* $\tau$ is increasing on $\R_-$ and decreasing on $\R_+$, and the norm of its derivative is upper bounded by $2 s^\inv$.

Graph of the function $\tau \colon \R \to [0,1]$

The parameters $t_i$ determine the area of the beginning of the isotopy in the strip, and the parameters $s$ determines the speed of the isotopy (Figure <ref>).
For $p \in B$, if $\sigma_t$ is the leaf of $\cG^{u, \iin} \cap B$ passing through $p$, we define

\begin{equation} \label{eq: psi}
\psi(p) = \l( \Psi_{t, \tau(t)} (p) \r)^\inv
\end{equation}

This expression defines a diffeomorphism $\psi \colon B \to B$.

Area of the start of the isotopy (light) and the end of the isotopy (dark) on the strip $B$

Let us show each item for the  $\psi \colon B \to B$ defined by Equation (<ref>).

* By definition of $\tau$ (Item <ref> above), the support of $\psi$ is a compact of the strip $B$ equal to :

$$\operatorname{Supp} (\psi) =  \operatorname{Supp}(\hat\psi) \ssm
\l( \l( \underset{-\infty}{\overset{t_1-s}{\bigcup}} \sigma_t \r)
\cup \l( \underset{t_2 +s}{\overset{+\infty}{\bigcup}} \sigma_t \r) \r)$$

* It is clear that this diffeomorphism preserves the foliation $\cG^{u, \iin}$ leaf-to-leaf.

* Let $(\partial_u, \partial_s)$ be a pair of two unit vector fields tangent to $\cG^{u, \iin}$ and $\cG^{s, \iin}$ respectively.
Let $p \in B$, let $t$ be a parameter such that $\sigma_t$ is the leaf of $\cG^{u, \iin} \cap B$ passing through $p$, let $q = \psi^\inv(p)$.
Then the image of $\partial_s$ by the differential of $\psi^\inv$ is
$d_p \psi^\inv (\partial_s) = \tau'(t) \, \partial_u + (H^{u, \iin}_{\varsigma_p, \varsigma_q})'(p) \, \partial_s. $
According to Lemma <ref>, the derivatives of these holonomy maps are uniformly bounded.
It suffices to choose a function $\tau$ such that the derivative is close enough to 0, in other words a parameter $s$ large enough (Item <ref> of the definition of $\tau$ above), so that
$\vert \tau'(t) \vert \leq \epsilon \alpha^\inv$
where $\alpha$ is the constant given by Lemma <ref> which bounds from below the derivative of $H^{u, \iin}_{\varsigma_p, \varsigma_q}$.
Thus the image $\psi^\inv_* \partial_s$ is in a $(\epsilon,\cG^{s, \iin}/\, \cG^{u, \iin})$-cone field (Definition <ref>). This proves Item <ref>.

* According to Equation (<ref>) of the barycentric isotopy in each leaf of $\cG^{u, \iin}$, the derivative of $\psi^\inv$ in the direction of $\cG^{u, \iin}$ is bounded by the upper and lower bounds of the derivatives of the family of  $\{ \hat\psi_t \}_t$, in other words by the upper and lower bounds of the derivative of $\hat\psi$ in the direction $\cG^{u, \iin}$.
Recall that $\hat\psi$ (Equation (<ref>)) is the conjugate by $H^{s, \iin}_{\sigma, \sigma'}$ of the  $h \colon \sigma \to \sigma$ (Lemma <ref>).
Since the distortion of the holonomies $H^{s, \iin}_{\sigma', \sigma}$ is uniformly bounded by a constant $\vartheta>0$, (Lemma <ref>), we deduce that the derivative of $\hat\psi$ in the direction of $\cG^{u, \iin}$ is uniformly controlled by the derivative of $h$ :

$$ \forall v \in T \cG^{u, \iin}, \qq \vartheta^\inv \vert h' \vert \leq \Vert \hat\psi_* v \Vert \leq \vartheta \vert h' \vert $$

Since $h$ is a diffeomorphism with compact support, we conclude that the differential of $\psi$ in the direction $\cG^{u,\iin}$ is uniformly bounded by a constant which only depends on $\lambda$.
We thus obtain Item <ref>.

* Let us show that we can choose $t_1, t_2$ in order to obtain Item <ref>.
Let $c>0$ be a constant which bounds from below the derivative of $\psi$ in the direction $\cG^{u, \iin}$.
This constant exists according to the previous Item <ref> and is independent of $t_i$.
Denote $B_1$ the  of the end of $B$ corresponding to $t\leq t_1$, and $B_2$ the  of the end of $B$ corresponding to $t \geq t_2$.

* If the end $B_i$ accumulates on a periodic orbit $\cQ_i \in \cO_*$, then choose $t_i$ such that $\sigma_{t_i}$ is contained in a $\delta$-neighborhood $\cV_\delta \subset \cV_*$ of $\cQ_i$, where we recall that $\cV_*$ is the union of the normalized neighborhoods of the periodic orbits of $\cO_*$.

* If the end $B_i$ accumulates on a compact leaf $\cQ_i$ of $\cL^s$ then choose $t_i$ such that $\sigma_{t_i}$ is contained in the neighborhood $\cW^\iin_{\lambda c^\inv}$ given by Proposition <ref>.[we can have $\cQ_1 = \cQ_2$ in the non-orientable case]

Such a choice is always possible because $\sigma_{t} \to \cQ_i$ when $t \to \pm \infty$, and because the neighborhoods $\cW^\iin_\lambda$ of Proposition <ref> are neighborhoods of compact leaves of $\cL^s$.
Then we have

* On $\underset{[t_1, t_2]}{\bigcup} \sigma_t$, we have $\psi = \hat\psi$ and the expansion of the composition $\foutin \circ \psi$ in the direction $\cG^{u, \iin}$ is satisfied according to Fact <ref>, Item <ref>.

* By construction, we have
$B \ssm \l( \l( \underset{[t_1, t_2]}{\bigcup} \sigma_t \r) \cup \cV_\delta \r)  = \l(\underset{\R \ssm [t_1, t_2]}{\bigcup} \sigma_t \r) \ssm \cV_\delta \ \subset \ \cW_{\lambda c}.$
According to the previous Item <ref>, the foliation $\cG^{u, \iin}$ is preserved by $\psi$.
Hence for any $v \in T \cG^{u, \iin}$ at a point of this set, we have by Proposition <ref> and the previously shown Item <ref> of Lemma <ref>:
$\Vert (\foutin \circ \psi)_*v \Vert > \lambda c^\inv \Vert \psi_*v \Vert > \lambda \Vert v \Vert$.

* By construction the  $\psi$ coincides with $\hat \psi$ on the complementary of the normalized $\cV_*$ neighborhoods of the tangent $\cO_*$ periodic orbits in $B$, hence is independent of $\delta$ in this region.
Item <ref> is thus verified.

§ CHOICE OF THE PARAMETERS AND CONE CRITERION

In this section and for the rest of the proof, $\varphi$ denotes a normalized gluing map of $(P,X)$ (Definitions <ref>).
We refer to the beginning of Section <ref> for a reminder of the notations.
Let $\cU$ be a hyperbolic filtrating neighborhood of $\Lambda$ in $P$ given by Lemma <ref>, where $\Lambda$ is the maximal invariant set of $(P,X)$, and $\res{TP}{\Lambda} = E^\ss \oplus \R.X \oplus E^\uu$ the hyperbolic splitting over $\cU$.
The Riemannian metric $g$ on $P$ defined in the previous section is a metric adapted to the hyperbolic splitting on $\cU$.
By definition of a normalized block (Definition <ref>, Item <ref>), the block $(P,X)$ is provided with an affine section $\Sigma$[Such a section is defined in an extension of $(P,X)$, but we consider in this chapter its restriction to $P$], in other words a transverse local section of $\Lambda$ such that the first return map of the flow of $X$ is affine in the neighborhood of $\Sigma \cap \Lambda$.

In the previous section we constructed a family of  $\psi^\iin_\led \colon \pP \to \pP$ with support in $\Pin$ and $\psi^\out_\led \colon \pP \to \pP$ with support in $\Pout$ indexed by parameters $\lambda>1$, $\epsilon>0$, $\delta>0$, which satisfy the hypotheses of Proposition <ref> and Proposition <ref>.
Recall that:

* $\lambda>1$ is a lower bound for the expansion factor of the composition $\foutin \circ \psi^\iin$ in the direction $\cG^{u, \iin}$ outside a neighborhood of the orbits $\cO_*$,

* $\epsilon>0$ measures the perturbation of the foliation $\cG^{s, \iin}$ under the action of $(\psi^\iin)^\inv$,

* $\delta>0$ is the size of the neighborhood of the orbits $\cO_*$ on which we no longer control the dilation of the composition $\foutin \circ \psi^\iin$.

* $\lambda>1$ is a lower bound for the expansion factor of the composition $(\psi^\out \circ \foutin)^\inv$ in the direction $\cG^{s, \out}$ outside a neighborhood of the orbits $\cO_*$,

* $\epsilon>0$ measures the perturbation of the foliation $\cG^{s, \iin}$ under the action of $\psi^\out$,

* $\delta>0$ is the size of the neighborhood of the $\cO_*$ orbits on which we no longer control the dilation of the composition $(\psi^\out \circ \foutin)^\inv$ on $\cG^{s, \out}$.

§.§ Statement of the main proposition and summary of the proof

§.§.§ Gluing map

For any parameter , define the composition

$$\psi_\led := \psi^\iin_\led \, \varphi \, \psi^\out_\led \, \colon \pP \to \pP$$

where $\varphi$ is a normalized gluing map of $(P,X)$.
For any , $\psi^\iin$ and $\psi^\out$ are equal to the identity on a neighborhood of $\cO_*$ in $\pP$, so the same holds for $\psi$ and $\psi$ is a dynamical gluing map of $(P,X)$ (Definition <ref> and Proposition <ref>).
Moreover, $\psi^\iin$ and $\psi^\out$ are isotopic to the identity via  equal to the identity on a neighborhood of the boundary lamination $\cL$, so $\psi_\led$ and $\varphi$ are isotopic via strongly transverse gluing maps.

