde \gamma $ in the positive sense. Analogous statements hold after changing the sign of the exponent of the first return map. Let us be more explicit, as this is crucial for our arguments.

Finally, let us note that the previous results hold independently of the sign of the eigenvalues of $\gamma $ .

4.4 Comparison of the holonomies in the quadrants: choosing the holonomies to be compared

Our next goal in this paper is to obtain the holonomies of $F^{s}_{Y}$ and $F^{u}_{Y}$ from the holonomies of $F^{s}_{X}$ and $F^{u}_{X}$ by using Proposition 4.2. In §4.5 we will first describe the change of holonomies for the unstable holonomies in the $C_{+,+}$ quadrants and then we will explain how to adapt the previous statement in all the other quadrants.

Before doing that, we need to explain which holonomies of $F^{u}_{Y}$ and $F^{u}_{X}$ will be compared. More precisely, consider a point $p_{X}\in {\mathcal P}_{X}$ , a point $q_{X}\in F^{u}_{+}(p_{X})$ and the unstable holonomy $h^{u}_{X,p_{X},q_{X}}$ from $F^{s}_{X,+}(p_{X})$ to $F^{s}_{X,+}(q_{X})$ . We want to describe the effect of the surgery on this holonomy and to compare the new holonomy with $h^{u}_{X,p_{X},q_{X}}$ .

Consider the path $\sigma _{X}$ obtained by the concatenation of:

• • the half stable leaf $F^{s}_{X,+}(p_{X})$ (with the negative orientation);

• • the unstable segment $[p_{X},q_{X}]^{u}$ ;

• • the half stable leaf $F^{s}_{X,+}(q_{X})$ .

We fix a parametrization of $\sigma _{X}$ so that the path $\sigma _{X}$ becomes a map $\sigma _{X}\colon {\mathbb R}\to {\mathcal P}_{X}$ .

4.4.1 The easy case: no point of $\tilde \Gamma $ on $\sigma _{X}$

Assume first that $\sigma _{X}$ is disjoint from $\tilde \Gamma _{X}$ . Consider a lift $p_{\Gamma }\in {\mathcal P}_{\Gamma }$ of $p_{X}$ and let $p_{Y}\in {\mathcal P}_{Y}$ be the projection of $p_{\Gamma }$ .

Now $\sigma _{X}$ has a well-defined lift $\sigma _{\Gamma }$ on ${\mathcal P}_{\Gamma }$ through $p_{\Gamma }$ . Consider $\sigma _{Y}$ the projection of $\sigma _{\Gamma }$ . In other words, we have

$$ \begin{align*}\sigma_{Y}=\Pi_{Y}(\Pi^{-1}_{X}(\sigma_{X})),\end{align*} $$

and $\sigma _{Y}(t)=\Pi _{Y}(\Pi ^{-1}_{X}(\sigma _{X}))$ will be called the corresponding point of $\sigma _{X}(t)$ in ${\mathcal P}_{Y}$ .

Thus $q_{Y}$ is the corresponding point of $q_{X}$ in ${\mathcal P}_{Y}$ and belongs to $F^{u}_{+}(p_{Y})$ . So the unstable holonomy $h^{u}_{Y,p_{Y},q_{Y}}$ from $F^{s}_{Y,+}(p_{Y})$ to $F^{s}_{Y,+}(q_{Y})$ is well defined.

As every point in $F^{s}_{+}(p_{X})$ (respectively, in $F^{s}_{+}(q_{X})$ ) has a corresponding point in $F^{s}_{+}(p_{Y})$ (respectively, in $F^{s}_{+}(q_{Y})$ ), it makes sense to compare $h^{u}_{X,p_{X},q_{X}}$ with $h^{u}_{Y,p_{Y},q_{Y}}$ .

4.4.2 The general case

In the next section we will need to compare holonomies of X and Y corresponding to stable leaves that intersect $\tilde \Gamma _{X}$ and $\tilde \Gamma _{Y}$ , in other words, we will consider the case where $\sigma _{X}$ is not disjoint from $\tilde {\Gamma }_{X}$ . In this case, $\sigma _{X}$ no longer lifts on ${\mathcal P}_{\Gamma }$ . We have seen in §4.2 that one may associate different paths $\gamma _{Y}$ in ${\mathcal P}_{Y}$ to a path $\gamma $ in ${\mathcal P}_{X}$ , depending, roughly speaking, on whether we choose to move at the right or at the left of x for every x in $\gamma _{X}\cap \tilde \Gamma _{X}$ .

The aim of this subsection is to fix our choices for the segment $\sigma _{X}$ .