For all $\led$, if $\psi = \psi_\led$, then
$(P,X,\psi)$ is a triple strongly isotopic to $(P,X,\varphi)$, and the quotient space $P_\psi := P/\,\psi$ is a closed manifold of dimension three with a vector field of class $\cC^1$ induced by $X$ on $P_\psi$ which is denoted by $X_\psi$

Indeed, $\psi = \psi_\led$ is a gluing map of $(P,X)$ isotopic to $\varphi$.
Moreover, there is an isotopy $h_t \colon \pP \to \pP$ that joins $h_0 = \Id$ and $h_1 = \psi^\out \circ (\psi^\iin)^\inv$, which preserves the boundary lamination $\cL$, and such that $h_1$ maps a leaf of $\psi_* \cL$ to a leaf of $\varphi_* \cL$.
The triples are thus strongly isotopic in the sense of Definition <ref>.
The last affirmation relies on the fact that $\psi$ is a dynamical gluing map and Fact <ref>.

Let $\pi_\psi$ be the projection $P \to P_\psi$.
In order not to complicate the notations, denote $\Pin_\psi = \pi_\psi(\Pin)$, $\Sigma_\psi = \pi_\psi(\Sigma)$, $\pi_\psi(\cO_*) = (\cO_*)_\psi$, etc.
More generally the index $_\psi$ indicates that one considers an element projected in the quotient manifold $P_\psi$.

§.§.§ Global section in $P_\psi$

Note that the set $\Sigma \cup \Pin \cup \Pout$ in $P$ intersects all orbits of the flow of $X$ in uniformly bounded time.
The projection

$$ \pi_\psi(\Sigma \cup \Pin \cup \Pout) = \pi_\psi(\Sigma \cup \Pin) = \pi_\psi(\Sigma \cup \Pout) $$

is a non-compact, immersed surface in $P_\psi$, transverse to the orbits of $X_\psi$, which intersects all orbits of $X_\psi$ in uniformly bounded time.
$S_{0, \psi} := \pi_\psi (\Pin \cup \Sigma)$.
Let $f_{0, \psi} \colon S_{0, \psi} \to S_{0, \psi}$ the first return map of the flow of $X_\psi$ on $S_{0, \psi}$.
Note that we have $S_{0, \psi} = \Pin_\psi \cup \Sigma_\psi$.
The section $\Pin_\psi$ is a union of closed surfaces transverse to $X_\psi$ and closed surfaces quasi-transverse to $X_\psi$ minus the periodic orbits $(\cO_*)_\psi$.
It follows that the section $S_{0, \psi}$ is not uniformly transverse to the vector field $X_\psi$.
Note that we have the equality $\Pin_\psi = \Pout_\psi$, but we will make the arbitrary choice to consider $\Pin_\psi$ in the following.

The section $S_{0, \psi}$ in the neighborhood of a periodic orbit of $(\cO_*)_\psi$

§.§.§ The cone fields condition

[Cone fields condition]

Let $f \colon S \to S$ be a .
We say that $f$ satisfies the cone fields condition if there exists a pair of cone fields $(C^u, C^s)$ on $S$ and constants $\cst>0$, $\lambda>1$ such that for all $p \in S$:

* $f_* (C^u(p)) \subset \intr C^u(f(p))$, and for all $v \in C^u(p)$, $n \geq 0$,
$\Vert f^n_* v \Vert \geq \cst \lambda^n \Vert v \Vert$ .

* $f^\inv_* (C^s(f(p))) \subset \intr C^s(p)$,
and for all $v \in C^s(f(p))$, $n \geq 0$,
$\Vert f^{-n}_* v \Vert \geq \cst \lambda^n \Vert v \Vert$.

Analogously, we say that $(C^u, C^s)$ is a pair of $f$-invariant cone fields.
We say that $C^u$ is an unstable cone field and $C^s$ is a stable cone field for $f$.

§.§.§ Goal

The goal of this section is to choose parameters  such that if $\psi = \psi_\led$, the first return map $f_{0, \psi}$ has hyperbolic properties.
More precisely, we show that an iterate (which depends on the starting point) of $f_{0, \psi}$ satisfies the cone fields condition in the  of certain points.
However, the non-uniform transversality of the section $f_{0, \psi}$ with the vector field $X_\psi$ prevents the first return (or an iterate) from being uniformly hyperbolic at any point.
In the following Section <ref>, we will show that for such a choice of parameters, we can truncate the section $S_{0, \psi}$ by removing a well-chosen neighborhood of the orbits $\pi_\psi (\cO_*)$ to make it compact and uniformly transverse to the  $X_\psi$, and so that the first return map of the flow of $X_\psi$ on this new section is uniformly hyperbolic, up to a uniformly bounded iterate.
We have to specify some technical aspects before stating the result.

§.§.§ Self-intersections

Notice that $S_{0, \psi}$ has double or triple self-intersection points (Figure <ref>).
Indeed the intersection $\Sigma \cap \Pin$ is a double point locus in $P$ so likewise for the image in $P_\psi$. Moreover, a point of $\pi_\psi(\Sigma)$ can be in the image of two disjoint disks of $\Sigma$ in the neighborhood of two periodic orbits $\cO_i$ and $\cO_j$ of $\cO_*$.
The projections by $\pi_\psi$ of these disks intersect at some points and are not necessarily equal everywhere.
Then the surface $S_{0, \psi}$ has two or three tangent planes at its self-intersection points.

* If $p \in \Pin_\psi \subset S_{0, \psi}$, we denote $T^\iin_p \subset T_p S_{0, \psi}$ the tangent plane induced by $T_{\tilde p} P^\iin$, where $\tilde p \in \Pin$ is the lift of $p$ in $\Pin$

* If $q \in \Sigma_\psi \subset S_{0, \psi}$, we denote $T^\Sigma_q \subset T_p S_{0, \psi}$ the tangent plane(s) induced by $T_{\tilde q} \Sigma$, where $\tilde q \in \Sigma$ is a lift of $q$ in $\Sigma$.

The differential of $f_{0, \psi}$ in $p$ is well defined for a choice of tangent plane over $p$ and $f_{0, \psi}(p)$.
More precisely, for $p \in S_{0, \psi}$, once a tangent plane $T_p^*$ to $S_{0, \psi}$ has been chosen at point $p$ and $T^*_{f(p)}$, the differential $d_p f_{0, \psi} \colon T_p^* \to T_{f(p)}^*$ is well defined.

One might want to adjust the section $\Sigma$ to avoid triple points of $S_{0, \psi}$.
However, the choice of the gluing maps $\psi = \psi_\led$ determined by the parameters $\led$ will depend on the data of $\Sigma$, and one cannot proceed in this way.

§.§.§ Metric on the global section

The main proposition of this section involves expansion results on non-invariant sets.
We must first choose a metric on $S_{0, \psi}$.
The metric $g$ on $P$ (see beginning of Section <ref>) induces a metric $g^\iin$ on $\Pin$ and a metric $g^\Sigma$ on $\Sigma$.
For technical reasons, we will consider the metric $\hat g^\iin := \psi^\iin_*g^\iin$ which is the push-forward by $\psi^\iin$ of the metric $g^\iin$.
We then consider the quotient of $\hat g^\iin$ and $g^\Sigma$ over $\pi_\psi(\Pin \cup \Sigma)$, denoted $g_\psi$.
These metrics do not coincide along self-intersections and the image $g_\psi$ does not define a continuous metric.
However, one can always lift everything in $P$ by the local homeomorphism $\pi_\psi$ and pretend that there are no intersections and all the pieces coming from $\Sigma$ and $\Pin$ are disjoint in $P_\psi$.
We don't write it but all these objects depend on the choice of the parameters $\led{}$.

§.§.§ Cone field

A cone field $C$ on $S_{0, \psi}$ is the data, for each $p \in S_{0, \psi}$ and for each tangent plane $T^*$ to $S_{0, \psi}$ at the point $p$, of a cone $C(p, T^*) \subset T^* $.

* When we specify that a point $p$ in $S_{0, \psi}$ belongs to $\Pin_\psi$, we consider the tangent space $T^\iin_p \subset T_p \Sigma_\psi$ induced by $T_{\tilde p}  \Pin$ in $p$, and the metric induced by $\hat g^\iin$, in other words we lift by the local homeomorphism $\pi_\psi$ in the neighborhood of the unique lift $\tilde p \in \Pin$.

* When we specify that a point $p \in S_{0, \psi}$ belongs to $\Sigma_\psi$, we consider the tangent space $T^\Sigma_p \subset T_p \Sigma_\psi$ induced by $T_{\tilde p} \Sigma$ in $p$, and the metric induced by $g^\Sigma$,
in other words we lift by the local homeomorphism $\pi_\psi$ in the neighborhood of an (arbitrary) lift $\tilde p \in \Sigma$.

§.§.§ Statement of the main proposition

The following proposition states that one can choose parameters  and cones on $\Sigma$ and on $\Pin$ such that a (non-uniform) iterate of the first return map $f_{0, \psi} \colon S_{0, \psi} \to S_{0, \psi}$ of the flow of $X_\psi$ satisfies the cone fields condition in the neighborhood of some points.
More precisely, we show:

There are parameters $\lambda_0>1$, $\epsilon_0>0$, $\delta_0>0$, with the following properties.
We denote
$\psi^\iin = \psi^\iin_{\lambda_0, \epsilon_0, \delta_0}$,
$\psi^\out = \psi^\out_{\lambda_0, \epsilon_0, \delta_0}$,
$\psi = \psi^\iin \varphi \psi^\out$.
Denote $\cV_0 := \psi^\iin( \cV_{\delta_0})$.
There exists a pair $(C^u_\psi, C^s_\psi)$ of cone fields on $S_{0, \psi}$, and integers $N_0$, $N_1$ and $N_\Sigma$ which satisfy the following properties.
Let $p \in S_{0, \psi}$.

Suppose that $p \in (\Pin \cap \cV_0)_\psi$. Then $f^k_{0, \psi}(p) \in \Sigma_\psi$ for $k= -N_0, \dots, -1, 1, \dots, N_0$.

Let $n \geq 1$ and let $q := f_{0, \psi}^n(p)$.

Suppose $p \in (\Pin \ssm \cV_0)_\psi$, $q \in \Sigma_\psi$, $n \geq N_1$, and $f^k_{0,\psi}(p) \in \Sigma_\psi$ for $k=1, \dots, n-1$.

* $(f_{0, \psi}^n)_* C^u_\psi(p, T^\iin_p) \subset \intr C^u_\psi(q, T^\Sigma_q)$,
and for all $v \in C^u_\psi(p, T_p^\iin)$, $\Vert (f_{0, \psi}^n)_* v \Vert \geq 2 \Vert v \Vert$,

* $(f_{0, \psi}^n)^\inv_* C^s_\psi(q, T_q^\Sigma) \subset \intr C^s_\psi(p, T_p^\iin)$,
and for all $v \in C^s_\psi(q, T_q^\Sigma)$, $\Vert (f_{0, \psi}^n)^\inv_* v \Vert \geq 2 \Vert v \Vert$.

Suppose $p \in (\Pin \ssm \cV_0)_\psi$, $q \in (\ssm \cV_0)_\psi$ and $f^k_{0, \psi}(p) \in \Sigma_\psi$ for $k=1, \dots, n-1$.

* $(f_{0, \psi}^n)_* C^u_\psi(p, T_p^\iin) \subset \intr C^u_\psi(q, T_q^\iin)$,
and for all $v \in C^u_\psi(p, T_p^\iin)$, $\Vert (f_{0, \psi}^n)_* v \Vert \geq 2 \Vert v \Vert$,

* $(f_{0, \psi}^n)^\inv_* C^s_\psi(q, T_q^\iin) \subset \intr C^s_\psi(p, T_p^\iin)$,
and for all $v \in C^s_\psi(q, T_q^\iin)$, $\Vert (f_{0, \psi}^n)^\inv_* v \Vert \geq 2 \Vert v \Vert$.

Suppose that $p \in (\Pin \ssm \cV_0)_\psi$, $q \in \Sigma_\psi$, $n \geq N_0 +1$,
there exists $1 \leq m \leq n-N_0$ such that $f^m_{0, \psi}(p) \in (\Pin \cV_0)_\psi$ and $f^k_{0, \psi}(p) \in \Sigma_\psi$ for $k =1, \dots, m-1, m+1, \dots, n$.

* $(f_{0, \psi}^{n})_* C^u_\psi(p, T_p^\iin) \subset \intr C^u_\psi(q, T_q^\Sigma)$,
and for all $v \in C^u_\psi(p, T_p^\iin)$, $\Vert (f_{0, \psi}^n)_* v \Vert \geq 2 \Vert v \Vert$,

* $(f_{0, \psi}^{n})^\inv_* C^s_\psi(q, T_q^\Sigma) \subset \intr C^s_\psi(p, T_p^\iin)$,
and for all $v \in C^s_\psi(q, T_q^\Sigma)$, $\Vert (f_{0, \psi}^n)^\inv_* v \Vert \geq 2 \Vert v \Vert$.

Suppose $p \in \Sigma_\psi$, $q \in \Sigma_\psi$, and $f^k_{0, \psi}(p) \in \Sigma_\psi$ for $k=1, \dots, n-1$.

* $(f_{0, \psi}^n)_* C^u_\psi(p, T_p^\Sigma) \subset \intr C^u_\psi(q, T_q^\Sigma)$,

* $(f_{0, \psi}^n)_*^\inv C^s_\psi(q, T_q^\Sigma) \subset \intr C^s_\psi(p, T_p^\Sigma)$.

Moreover if $n \geq N_\Sigma$, then :

* for all $v \in C^u_\psi(p, T_p^\Sigma)$, $\Vert (f_{0, \psi}^n)_* v \Vert \geq 2 \Vert v \Vert$,

* for all $v \in C^s_\psi(q, T_q^\Sigma)$, $\Vert (f_{0, \psi}^n)^\inv_* v \Vert \geq 2 \Vert v \Vert$.

Suppose $p \in \Sigma_\psi$, $q \in \Pin_\psi$ and $f^k_{0, \psi}(p) \in \Sigma_\psi$ for $k=1, \dots, n-1$.

* $(f_{0, \psi}^n)_* C^u_\psi(p, T_p^\Sigma) \subset \intr C^u_\psi(q, T_q^\iin)$ and,

* $(f_{0, \psi}^n)_*^\inv C^s_\psi(q, T_q^\iin) \subset \intr C^s_\psi(p, T_p^\Sigma)$.

Suppose that $p \in \Sigma_\psi$, $q \in \Sigma_\psi$, $n \geq N_0+1$, there exists $1 \leq m \leq n-N_0$ such that $f_{0, \psi}^m(p) \in \Pin_\psi$, and $f^k(p) \in \Sigma_\psi$ for $k=1, \dots, m-1, m+1, \dots, n-1$.

* $(f_{0, \psi}^n)_* C^u_\psi(p, T_p^\Sigma) \subset \intr C^u_\psi(q, T_q^\Sigma)$,
and for all $v \in C^u_\psi(p, T_p^\Sigma)$, $\Vert (f_{0, \psi}^n)_* v \Vert \geq 2 \Vert v \Vert$,

* $(f_{0, \psi}^n)^\inv_* C^s_\psi(q, T_q^\Sigma) \subset \intr C^s_\psi(p, T_p^\Sigma)$, and
for all $v \in C^s_\psi(q, T_q^\Sigma)$, $\Vert (f_{0, \psi}^n)^\inv_* v \Vert \geq 2 \Vert v \Vert$.

Suppose $p \in \Sigma_\psi$, $q \in (\Pin \ssm \cV_0)_\psi$, and there exist $m \geq N_1$, and integers $1 \leq k_1 < \dots <k_m = n$ such that
and $f^{k_1}(p), \dots, f^{k_m}(p) \in (\Pin \ssm \cV_0)_\psi$
and $f^k(p) \in \Sigma_\psi$ for $k \neq k_1, \dots, k_m$, $1 \leq k \leq n-1$.

* $(f_{0, \psi}^n)_* C^u_\psi(p, T_p^\Sigma) \subset \intr C^u_\psi(q, T_q^\iin)$, and
for all $v \in C^u_\psi(p, T_p^\Sigma)$, $\Vert (f_{0, \psi}^n)_* v \Vert \geq 2 \Vert v \Vert$,

* $(f_{0, \psi}^n)^\inv_* C^s_\psi(q, T_q^\iin) \subset \intr C^s_\psi(p, T_p^\Sigma)$, and
for all $v \in C^s_\psi(q, T_q^\iin)$, $\Vert (f_{0, \psi}^n)^\inv_* v \Vert \geq 2 \Vert v \Vert$.

We refer to Figure <ref> and the associated subfigures for a diagram of the orbit corresponding to each of the items in the proposition.

Item <ref>

Item <ref>

Item <ref>

Action of $f^n_{0,\psi}$ for each orbit of Proposition <ref>

Item <ref>

Item <ref>

Item <ref>

Action of $f^n_{0,\psi}$ for each orbit of Proposition <ref>

Item <ref>

Item <ref>

Action of $f^n_{0,\psi}$ for each orbit of Proposition <ref>

§.§.§ Lifting the return map in $P$

For the proof, we will study lifts in $P$ of the iterates $f_{0, \psi}^n$.
If a lift $\tilde f_{0, \psi}^n$ maps a cone field $C$ on $\Sigma \cup \Pin$ on its interior in the neighborhood of a point $\tilde p$, and expands the norm of the vectors of $C$ by a factor $2$ for the metric $g$, then we can project into $P_\psi$ and we have the same result for the action $f_{0, \psi}^n$ on the cone field $C_\psi$ for the metric $g_\psi$ in $P_\psi$, in the neighborhood of $p= \pi_\psi(\tilde p)$.
The lifts $\tilde f_{0, \psi}^n$ are compositions of the crossing and return maps of the $X$-flow and the gluing map $\psi$.

* $f_\Sigma \colon \Sigma \to \Sigma$ the first return map of the flow of $X$ on $\Sigma$

* $f_{\Sigma, \iin} \colon \Pin \to \Sigma$ the crossing map of the flow of $X$ from $\Pin$ to $\Sigma$

* $f_{\out, \Sigma} \colon \Sigma \to \Pout$ the crossing map of the flow of $X$ from $\Sigma$ to $\Pout$

* $\foutin \colon \Pin \to \Pout$ the crossing map of the flow of $X$ from $\Pin$ to $\Pout$

We have the following

* If $p \in \Sigma_\psi$ and $f_{0, \psi}(p) \in \Sigma_\psi$ then a lift of $f_{0, \psi}$ is $f_\Sigma \colon \Sigma \to \Sigma$.

* If $p \in \Sigma_\psi$ and $f_{0, \psi}(p) \in \Pin_\psi$ then a lift of $f_{0, \psi}$ is $\psi \, f_{\out, \Sigma}: \Sigma \to \Pin$.

* If $p \in \Pin_\psi$ and $f_{0, \psi}(p) \in \Sigma_\psi$ then a lift of $f_{0, \psi}$ is $f_{\Sigma, \iin}: \Pin \to \Sigma$.

* If $p \in \Pin_\psi$ and $f_{0, \psi}(p) \in \Pin_\psi$ then a lift of $f_{0, \psi}$ is $\psi \, \foutin: \Pin \to \Pin$

§.§.§ Section summary

We will look for cones on $\Pin$ and on $\Sigma$ such that the lifts of the iterates $f^n_{0, \psi}$ satisfy the cone fields condition in the  of some points.
The cones of Proposition <ref> will be the projections in $P_\psi$ of the cones on $\Pin$ and on $\Sigma$.
The difficulty of the proof is to fix the parameters and the cones in the right order according to their relative dependence, in order to have the different compatibility conditions.
The steps are done in the following order.
Denote $\hat \cV_\delta = \psi^\iin(\cV_\delta)$.