Consider a point $p_{X}\in {\mathcal P}_{X}$ and $p_{Y}\in {\mathcal P}_{Y}$ , which are obtained in one of the following ways:

• • either as the projections of a same point $p_{\Gamma } \in {\mathcal P}_{\Gamma }$ , which means, in particular, that $p_{X}\notin \tilde \Gamma _{X}$ ;

• • or, if $p_{X}\in \tilde \Gamma _{X}$ , we consider a small rectangle $R_{X,+,+}$ admitting $p_{X}$ as its lower left corner and such that $R_{X,+,+} \cap \tilde {\Gamma }_{X}=\{p_{X}\}$ . We lift $R_{X,+,+}\setminus \{p_{X}\}$ on ${\mathcal P}_{\Gamma }$ and we project this lift on ${\mathcal P}_{Y}$ . One gets a rectangle $R_{Y,+,+}$ punctured at its lower left corner, which is denoted by $p_{Y}\in \tilde \Gamma _{Y}$ .

Given a point $q_{X}$ in $F^{u}_{X,+}(p_{X})$ , we previously defined a path $\sigma _{X}$ obtained by the concatenation of $\sigma ^{1}_{X}=F^{s}_{X,+}(p_{X})$ (with negative orientation), $\sigma ^{2}_{X}=[p_{X},q_{X}]^{u}$ and $\sigma ^{3}_{X}=F^{s}_{X,+}(q_{X})$ .

The point $q_{Y}\in {\mathcal P}_{Y}$ corresponding to $q_{X}$ will be the end point of the path $\sigma ^{2}_{Y,+}$ , defined in §4.2 and whose origin is $p_{Y}$ .

We want to compare the unstable holonomy $h^{u}_{X,p_{X},q_{X}}$ with the unstable holonomy $h^{u}_{Y,p_{Y},q_{Y}}$ for Y. In order to do that, we consider the path $\sigma _{Y}$ obtained by the concatenation of three paths (see Figure 7):

• • the half stable leaf $\sigma ^{1}_{Y,+}$ (which corresponds to $F^{s}_{Y,+}(p_{Y})$ , negatively oriented);

• • the unstable segment $\sigma ^{2}_{Y,+}$ (joining $p_{Y}$ to $q_{Y}$ );

• • the half stable leaf $\sigma ^{3}_{Y,-}$ (which corresponds to $F^{s}_{Y,+}(q_{Y})$ , positively oriented)

Figure 7

The dotted curves correspond to the approximations used in order to construct $\sigma _{Y}$ .

The construction of the paths $\sigma ^{i}_{Y,\pm }$ (see §4.2) induces a homeomorphism from $\sigma ^{i}_{X}$ to $\sigma ^{i}_{Y}$ , mapping $\sigma ^{i}_{X}(t)$ on $\sigma ^{i}_{Y}(t)$ . By gluing these homeomorphisms together, we get a homeomorphism between $\sigma _{X}$ and $\sigma _{Y}$ , mapping $\sigma _{X}(t)$ on $\sigma _{Y}(t)$ .

For any point $z_{X}=\sigma _{X}(t)$ , we define its corresponding point as $z_{Y}:=\sigma _{Y}(t)$ .

Remark 4.3. Our choice to define $\sigma _{Y}$ as the concatenation of the paths $\sigma ^{1}_{Y,+} \sigma ^{2}_{Y,+}$ and $\sigma ^{3}_{Y,-}$ may look arbitrary. According to the previous definition, $\sigma _{Y}$ corresponds to the projection on ${\mathcal P}_{Y}$ of the lifts of a sequence of (continuous) paths $\sigma _{X,n}$ disjoint from $\tilde \Gamma _{X}$ , converging to $\sigma _{X}$ and approaching $F^{s}_{X,+}(p_{X})$ from above, $[p_{X},q_{X}]^{u}$ from the right and $F^{s}_{X,+}(q_{X})$ from above. In particular, the paths $\sigma _{X,n}$ are contained in $C_{+,+}(p_{X})$ and intersect $F^{s}_{X,+}(q_{X})$ .

In the terms of the holonomy $h^{u}_{Y,p_{Y},q_{Y}}$ , this means that we will suppose that the segments $[y,h^{u}_{Y,p_{Y},q_{Y}}(y)]^{u}$ do not ‘cross’ $F^{s}_{Y,+}(p_{Y})$ , but they do ‘cross’ $F^{s}_{Y,+}(q_{Y})$ . (Another choice of $\sigma ^{i}_{Y}$ would not change the definition of the holonomy $h^{u}_{Y,p_{Y},q_{Y}}$ , but would change the parametrization of the path $\sigma _{Y}$ , thus interfering in the comparison of holonomies.)