* In Subsection <ref>,
we show the existence of the parameters $\epsilon=\epsilon_0$, $\lambda = \lambda_0$ and the cone fields $(C^u_\iin, C^s_\iin)$ on $\Pin$ which will allow us to obtain Item <ref> of Proposition <ref>, and this uniformly in $\delta$.

This item concerns the points $p \in (\Pin \ssm \hat \cV_\delta)_\psi$ whose future orbit by $X_\psi$ intersects again $(\Pin \ssm \hat \cV_\delta)_\psi$ in $q = f^n_{0,\psi}(p)$ (Figure <ref>).
We therefore consider a well chosen restriction of the return map of the flow of $X_\psi$ on the section $(\Pin)_\psi$.
A lift in $P$ of $f^n_{0, \psi}$ in the neighborhood of $p$ is $\psi \foutin \colon \Pin \to \Pin$, but for practical reasons we will study the composition
$ f_{\iin, \psi} = \varphi \psi^\out \foutin \psi^\iin \colon \Pin \to \Pin $
which is its conjugate by $\psi^\iin$.
We show that $f_{\iin, \psi}$ satisfies the cone fields condition for a good choice of cones $(C^u_\iin, C^s_\iin)$ on $\Pin$ and of parameters $\lambda$ and $\epsilon$ on the complementary of $\cV_\delta$, and this for all $\delta$.
The result is summarized in Proposition <ref>.
The key point is that one can choose $\lambda$ large enough to have arbitrarily strong expansion and contraction of two transverse directions by $f_{\iin, \psi}$, uniformly in the parameters $\epsilon$ and $\delta$.

The cone fields $(C^u_\iin, C^s_\iin)$ on $\Pin$ and the parameters $\epsilon=\epsilon_0$, $\lambda = \lambda_0$ will be fixed for the rest of the proof.

* In Subsection <ref>, we show the existence of cone fields $(C^u_\Sigma, C^s_\Sigma)$ on $\Sigma$ which will allow us to obtain Item <ref>, and which determine an integer $N_\Sigma$ which satisfies Item <ref> of Proposition <ref>, uniformly in $\delta$.

* Item <ref> concerns points $p \in \Sigma_\psi$ whose future orbit by $X_\psi$ intersects a first time $P^\iin_\psi$ in $q = f^n_{0,\psi}(p)$ (Figure <ref>).
A lift in $P$ of $f^n_{0, \psi}$ in the neighborhood of $p$ is the composition $\psi f_{\out, \Sigma} f_\Sigma^i \colon \Sigma \to \Pin$.
We will study
$\varphi \psi^\out f_{\out, \Sigma} f_\Sigma^i$
which is its conjugate by $\psi^\iin$ ($\psi^\iin$ is the identity outside $\Pin$).
We show the existence of two cone fields $(C^u_\Sigma, C^s_\Sigma)$ on $\Sigma$ such that a compatibility criterion between the cones on $\Sigma$ and the cones on $\Pin$ is satisfied for the composition $\varphi \psi^\out f_{\out, \Sigma} f_\Sigma^i$, and this uniformly in $\delta$.
This is Lemma <ref>.
The key point is that the projection $\pi \colon T\Sigma \to T\Pout$ parallel to $\R.X$ between $\Sigma$ and $\Pout$ (or $\Pin$) has a uniformly bounded effect on the slope of the cones (Lemma <ref>).

* Item <ref> concerns the points $p$ in $\Sigma_\psi$ whose future orbit by $X_\psi$ intersects $N_\Sigma$ consecutive times $\Sigma_\psi$ (Figure <ref>).
A lift in $P$ of $f^n_{0, \psi}$ in the neighborhood of $p$ is $f^n_\Sigma \colon \Sigma \to \Sigma$.
Once the cones are fixed on $\Sigma$ containing the suitable tangent directions, the hyperbolicity of $f_\Sigma$ ensures that an iterate greater than a uniform integer $N_\Sigma$ satisfies the cone fields condition.

The cones $(C^u_\Sigma, C^s_\Sigma)$ on $\Sigma$ and the integer $N_\Sigma$ will be fixed for the rest of the proof.

* In Subsection <ref>, we show the existence of an integer $N_0$ which will allow us to obtain Items <ref> and <ref> of Proposition <ref>, and this uniformly in $\delta$.

* Item <ref> concerns points $p \in \Sigma_\psi$ whose future orbit through $X_\psi$ intersects $\Pin_\psi$ once and then $N_0$ consecutive times $\Sigma_\psi$ (Figure <ref>).
A lift in $P$ of $f^n_{0, \psi}$ in the neighborhood of $p$ is the composition map
$f_\Sigma^j , f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^i \colon \Sigma \to \Sigma$.
We show that the pair $(C^u_\Sigma, C^s_\Sigma)$ on $\Sigma$ fixed in the previous section form a pair of strictly invariant cone fields under this map, for an integer $j$ greater than a certain uniform integer $N_0$, uniformly in $\delta$.
This property is summarized in Corollary <ref>.
The key point is that the contraction of the composition map
$ f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} $
is bounded below uniformly in $\delta$ (Lemma <ref>).

* Item <ref> is shown uniformly in $\delta$.
It concerns points $p \in (\Pin \ssm \hat \cV_\delta)_\psi$ whose future orbit by $X_\psi$ intersects once $(\Pin \cap \hat \cV_\delta )_\psi$ and then $N_0$ consecutive times $\Sigma_\psi$ (Figure <ref>).
A lift in $P$ of $f^n_{0, \psi}$ in the neighborhood of $p$ is the composition map
$f_\Sigma^j \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^i \, f_{\Sigma, \iin} \colon \Pin \ssm \cV_\delta \to \Sigma$.
We will study the composition
$f_\Sigma^j \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^i \, f_{\Sigma, \iin} \, \psi^\iin$
which is its conjugate by $\psi^\iin$.
We show that it satisfies the cone fields condition for the pair $(C^u_\iin \cup C^u_\Sigma, C^s_\iin \cup C^s_\Sigma)$ for an integer $j$ greater than a certain uniform integer $N_0$, and this uniformly in $\delta$.
This is Corollary <ref>.
The key result is that the contraction along an orbit of the flow of $X$ from $\Pin \ssm \cV_\delta$ to $\Sigma \cap \cV_\delta$ is bounded below, uniformly in $\delta$ (Lemma <ref>).

The integer $N_0$ is now fixed for the rest of the proof.

* In Subsection <ref>, we show the existence of a parameter $\delta_0$ which will allow us to obtain Item <ref> of Proposition <ref>.

The integer $N_0$ is fixed in the previous section.
We look for a parameter $\delta = \delta_0$ such that a point $p \in (\cV_0)_\psi= (\psi^\iin (\cV_{\delta_0}))_\psi$ has an orbit which intersects $N_0$ consecutive times the section $\Sigma_\psi$ in the past and in the future (Figure <ref>).
We will study the composition maps
$f_\Sigma^{k} \, f_{\Sigma, \iin} \, \psi^\iin$ and $f_\Sigma^{-k} \, f_{\out, \Sigma}^\inv \, (\varphi \psi^\out)^\inv$ restricted to $\Pin \cap \cV_\delta$.
We look for a parameter $\delta_0$ such that these maps are well defined for $k \geq N_0$.
A key result is that a point $p$ arbitrarily close to $\Lambda$ in $P$ has an orbit which intersects the section $\Sigma$ an arbitrarily large number of times in the past and in the future, if it is well defined.
We will use a key lemma which states that the action of $\psi^\iin$ and $\psi^\out$ on the distance to the set $\cO_*$ is bounded uniformly in $\delta$ (Lemma <ref>).

The parameter $\delta= \delta_0$ will be fixed for the rest of the proof.

* In Subsection <ref>, we show the existence of an integer $N_1$ which will allow us to have Items <ref> and <ref> of Proposition <ref>.

At this point, the parameters $(\led{}) =( \lambda_0,\epsilon_0,\delta_0)$ are now fixed as well as the cone fields $(C^u_\Sigma, C^s_\Sigma)$ and $(C^u_\iin, C^s_\iin)$ on $\Sigma$ and on $\Pin$.
Denote $\cV_0= \psi^\iin (\cV_0)$.

* Item <ref> concerns the points $p \in (\Pin \ssm \cV_0)_\psi$ whose future orbit by $X_\psi$ intersects $N_1$ consecutive times the section $\Sigma_\psi$ (Figure <ref>).
A lift in $P$ of $f^n_{0, \psi}$ in the neighborhood of $p$ is the composition map $f^j_\Sigma , f_{\Sigma, \iin} \colon \Pin \ssm \cV_\delta \to \Sigma$.
We will study the composition map
$f^j_\Sigma \, f_{\Sigma, \iin} \, \psi^\iin$
which is its conjugate by $\psi^\iin$.
We show that it satisfies the cone fields condition for the pair $(C^u_\iin \cup C^u_\Sigma, C^s_\iin \cup C^s_\Sigma)$ for an integer $j$ greater than some uniform integer $N_1$.
The key points are the uniform transversality between the vector field $X$ and the surface $\Pin \ssm \cV_0$, the hyperbolicity of $f_\Sigma$, and the fact that the cone fields $(C^u_\iin, C^s_\iin)$ and $(C^u_\iin \cup C^s_\iin)$ contain compatible directions.

* Item <ref> concerns the points $p \in \Sigma_\psi$ whose future orbit by $X_\psi$ intersects $(\Pin \ssm \cV_0)_\psi$ a number $N_1$ of times (Figure <ref>).
A lift in $P$ of $f^n_{0, \psi}$ in the neighborhood of $p$ is the composition maps $(\foutin \psi)^j f_{\out, \Sigma} f_\Sigma^i \colon \Sigma \to \Pin{}$.
We will study the conjugate by $\psi^\iin$.
We show that it satisfies the cone fields condition for the pair $(C^u_\iin \cup C^u_\Sigma, C^s_\iin \cup C^s_\Sigma)$, for an integer $j$ greater than some uniform integer $N_1$.
The key points are the hyperbolicity of $f_{\iin, \psi}$, and the bounded effect of the composition map $\varphi \psi^\out f_{\out, \Sigma} f_\Sigma^i$ on the slope of the cones and on the norm of the vectors.

The integer $N_1$ will be fixed for the rest of the proof.

* In Subsection <ref>, we show each item of Proposition <ref> using the results of the previous subsections.
Each previous result is true for the parameter choice $\lambda=\lambda_0$, $\epsilon = \epsilon_0$ and $\delta= \delta_0$.
We have to restore the conjugation defect by $\psi^\iin$ and then project into $P_\psi$.
In other words, we push by $\psi^\iin$ the cones and the metric on $\Pin$ and project on $P_\psi$ these cones and this metric, as well as the cones and the metric on $\Sigma$.
All the results follow afterwards.

§.§ Parameters λ_0, ε_0 and cone fields on Pîn̂

In this subsection
we show the existence of the parameters $\lambda = \lambda_0$, $\epsilon = \epsilon_0$ and the cone fields on $\Pin$ which will allow us to obtain Item <ref> of Proposition <ref>, and this uniformly in $\delta$.
This item concerns the points $p \in (\Pin \ssm \hat \cV_\delta)_\psi$ whose future orbit by $X_\psi$ intersects again $(\Pin \ssm \hat \cV_\delta)_\psi$.
Define the following composition map:

\begin{equation} \label{eq: f_in,psi}
f_{\iin, \psi} := \varphi \psi^\out \foutin \psi^\iin \colon \Pin \to \Pin
\end{equation}

Let us show the following

There exist parameters $\lambda_0 >1$, $\epsilon_0>0$,
a pair of continuous (closed) cone fields $(C^u_\iin,C^s_\iin)$ on $\Pin$,
there exist $K>0$, $k>0$, $k'>0$
such that for all $\delta >0$,
if $\psi^\iin = \psi_{\lambda_0, \epsilon_0, \delta}^\iin$ and $\psi^\out = \psi_{\lambda_0, \epsilon_0, \delta}^\out$ are the diffeomorphisms given by Proposition <ref> and <ref>, then

*  (directions)
the cone field $C^u_\iin$:

* is contained in a $(K, \cG^{u, \iin} /\, (\psi^\iin)^\inv_* \cG^{s, \iin})$-cone field,

* contains a $(k, \cG^{u, \iin} /\, (\psi^\iin)^\inv_* \cG^{s, \iin})$-cone field,

* contains a $(k', \ (\varphi \psi^\out)_* \cG^{u, \out} /\,\varphi_* \cG^{s, \out})$-cone field.

and the cone field $C^s_\iin$:

* is contained in a $(K,\ (\psi^\iin)^\inv_* \cG^{s, \iin}/\, \cG^{u, \iin})$-cone field.

* contains a $(k,\ (\psi^\iin)^\inv_* \cG^{s, \iin}/\, \cG^{u, \iin})$-cone field.

* contains a  $(k',\ \varphi_* \cG^{s, \out}/\,(\varphi \psi^\out)_* \cG^{u, \out})$-cone field.

*  (invariant cones)
For any $p \in (\Pin \ssm \cV_\delta) \cap f_{\iin, \psi}^\inv (\Pin \ssm \cV_\delta)$,

* $\l(f_{\iin, \psi}\r)_*C^u_\iin(p) \subset \intr  C^u_\iin (f_{\iin, \psi}(p))$, and for all $v \in C^u_\iin(p)$, $\Vert (f_{\iin, \psi})_* v \Vert \geq 2 \Vert v \Vert$

* $\l(f_{\iin, \psi}^\inv\r)_*C^s_\iin( f_{\iin, \psi} (p) ) \subset \intr  C^s_\iin (p)$, and for all $v \in C^s_\iin(f_{\iin, \psi} (p))$, $\Vert (f_{\iin, \psi})^\inv_* v \Vert \geq 2 \Vert v \Vert$

Decomposition of the map $f_{\iin, \psi}$ and action on the cone field $C^u_\iin$ on $(\Pin \ssm \cV_\delta)$

See Figure <ref>.
We will only show the existence of the unstable cone field $C^u_\iin$.
The proof to find a stable cone field is symmetric up to reversing the flow.
We do not require that these cones are disjoint, or complementary, so the two proofs are independent.
To show the existence of an unstable cone field for $f_{\iin, \psi}$ it is enough to find two transverse directions (not necessarily invariant), one uniformly expanded and the other uniformly contracted by $f_{\iin, \psi}$.
A cone field $C$ which contains the expanded direction and its image by $f_{\iin, \psi}$ in its interior, and whose adhesion does not contain the contracted direction is a good candidate.
A strong enough expansion and contraction factor ensures that such a cone field will be an unstable cone field, in other words strictly invariant and expanded by $f_{\iin, \psi}$.

The following lemma gives us the two transverse directions which are respectively expanded and contracted by a factor proportional to $\lambda$ (respectively $\lambda^\inv)$, on the complementary of a $\delta$-neighborhood of $\cO_*$ and its reciprocal image, and this uniformly in .

There exists a constant $\cst{}>0$ such that for any ,

* for all $p \in \Pin \ssm \cV_\delta$, for all $v \in T_p\cG^{u, \iin}$,
$\Vert (f_{\iin, \psi})_* v \Vert > \cst \lambda \Vert v \Vert $;

* for all $p \in f_{\iin, \psi}^\inv(\Pin \ssm \cV_\delta)$, for all $v \in T_p (\psi^\iin)^\inv_* \cG^{s, \iin}$,
$\Vert (f_{\iin, \psi})_* v \Vert < (\cst \lambda)^\inv \Vert v \Vert $.

We deduce the following corollary

There exists a constant $\cst>0$, such that for any , for any
$p \in (\Pin \ssm \cV_\delta) \cap f_{\iin, \psi}^\inv (\Pin \ssm \cV_\delta)$,
if $C \subset T_p\Pin$ is a $(K,\cG^{u, \iin}/\, (\psi^\iin)^\inv_* \cG^{s, \iin})$-cone, and $q = f_{\iin, \psi}(p)$, then:

* $(f_{\iin, \psi})_* (C) \subset \intr C'$ where $C'$ is a $\l(\frac{K}{(\cst \, \lambda)^2},\   \varphi\psi^\out_* \cG^{u, \out}/\varphi_* \cG^{s, \out}\r)$-cone in $T_q\Pin$,

* $\Vert (f_{\iin, \psi})_* v \Vert \geq \cst \lambda K^\inv \Vert v \Vert$ for all $v \in C$.

It is enough to check that the mapped directions are indeed those announced.
Recall that $\psi^\iin$ preserves $\cG^{u, \iin}$ (Proposition <ref>, Item <ref>), $\foutin \colon \Pin \to \Pout$ maps the pair $(\cG^{u, \iin}, \cG^{s, \iin})$ to the pair $(\cG^{u, \out}, \cG^{s, \out})$, and $\psi^\out$ preserves $\cG^{s, \out}$.
Thus we have:

\begin{align*}
(f_{\iin, \psi})_* \cG^{u, \iin} &= (\varphi \psi^\out \foutin \psi^\iin)_* \cG^{u,\iin} \\
&= (\varphi \psi^\out \foutin)_* \cG^{u,\iin}\\
&= (\varphi\psi^\out)_* \cG^{u,\out}
\end{align*}

\begin{align*}
(f_{\iin, \psi})_* (\psi^\iin)^\inv_* \cG^{s, \iin} &= (\varphi \psi^\out \foutin \psi^\iin)_* \, (\psi^\iin)^\inv_* \cG^{s, \iin} \\
&= (\varphi \psi^\out \foutin)_* \cG^{s,\iin}\\
&= \varphi_* \cG^{s,\out}
\end{align*}

The rest follows directly from Lemma <ref> when standing on the intersection of $\Pin \ssm \cV_\delta$ and its reciprocal image by $f_{\iin, \psi}$.

To show Lemma <ref>, we need the following lemma, which states that the derivative of $\psi_\led^\out$ in the direction tangent to $\cG^{u, \out}$ is uniformly (in ) bounded, and symmetrically, the derivative of $\psi_\led^\iin$ in the direction tangent to $\cG^{s, \iin}$ is uniformly (in ) bounded.

There exists a constant $\cst>0$, such that for any , if $\psi^\out = \psi^\out_\led$ and $\psi^\iin = \psi^\iin_\led$, then

* for all $p \in \Pout$ and $v \in T_p \cG^{u, \out}$, we have
$\cst^\inv \Vert v \Vert \leq \Vert \psi^\out_* v \Vert \leq \cst \Vert v \Vert$;

* for all $p \in \Pin$ and $v \in T_p \cG^{s, \iin}$, we have
$\cst^\inv \Vert v \Vert \leq \Vert \psi^\iin_* v \Vert \leq \cst \Vert v \Vert$.