This particular choice is convenient for composing holonomies.

4.5 Comparison of the holonomies in the quadrants: the formula

We are now ready to compare the holonomies $h^{u}_{X,p_{X},q_{X}}$ and $h^{u}_{Y,p_{Y},q_{Y}}$ .

Theorem 13. With the notation above, let $x_{Y}\in F^{s}_{Y,+} (p_{Y})$ . Then

$$ \begin{align*}h^{u}_{Y,p_{Y},q_{Y}}(x_{Y})=y_{Y}\end{align*} $$

if and only if there exist $\ell \in {\mathbb N}$ and two finite sequences $t_{i}\in {\mathbb R}$ and $x_{i}\in {\mathcal P}_{X}$ with $i\in \{0,\ldots , \ell \}$ such that the following statements hold.

1. (1) $x_{0}=x_{X}$ and $x_{l}=y_{X}$ .

2. (2) $\sigma _{X}(t_{0})=p_{X}$ , $\sigma _{X}(t_{\ell })=q_{X}$ .

3. (3) $t_{i}<t_{i+1}$ for $i\in \{0,\ldots , \ell -2\}$ and $t_{\ell -1} \leq t_{\ell }$ , therefore $\sigma _{X}(t_{i})\in [p_{X},q_{X}]^{u}$ . We denote $q_{X,i}=\sigma _{X}(t_{i})$ .

4. (4) For $i\in \{1,\ldots ,\ell -1\}$ there exists $\mu _{i}\in \tilde \Gamma _{X}$ (see Figure 8) such that the point $q_{X,i}$ belongs to $F^{s}_{X,-}(\mu _{i})$ and the point $x_{i}$ belongs to $F^{s}_{X,+}(\mu _{i})$ . We denote by $k_{i}$ the corresponding characteristic number of the surgery and we take $k_{0}=0$ .

   Figure 8

   In this figure we performed negative surgeries along the red periodic points ( ) and positive along the blue ones ( ). Every time we hit a stable manifold of either a blue or red point the holonomy is respectfully contracted or expanded. Colour available online.

   

5. (5) $\{x_{1}\}=F_{X}^{u}(x_{0})\cap F_{X}^{s}(q_{X,1})$ and $\{x_{i+1}\}=F_{X}^{u}(P^{k_{i}}_{\mu _{i}}(x_{i}))\cap F_{X}^{s}(q_{X,i+1})$ (where $P_{\mu _{i}}$ is the first return map of X associated to $\mu _{i}$ see Lemma 3.2) for $i \in \{1,\ldots ,\ell -1\}$ .

6. (6) Let $R_{i}$ for $i\in \{0,\ldots ,\ell -1\}$ be the rectangle ( $R_{\ell -1}$ can be degenerated) bounded by the segments $[q_{X,i},q_{X,i+1}]^{u}$ , $[q_{X,i+1}, x_{i+1}]^{s}$ , $[q_{X,i}, P^{k_{i}}_{\mu _{i}}(x_{i})]^{s}$ and $[P^{k_{i}}_{\mu _{i}}(x_{i}),x_{i+1}]^{u}$ . Then the interior of $R_{i}$ is disjoint from $\tilde \Gamma _{X}$ .

Let us make some remarks about the previous theorem.

4.6 The special case where X is a suspension: calculating the holonomies as a dynamical game

In this section we assume that X is the suspension flow of a hyperbolic Anosov diffeomorphism $f_{A}$ , where $A\in SL(2,{\mathbb Z})$ is a hyperbolic matrix with positive eigenvalues $0<\lambda ^{-1}<1<\lambda $ . We will explain how our arguments can be adapted for the case of matrices in $SL(2,{\mathbb Z})$ with negative eigenvalues in §4.6.4.

In this particular case, the bifoliated plane is trivial, so the holonomies are also trivial in ${\mathcal P}_{X}$ and the first return maps are simple to understand. This will simplify significantly the statement of Theorem 13.

We perform a linear change of coordinates on ${\mathcal P}_{X}={\mathbb R}^{2}$ so that $F^{s}_{X}$ is the horizontal foliation and $F^{u}_{X}$ is the vertical foliation. In these coordinates, A is the linear map

$$ \begin{align*}{\mathcal A}=\left(\begin{array}{@{}cc@{}} \lambda^{-1}&0\\ 0&\lambda \end{array}\right)\end{align*} $$

We now consider:

• • two finite sets ${\mathcal X}$ and ${\mathcal Y}$ of ${\mathbb T}^{2}$ which are disjoint and $f_{A}$ -invariant. Every point x in ${\mathcal X}\cup {\mathcal Y}$ is periodic and we denote by $\tau (x)$ its period. Notice that $\tau $ is invariant by $f_{A}$ .