Let us show the lemma for $\psi^\out$.
Let $v^u$ be a tangent vector to $\cG^{u, \out}$ and
$w := \psi^\out_*(v^u) = w^s + w^u \in T\cG^{s, \out} \oplus T \cG^{u, \out}$.
From Proposition <ref>, Item <ref>, we know that $w$ is in a $(\epsilon,\cG^{u, \out}/, \cG^{s, \out})$-cone, which means that
$\Vert w^s \Vert \leq \epsilon \Vert w^u \Vert$.
From Item <ref>, Proposition <ref>, we know that $\psi^\out$ preserves the foliation $\cG^{s, \out}$ leaf-to-leaf, and it follows that $w^u = (H^{s, \out}_{\varsigma_{q}, \varsigma_p})_* (v^u)$
where $H^{s, \out}_{\varsigma_{q}, \varsigma_p}$ is the holonomy map of $\cG^{s,\out}$ between the leaf $\varsigma_p \in \cG^{u, \out}$ passing through $p$ and the leaf $\varsigma_{q} \in \cG^{u, \out}$ passing through $q = \psi^\out(p)$.
As the support of $\psi^\out$ is contained in strips $B_i \subset \Pout \ssm \cL^u$ (Item <ref>, Proposition <ref>), we have $p, q \in B_i$ and the maps $H^{s, \out}_{\varsigma_{q}, \varsigma_p}$ are holonomy maps in the strips $B_i$, we can apply Lemma <ref> which states that the ratio $\frac{\Vert w^u \Vert}{\Vert v^u \Vert}$ is uniformly bounded.
It follows that the ratio $\frac{\Vert w \Vert}{\Vert v^u \Vert}$ is bounded uniformly in the parameters  for $\epsilon$ small enough.

Recall that $f_{\iin, \psi} = \varphi \psi^\out \foutin \psi^\iin$.
From Proposition <ref>, Item <ref> and <ref>, the composition $\foutin \psi^\iin$ expands by a factor $\lambda$ the vectors tangent to $\cG^{u, \iin}$ on $\Pin \ssm \cV_\delta$ and maps the direction tangent to $\cG^{u, \iin}$ to the direction tangent to $\cG^{u, \out}$.
According to the previous Lemma <ref>, the derivative of $\psi^\out$ in the direction $\cG^{u, \out}$ is bounded by a constant uniform in .
Moreover, $\varphi$ is a  with compact support, its differential is therefore uniformly bounded.
We deduce that there exists a constant $\cst>0$ uniform in  which satisfies Item <ref> of Lemma <ref>.

Let us show Item <ref>.
Let $p \in f_{\iin, \psi}^\inv (P^\iin \ssm \cV_\delta)$ and $v \in T_p (\psi^\iin)^\inv \cG^{s, \iin}$.
Then $w = (f_{\iin, \psi})_* v$ is a vector tangent to $T_q \varphi_* \cG^{s, \out}$ at the point $q = f_{\iin, \psi}(p) \in \Pin \ssm \cV_\delta$.
This item is equivalent to showing that $(f_{\iin, \psi}^\inv)_*$ expands the vector $w$ by a factor uniformly proportional to $\lambda$.
We have $f_{\iin, \psi}^\inv = (\psi^\iin)^\inv \psi^\out (\psi^\out \foutin)^\inv \varphi^\inv$.
We study the effect of each map on the vector $w \in T_q \varphi_* \cG^{s, \out}$.

* $\varphi^\inv$ is a compactly supported , so its differential is uniformly bounded and maps the direction tangent to $\varphi_* \cG^{s, \out}$ to the direction tangent to $\cG^{s, \out}$.
As $\varphi$ is a normalized gluing map, it maps $\Pin \ssm \cV_\delta$ over $\Pout \ssm \cV_\delta$ (Definition <ref>, Item <ref>).

* According to Proposition <ref>, Item <ref> and <ref>, the composition $(\psi^\out \foutin)^\inv$ expands by a factor $\lambda$ the vectors tangent to $\cG^{s, \out}$ on $\Pout \ssm \cV_\delta$ and maps the direction tangent to $\cG^{s, \out}$ to the direction tangent to $\cG^{s, \iin}$.

* By the previous Lemma <ref>, the derivative of $(\psi^\iin)^\inv$ in the direction $\cG^{s, \iin}$ is bounded by a constant uniform in ,

We deduce that $(f_{\iin, \psi}^\inv)_*$ expands the vector $w$ by a factor uniformly proportional to $\lambda$, so there exists a constant $\cst>0$ uniform in  which satisfies Item <ref> of Lemma <ref>.

Let us show the following lemma, which allows us to separate the suitable directions by a cone field (Figure <ref>).

There exists a closed cone field $C^u_\iin$ on $\Pin$ and a parameter $\epsilon_0$ such that for any $\lambda>1$, $\delta>0$, if $\psi^{\iin} = \psi_{\lambda, \epsilon_0, \delta}^{\iin}$ and $\psi^{\out} = \psi_{\lambda, \epsilon_0, \delta}^{\out}$,

* $\intr C^u_\iin$ contains the directions tangent to ${\psi^\iin}^\inv \cG^{u, \iin}$ and $\varphi \psi^\out \cG^{u, \out}$,

* $C^u_\iin$ does not contain the direction tangent to $(\psi^\iin)^\inv_* \cG^{s, \iin}$,

* $C^u_\iin$ is the bisector $(\cG^{u, \iin}/\, \cG^{s, \iin})$-cone field on $\Pin \cap \cV_*$.

Cone field $C^{u, \iin}$ on $\Pin$ in the neighborhood of $\cO_*$

Let us start by showing the following

There exists a closed cone field $C^u_\iin$ on $\Pin$ such that :

* $\intr (C^u_\iin)$ contains the directions tangent to $\cG^{u, \iin}$ and $\varphi_* \cG^{u, \out}$,

* $C^u_\iin$ does not contain the direction tangent to $\cG^{s, \iin}$,

* $C^u_\iin$ is the bisector $(\cG^{u, \iin}/ \, \cG^{s, \iin})$-cone field on $\Pin \cap \cV_*$.

Recall that the gluing map $\varphi$ is normalized for the normalized block $(P,X)$ equipped with the  $(\cG^s, \cG^u)$, hence $\varphi_*(\cG^{u,\out})$ is transverse to $\cG^{s, \iin}$ on $\Pin$ (Item <ref>, Definition <ref>).
Moreover $\varphi$ is the reflection in the  $(\cV_i, \xi_i)$, which implies that we have the relation $\varphi_* \cG^{u, \out} = \cG^{u, \iin}$ and $\varphi_* \cG^{s, \out} = \cG^{s, \iin}$ on $\Pin \cap \cV_*$.
According to this remark and by transversality of $\cG^{u, \iin}$ and $\cG^{s, \iin}$ on , there exists an open (thin) cone field around the direction tangent to $\cG^{s, \iin}$, which does not contain either the direction tangent to $\cG^{u, \iin}$ or the direction tangent to $\varphi_* \cG^{u, \out}$.
Moreover, we can choose this cone field as a bisector of $(\cG^{s, \iin}/\, \cG^{u, \iin})$ on $\Pin \cap \cV_*$.
The cone field $C^u_\iin$ is the complementary cone field.

There exists $\hat K>0$, $\hat k>0$, $\hat k'>0$, such that the field $C^u_\iin$:

* is contained in a $(\hat K,\ \cG^{u, \iin}/\, \cG^{s, \iin})$-cone field,

* contains a $(\hat k,\ \cG^{u, \iin}/\, \cG^{s, \iin})$-cone field.

* contains a $(\hat k',\ \varphi_* \cG^{u, \out}/\, \varphi_* \cG^{s, \out})$-cone field.

$\Pin$ is not compact, but on $\cV_* \cap \Pin$, the cone field $C^u_\iin$ coincides with the $(1,\cG^{u, \iin}/, \cG^{s, \iin}) = (1,\ \varphi_* \cG^{u, \out}/, \varphi_* \cG^{s, \out})$-cone field.
The complementary of these neighborhoods is compact in , which allows us to conclude.

We can now show Lemma <ref>.

According to Propositions <ref> and <ref>, we have $({\psi^\iin})_*^\inv\cG^{u, \iin} = \cG^{u, \iin}$, and the direction tangent to $({\psi^\iin})_*^\inv\cG^{s, \iin}$ is in a $(\epsilon,\cG^{s, \iin}/\,\cG^{u, \iin})$-cone field.
Similarly $(\varphi \psi^\out)_* \cG^{s, \out} = \varphi_* \cG^{s, \out}$, and the direction tangent to $(\psi^\out)_* \cG^{u, \out}$ is in a $(\varphi_* \cG^{u, \out}, \varphi_* \cG^{s, \out})$-cone field of slope uniformly proportional to $\epsilon$ (the uniform constant comes from the action of $\varphi$ which is a compactly supported ).
Let $C^u_\iin$ be the cone field given by Lemma <ref>.
According to Corollary <ref>,
it suffices to choose $\epsilon_0$ small before $\hat k$ and $\hat k'$.

Let $\epsilon=\epsilon_0$ satisfying Lemma <ref>.

There exists $K>0$, $k>0$, $k'>0$, such that for any $\lambda>1, \delta>0$, if $\psi^{\iin} = \psi_{\lambda, \epsilon_0, \delta}^{\iin}$ and $\psi^{\out} = \psi_{\lambda, \epsilon_0, \delta}^{\out}$, then
the cone field $C^u_\iin$:

* is contained in a $(K,\ \cG^{u, \iin}/\, (\psi^\iin)^\inv_* \cG^{s, \iin})$-cone field,

* contains a $(k,\ \cG^{u, \iin}/\, (\psi^\iin)^\inv_* \cG^{s, \iin})$-cone field,

* contains a $(k',\ (\varphi^\iin)_* \cG^{u, \out}/\, \varphi_* \cG^{s, \out})$-cone field.

This is a direct consequence of the first two items of Lemma <ref>, which states that the directions tangent to $\cG^{u, \iin} = (\psi^\iin)^\inv_* \cG^{u, \iin}$ and to $(\varphi \psi^\out)_* \cG^{u, \out}$ are contained in the interior of $C^u_\iin$ and the direction tangent to $(\psi^\iin)^\inv_* \cG^{s, \iin}$ is everywhere disjoint from $C^u_\iin$.

The cone field $C^u_\iin$ contains the suitable directions for $\epsilon= \epsilon_0$, and this is true for any choice of $\lambda$ and $\delta$.
It remains to show that we can choose $\lambda$ large enough so that the cone field $C^u_\iin$ is $f_{\iin, \psi}$-invariant and its vectors are expanded by $f_{\iin, \psi}$ on the well-chosen domain, and this uniformly in $\delta$.
We refer to Figure <ref> for an image of the action of $f_{\iin, \psi}$ on the cone field $C^u_\iin$.

Directions in the cone field $C^u_\iin$ and $(f_{\iin, \psi})_* C^u_\iin$

We fix $C^u_\iin$ given by Lemma <ref>.
Item <ref> of Proposition <ref> is satisfied by Corollary <ref> and let $ K>0$ and $ k'>0$ be the constants of the corollary.
Let $\cst>0$ be a constant which satisfies Corollary <ref>.
Then for $\lambda$ large enough compared to $\cst^\inv K k^\inv$, the field $C^u_\iin$ is strictly invariant by $f_{\iin, \psi}$ and its vectors are uniformly expanded on $(\Pin \ssm \cV_\delta) \cap f_{\iin, \psi}^\inv(\Pin \ssm \cV_\delta)$, and this uniformly in $\delta$.
The proof for the existence of a stable cone field $C^s_\iin$ is symmetric.

We fix the parameters $\lambda_0$, $\epsilon_0$ and a pair of cone fields $(C^u_\iin, C^s_\iin)$ satisfying Proposition <ref>.

§.§ Cone fields on Σ and integer N_Σ

In this section we show the existence of cone fields on $\Sigma$ which will allow us to obtain Item <ref>, and which determine an integer $N_\Sigma$ which satisfies Item <ref> of Proposition <ref>.
Recall that Item <ref> concerns the points $p \in \Sigma_\psi$ whose future orbit by $X_\psi$ intersects a first time $P^\iin_\psi$ in $q = f^n_{0,\psi}(p)$,
and Item <ref> concerns the points $p \in \Sigma_\psi$ whose future orbit by $X_\psi$ intersects $n$ consecutive times $\Sigma_\psi$.
Recall that the parameters $\lambda = \lambda_0$ and $\epsilon= \epsilon_0$ were fixed in the previous Subsection <ref>, as well as the cone fields $(C^u_\iin, C^s_\iin)$ on $\Pin$. The results will be uniform in $\delta$.
We study the maps:

$$f^i_\Sigma \colon \Sigma \to \Sigma, \qq \mb{ and } \qq \varphi \psi^\out f_{\out, \Sigma} f_\Sigma^i \colon \Sigma \to \Pin$$

Recall that $\Sigma$ is an affine section of $(P,X)$ (Definition <ref>, Item <ref>).
Up to restrict $\Sigma$, there exists a coordinate system on $\Sigma$ in which the map $f_\Sigma \colon \Sigma \to \Sigma$ is an hyperbolic diagonal affine map, wherever it is well defined.
Let $T\Sigma = E^\uu \oplus E^\ss$ be the hyperbolic splitting of $f$ on $\Sigma$, and
$(\zeta^s, \zeta^u)$ the pair of stable and unstable foliation of $f_\Sigma$ on $\Sigma$.
The metric $g_\Sigma$ induced by the ambient metric $g$ on $\Sigma$ is adapted to the hyperbolic splitting of $f_\Sigma$.

Let $\mu>1$ be a uniform factor of expansion of $f_\Sigma$, in other words such that

* for all $v \in E^\uu$, $n \geq 0$, $\Vert (f_\Sigma^n)_* v \Vert \geq \mu^n \Vert v \Vert$

* for all $v \in E^\ss$, $n \geq 0$, $\Vert (f_\Sigma^{-n})_* v \Vert \geq \mu^n \Vert v \Vert$

It follows that any closed cone field $C^u$ of constant slope whose interior $\intr C^u$ contains the direction $E^\uu$ and is disjoint from the direction $E^\ss$ is strictly invariant under the action of $f_\Sigma$ (at the first iteration) and its vectors are uniformly expanded by a (uniformly) large enough iteration $f^n_\Sigma$, which depends on the slope of the cone.
In other words, we have:

For any $K$, if $C^u$ is a $(K,\ \zeta^u/\,\zeta^s)$-cone field on $\Sigma$, then

* $(f_\Sigma)_* C^u$ is contained in a $(K \mu^{-2},\ \zeta^u/\, \zeta^s)$-cone field,

* there exists a constant $\cst = \cst(K) >0$ such that $\forall v \in C^u$, $\forall n \geq 0$, $\Vert (f^n_\Sigma)_* v \Vert \geq \cst \mu^n \Vert v \Vert$.

Similarly, if $C^s$ is a $(K,\ \zeta^s/\, \zeta^u)$-cone field on $\Sigma$, then

* $(f_\Sigma)^\inv_* C^s$ is contained in a $(K \mu^{-2},\ \zeta^s/\, \zeta^u)$-cone field,

* there exists a constant $\cst = \cst(K) >0$ such that $\forall v\in C^s$, $\forall n \geq N$, $\Vert (f^{-n}_\Sigma)_* v \Vert \geq \cst \mu^n \Vert v \Vert$.

We will choose the slope $K$ of a cone field $C^u_\Sigma$ of $(\zeta^u/\,\zeta^s)$-cone field such that we have a compatibility of cones on $\Sigma$ and cones on $\Pin$ for the composition map $\varphi, \psi^\out \, f_{\out, \Sigma} \, f_\Sigma^n : \Sigma \to \Pin$, for any $n \geq 0$, when it is well defined (Figure <ref>).
It will then suffice to choose an iterate of $f_\Sigma$ large enough for the vectors of $C^u_\Sigma$ to be uniformly expanded.
Denote $\psi^\iin_\delta = \psi^\iin_{\lambda_0, \epsilon_0, \delta}$ and $\psi^\out_\delta = \psi^\out_{\lambda_0, \epsilon_0, \delta}$.

There exist $K^u>0$ and $K^s>0$ such that if $C^u_\Sigma$ is a $(K^u,\ \zeta^u/\,\zeta^s)$-cone field and $C^s_\Sigma$ is a $(K^s,\ \zeta^s/\,\zeta^u)$-cone field on $\Sigma$, then they satisfy the following property. For all $\delta>0$, if $\psi^\out = \psi^\out_\delta$, and for all $p \in \Sigma$ such that the point $q = \varphi \, \psi^\out \, f_{\out, \Sigma} \, f_\Sigma^n \, (p) \in \Pin$ is well defined for some integer $n \geq 0$, then:

* $(\varphi \, \psi^\out \, f_{\out, \Sigma} \, f_\Sigma^n \,)_* C^u_\Sigma(p) \subset \intr C^u_\iin(q)$, and

* $(\varphi \, \psi^\out \, f_{\out, \Sigma} \, f_\Sigma^n \,)^\inv_* C^s_\iin(q) \subset \intr C^s_\Sigma(p) $

Trajectory (in dark) of a cone $C^u_\Sigma$ under the map $\varphi, \psi^\out \, f_{\out, \Sigma} \, f_\Sigma^n$ on $\Pin$

There is no requirement that the fields $C^s_\Sigma$ and $C^u_\Sigma$ be complementary to each other.
For the lemma to be true, the slope $K^s$ of $C^s_\Sigma$ will be large in front of 1, and the two cone fields can have a large intersection.
The important point of this lemma is that the slope $K^s$ and $K^u$ is independent of $\delta$.

Recall that the map $f_{\out, \Sigma}: \Sigma \to \Pout$ is the flow of $X$ from the section $\Sigma$ to the boundary $\Pout$.
In order to show this lemma we need to show that we keep a control on the slope of the cones when we apply $f_{\out, \Sigma} \colon \Sigma \to \Pout$.
This is the following lemma, which is symmetric for $f_{\out, \Sigma}$ and $f_{\Sigma, \iin}$.

There exists a constant $\cst>0$ such that for all $K>0$,

* if $C^u$ is a $(K,\ \zeta^u/\,\zeta^s)$-cone field on $\Sigma$, then $(f_{\out, \Sigma})_* C^u$ is contained in a $(\cst K,\ \cG^{u, \out}/\,\cG^{s, \out} )$-cone field on $\Pout$.

* if $C^s$ is a $(K,\ \cG^{s, \out}/\,\cG^{u, \out} )$-cone field on $\Pout$, then $(f_{\out, \Sigma})^\inv_* C^s$ is contained in a $(\cst K,\ \zeta^s /\, \zeta^u)$-cone field on $\Sigma$.

* if $C^u$ is a $(K,\ \cG^{u, \iin}/\,\cG^{s, \iin} )$-cone field on $\Pin$, then $(f_{\Sigma, \iin})_* C^u$ is contained in a $(\cst K,\ \zeta^u/\,\zeta^s)$-cone field on $\Sigma$.

* if $C^s$ is a $(K,\ \zeta^s/\,\zeta^u)$-cone field on $\Sigma$, then $(f_{\Sigma, \iin})^\inv_* C^s$ is contained in a $(\cst K,\ \cG^{s, \iin}/\,\cG^{u, \iin} )$-cone field on $\Pout$.

We will just show the result for $f_{\out, \Sigma}$.
This result is not trivial because $\Pout$ is not uniformly transverse to the vector field, so the projection $T P \to T\Pout$ parallel to $\R.X$ has an unbounded effect on the norm of vectors.
The key point is that the surface $\Pout$ is invariant by a symmetry which exchanges the strong stable direction $E^\ss$ and strong unstable direction $E^\uu$ in a neighborhood of $\cO_*$.
As a result, the effect of the projection on the norm of the vectors of $E^\uu$ and $E^\ss$ is the same, which ensures that the slope of a $(E^\uu/\,E^\ss)$-cone is preserved.

Let us show the first item. The proof of the others are analogous.
This lemma concerns the orbits which intersect $\Sigma$, which are thus contained in the invariant neighborhood $\hat \cU$ on which we have a hyperbolic splitting $E^\ss \oplus \R.X \oplus E^\uu$ for the flow of $X$ (Corollary <ref>).
Note that the lemma is true for all orbits intersecting $\Pout$ outside $\cV_*$.
Indeed the sections $\Sigma$ and $\Pout \ssm \cV_*$ are uniformly transverse to the vector field $X$, the flow of $X$ is hyperbolic on $\hat \cU$, and the cones in question are uniformly far from the center-stable bundle.
It remains to check this for the orbits intersecting $\Pout$ in a  $(\cV_i, \xi_i)$.