• • two functions $m\colon {\mathcal X}\to {\mathbb N}$ and $n\colon {\mathcal Y}\to {\mathbb N}$ which are $f_{A}$ -invariant.

By a convenient abuse of language, we will still denote by ${\mathcal X}$ and ${\mathcal Y}$ the periodic orbits of the vector field X. In this way, the functions m and n become integer functions on this finite set of orbits of X.

We denote by $\tilde {\mathcal X}$ and $\tilde {\mathcal Y}$ the lifts of ${\mathcal X}$ and ${\mathcal Y}$ on ${\mathcal P}_{X}$ , which is canonically identified with the universal cover of the torus ${\mathbb T}^{2}$ . We still denote by $\tau $ , m and n the lifts of the previous functions.

In the previous section we defined the first return map $P_{ x}$ associated to a point $x\in {\mathcal P}_{X}$ corresponding to a periodic orbit of X. In our setting, the first return map associated to a point $ x\in \tilde {\mathcal X}\cup \tilde {\mathcal Y}$ is the affine map having $ x$ as its unique fixed point and ${\mathcal A}^{\tau (x)}$ as its linear part:

$$ \begin{align*}P_{x}= p\mapsto {\mathcal A}^{\tau(x)}(p-x) + x.\end{align*} $$

We denote by Y the vector field obtained from X by performing surgeries with characteristic numbers m on the orbits in ${\mathcal X}$ and $-n$ on the orbits in ${\mathcal Y}$ .

We denote

$$ \begin{align*}\mu= m\cdot \tau \quad\mbox{and}\quad \nu=n\cdot \tau.\end{align*} $$

The aim of this section is to express Theorem 13 in this particular setting.

Consider a point $p=p_{X}=(p^{s},p^{u})\in {\mathbb R}^{2}$ . We want to describe the holonomies of $F^{s}_{Y}$ and $F^{u}_{Y}$ in the quadrants $C_{\pm ,\pm }(p_{Y})$ , where $p_{Y}$ is the projection on ${\mathcal P}_{Y}$ of a lift of $p_{X}$ on the universal cover of ${\mathcal P}_{X}\setminus (\tilde {\mathcal X}\cup \tilde {\mathcal Y})$ . Let us start from the $C_{+,+}$ quadrant, in order to avoid useless formalism.

4.6.1 In the $C_{+,+}$ quadrants

Consider $r>0$ , $t_{0}>0$ , a point $q=(p^{s}, p^{u}+r)$ in the positive unstable manifold of p, a point $z_{0}=(p^{s}+t_{0}, p^{u})$ in the positive stable manifold of p, and their corresponding points $p_{Y},z_{0,Y}$ in ${\mathcal P}_{Y}$ . One would like to know if the holonomy $h^{u}_{p,q}$ of $F^{u}_{Y}$ from the positive stable manifold of $p_{Y}$ to the positive stable manifold of $q_{Y}$ is defined on $z_{0,Y}$ , and if this is the case, what its value is.

In order to answer the previous question, one considers the set of points $z_{s}=(p^{s}+t_{0}, p^{u}+s)$ in ${\mathcal P}_{X}$ , with $0<s < s_{1} \leq r$ , where $s_{1}$ is the smallest positive real for which there exists a point $\gamma _{1}=(p^{s}+u_{1},p^{u}+s_{1})\in \tilde {\mathcal X}\cup \tilde {\mathcal Y}$ , with $0<u_{1}<t_{0}$ . If such an $s_{1}$ does not exist, then the holonomy from $W^{s}_{+}(p)$ to $W^{s}_{+}(q)$ is defined on $(p^{s}+t_{0}, p^{u})$ and its value is $(p^{s}+t_{0}, p^{u}+r)$

If such an $s_{1}$ exists (see Figure 9), then one defines $z_{s}=(p^{s}+t_{1}, p^{u}+s)$ , with $s\in [s_{1},s_{2})$ , where

• • $t_{1}= u_{1}+\lambda ^{-\mu (\gamma _{1})}(t_{0}-u_{1})$ if $\gamma _{1}\in \tilde {\mathcal X}$ ,

• • $t_{1}= u_{1}+\lambda ^{\nu (\gamma _{1})}(t_{0}-u_{1})$ if $\gamma _{1}\in \tilde {\mathcal Y}$ , and

• • $s_{2}$ is the smallest positive number in $(s_{1},r]$ such that $z_{t}$ crosses the positive stable manifold of a point $\gamma _{2}=(p^{s}+u_{2},p^{u}+s_{2})\in \tilde {\mathcal X}\cup \tilde {\mathcal Y}$ , with $0<u_{2}<t_{1}$ .

Figure 9

In this figure the periodic points on which we performed positive surgery are represented by blue ( ) and the others by red ( ). Colour available online.

Analogously, if such an $s_{2}$ does not exist then the holonomy from $W^{s}_{+}(p)$ to $W^{s}_{+}(q)$ is defined on $(p^{s}+t_{0}, p^{u})$ and its value is $(p^{s}+t_{1}, p^{u}+r)$ . If $s_{2}$ exists, then one defines $z_{s}=(p^{s}+t_{2}, p^{u}+s)$ , $s\in [s_{2},s_{3})$ , where

• • $t_{2}= u_{2}+\lambda ^{-\mu (\gamma _{2})}(t_{1}-u_{2})$ if $\gamma _{2}\in \tilde {\mathcal X}$ ,

• • $t_{2}= u_{2}+\lambda ^{\nu (\gamma _{2})}(t_{1}-u_{2})$ if $\gamma _{2}\in \tilde {\mathcal Y}$ , and

• • $s_{3}$ is the smallest positive number in $(s_{2},r]$ such that $z_{t}$ crosses the positive stable manifold of a point $\gamma _{3}=(p^{s}+u_{3},p^{u}+s_{3})\in \tilde {\mathcal X}\cup \tilde {\mathcal Y}$ , with $0<u_{3}<t_{2}$ .

We define by induction the sequences $t_{i}$ , $s_{i+1}$ , $u_{i+1}$ , $\gamma _{i+1}$ : $z_{s}=(p^{s}+t_{i}, p^{u}+s)$ , $s\in [s_{i},s_{i+1})$ , where

• • $t_{i}= u_{i}+\lambda ^{-\mu (\gamma _{i})}(u_{i}-t_{i-1})$ if $\gamma _{i}\in \tilde {\mathcal X}$ ,

• • $t_{i}= u_{i}+\lambda ^{\nu (\gamma _{i})}(u_{i}-t_{i-1})$ if $\gamma _{i}\in \tilde {\mathcal Y}$ , and

• • $s_{i+1}$ is the smallest positive number in $(s_{i},r]$ such that $z_{t}$ crosses the positive stable manifold of a point $\gamma _{i+1}=(p^{s}+u_{i+1},p^{u}+s_{i+1})\in \tilde {\mathcal X}\cup \tilde {\mathcal Y}$ , with $0<u_{i+1}<t_{i}$ .

Now,

• • either this process is repeated infinitely many times, in which case the holonomy $h^{u}_{p,q}$ is not defined at the point $z_{0}$ ,

• • or the process ends when, for some $i\in \mathbb {N}$ , $s_{i}$ is not defined, in which case $h^{u}_{p,q}$ is defined at the point $z_{0}$ and

  $$ \begin{align*}h^{u}_{p,q}(z_{0})= z_{r}.\end{align*} $$

In this game, one sees that:

• • the points in $\tilde {{\mathcal X}}$ induce a contraction of the horizontal coordinate of $z_{s}$ , increasing the chances of the holonomy being defined on $z_{0}$ ;

• • a contrario the points in $\tilde {{\mathcal Y}}$ induce an expansion of the horizontal coordinate of $z_{s}$ , making it in this way more likely to meet the positive stable manifold of new points in $\tilde {\mathcal X}\cup \tilde {\mathcal Y}$ . If the new points are in $\tilde {{\mathcal Y}}$ , the expansion continues. This explains why, after surgeries, the quadrant $C_{+,+}(p)$ may be no more complete for Y. This is what happens if $\tilde {{\mathcal X}}$ is empty, which was already shown in [Reference FenleyFe1].

When playing the previous game, an important tool emerges: if two successive points $\gamma _{i}$ and $\gamma _{i+1}$ both belong to $\tilde {\mathcal X}$ (respectively, $\tilde {\mathcal Y}$ ), then there is a rectangle admitting $\gamma _{i}$ and $\gamma _{i+1}$ as corners, which is disjoint from $\tilde {\mathcal Y}$ (respectively, from $\tilde {\mathcal X}$ ). This rectangle will be the main object of §7. The existence or not of such rectangles is what determines the different cases that we consider in our study.