Let $C^u$ be a $(K,\ \zeta^u/\,\zeta^s)$-cone on $\Sigma$, and $v$ a vector tangent to $\Sigma$ at a point $p$.
Then $v = v^\uu + v^\ss + 0 \in E^\uu \oplus E^\ss \oplus \R. X$
with $\Vert v^\ss \Vert \leq K \Vert v^\uu \Vert$.
The transport by the (linear) flow $X^t$ during a time $t>0$ contributes to expand exponentially the component $v^\uu$ and contract exponentially the component $v^\ss$, and has no effect on the component $v^X=0$.
So the action of $X^t$ contributes to close a $(E^\uu/\,E^\ss)$-cone field exponentially fast, in other words to contract the slope of the cones exponentially fast.

It is therefore sufficient to look at the effect of the projection on $T\Pout$ parallel to $X$ of a $(K,\ E^\uu/\,E^\ss)$-cone field transverse to the vector field $X$ (contained in the plane $E^\uu \oplus E^\ss$) in a  $(\cV_i, \xi_i = (x, y, \theta))$.
Let us note this projection $\pi \colon E^\uu \oplus E^\ss \to T \Pout$.
To say that $\pi$ maps a $(K,\ E^\uu/\,E^\ss)$-cone onto a $(K,\ \cG^{u, \out}/\,\cG^{s, \out})$-cone is equivalent to saying that
$\pi$ maps the decomposition $E^\uu \oplus E^\ss$ onto $T\cG^{u, \out} \oplus T\cG^{s, \out}$ and that
$||| \res{\pi}{E^\uu} ||| = ||| \res{\pi}{E^\ss} |||$.
The first assertion is true because by definition we have $T\cG^{u, \out} = (E^\uu \oplus \R.X) \cap T\Pout$, and $T\cG^{s, \out} = (E^\ss \oplus \R.X) \cap T\Pout$.
The second one is shown with the following argument.
By definition of a normalized block (Definition <ref>, Item <ref>), the sets $\Sigma$, $\Pout$ and $\R.X$ are invariant in $\cV_i$ by the symmetry $s \colon (x, y, \theta) \mapsto (y, x, \theta)$, so
we have the commutation relation $\pi \circ s_* = s_* \circ \pi$.
We have $\Vect (\partial_x) = E^\ss$ and $\Vect (\partial_y) = E^\uu$,
and $s_* \partial_x = \partial_y$, so $s_*$ permutes the directions $E^\uu$ and $E^\ss$.
Finally, $s$ is an isometry for the metric $g$ which coincides with $d x^2 + d y^2 + d \theta^2$ on $\cV_i$.
We conclude by writing that $\res{\pi}{E^\uu} = \res{\pi \circ s_*}{E^\ss} = s_* \circ \res{\pi}{E^\ss}$.

To measure the action of $\psi^\out_\delta$, we need the following lemmas.
The first lemma states that the action of $\psi^\out_\delta$ on the distance to the orbit is bounded uniformly in $\delta$.

There exists a constant $\cst>1$ such that for all $\delta>0$,

* $\forall \, p \in \Pout{}$,
$\ \cst^\inv \, \dist (p, \cO_*) \leq \dist (\psi^\out_\delta(p), \cO_*) \leq \cst\, \dist (p, \cO_*)$;

* $\forall \, p \in \Pin{}$,
$\ \cst^\inv \, \dist (p, \cO_*) \leq \dist (\psi^\iin_\delta(p), \cO_*) \leq \cst\, \dist (p, \cO_*)$

We show the lemma for $\psi^\out_\delta$.
The  $\psi^\out_\delta$ is independent of the parameter $\delta$ on the complementary of the union $\cV_*$ (Proposition <ref>, Item <ref>).
The complementary of $\cV_*$ in $\Pout$ is compact, so the property is true for any point $p \in \Pout \ssm \cV_*$.
It is sufficient to check this property in a  $(\cV_i, \xi_i = (x, y, \theta))$.
Let $(x, \theta)$ be the coordinates induced by $\xi_i$ on $\Pout$.
A point $p=(x, \theta)$ is in the closure of a strip $B$ which is a  of $\Pout \ssm \cL^u$.
According to Items <ref> and <ref> of Proposition <ref>, the diffeomorphism $\psi^\out$ moves the point $p$ along the leaf arc of $\cG^{s, \out}$ inside $B$ and this for any parameter $\led$.
Let $\sigma^s$ be the leaf of $\cG^{s, \iin} \cap B$ passing through $p = (x,\theta)$.
Then $\sigma^s$ intersects one of the boundary leave of strip $B$ at a point $(x_1,\theta_1)$ with $x_1 > x/\sqrt{2}$ and the other boundary leave of $B$ at a point $(x_2, \theta_2)$ with $x_2 < \sqrt{2} x$ (Remark <ref>)
We conclude that the variation of the coordinate $x$ of a point $p \in B$ under the action of $\psi^\out$ is uniformly bounded.
The same is true for the distance to the orbit $\cO_i$.

The second lemma states that the differentials of the maps $\psi^\out_\delta$ and $\psi^\iin_\delta$ are bounded uniformly in $\delta$.

There exists $\cst>0$, such that for all $\delta>0$ and for all $v \in T\Pout{}$,

* for all $v \in T\Pout$, $\cst^\inv \Vert v \Vert \leq \Vert (\psi^\out_\delta)_* v \Vert \leq \cst\,\Vert v \Vert$

* for all $v \in T\Pin$, $\cst^\inv \Vert v \Vert \leq \Vert (\psi^\iin_\delta)_* v \Vert \leq \cst\,\Vert v \Vert$.

We show the lemma for $\psi^\out$.
Let $(\partial_u, \partial_s)$ be a basis of unit vectors tangent to $\cG^{u, \out}$ and $\cG^{s, \out}$ respectively on .
The following fact is a direct consequence of Item <ref> of Proposition <ref> and of Lemma <ref>.

The norm of the matrix of $(\psi^\out_\delta)_*$ in the basis $(\partial_u, \partial_s)$ is uniformly (in $\delta$) bounded.

This fact is not enough to bound the action of the differential because the angle between the vectors $(\partial_u, \partial_s)$ tends to 0 as we approach a periodic orbit of the boundary.
If $B$ is a field of basis on $\Pout$, and $f \colon \Pout \to \Pout$ is a , we denote $M_B(d_pf)$ the matrix of the differential from the basis $B(p)$ on $T_p \Pout$ to the basis $B(f(p))$ on $T_{f(p)} \Pout$.
Let $M^{B'}_{B, p}$ be the change of basis matrix from basis $B$ to basis $B'$ on $T_p \Pout$.
Let $B := (\partial_u, \partial_s)$ and $B'$ be a field of orthonormal basis on $T\Pout{}$.
We want to bound uniformly in $\delta$ the operator
$\Pout{} \ni p  \mapsto M_{B'} \l( d_p \psi^\out_\delta \r) \in GL_2(\R) $.
We have the following change of basis relation in $GL_2(\R)$, where $q := \psi^\out_\delta(p)$:

\begin{align*}
{M_{B'}\l( d_p \psi^\out_\delta \r)}
= {M^{B'}_{B, q}} \times {M_{B}\l( d_p \psi^\out_\delta \r)} \times {M^{B}_{B', p}}
\end{align*}

As the operator
$ p \mapsto {M_{B} (d_p\psi^\out_\delta)} $
is bounded on $\Pout{}$ uniformly in $\delta$ (Fact <ref>), it is sufficient to show that
$p \mapsto {M^{B'}_{B, q}} \times {M^{B}_{B', p}}$
is bounded on $\Pout{}$.
We can assume without loss of generality that the vector $\partial_u$ is the first vector of the orthonormal basis $B'$.
Therefore the product $ {M^{B'}_{B, q}} \times {M^{B}_{B', p}} $ is a $GL_2(\R)$ matrix of determinant
$\frac{\sin \theta_{B}(p)}{\sin \theta_{B}(q)}$,
where $\sin \theta_{B}$ denotes the sine of the angle of the vectors of the basis $B$.
We want to bound this ratio uniformly.
We compare the angle
$\measuredangle(\cG^{u, \out}, \cG^{s, \out})$ at the point $p \in \Pout$ and the point $q= \psi^\out_\delta(p)$.

\begin{align*}
\frac{\measuredangle(\cG^{u, \out}, \cG^{s, \out})_p}{\measuredangle(\cG^{u, \out}, \cG^{s, \out})_q} &
= \frac{\measuredangle(\cG^{u, \out}, \cG^{s, \out})_p}{\dist(p, \cO_*)} \times \frac{\dist(q, \cO_*)}{\measuredangle(\cG^{u, \out}, \cG^{s, \out})_q} \times  \frac{\dist(p, \cO_*)}{\dist(\psi^\out_\delta(p), \cO_*)}
\end{align*}

According to Lemma <ref> and Lemma <ref>, each of these ratios is bounded uniformly in $\delta$.
As explained above, Lemma <ref> follows from this.

We deduce the following lemma:

There exists a constant $\hat K$, such that for any $\delta$,

* $(\psi^\iin_\delta)_* C^u_\iin$ is contained in a $(\hat K,\cG^{u, \iin}/, \cG^{s, \iin})$-cone field.

* $(\varphi \psi^\out_\delta)^\inv_* C^s_\iin$ is contained in a $(\hat K,\ \cG^{s, \out}/\, \cG^{u, \out})$-cone field.

Let us show the first item.
The differential of $\psi^\iin_\delta$ is bounded uniformly in $\delta$ (Lemma <ref>) and
$C^u_\iin$ is contained in a $(K,\cG^{u, \iin}/\, (\psi^\iin)_*^\inv(\cG^{s, \iin}))$-cone field (Proposition <ref>, Item <ref>).
As $\psi^\iin_\delta$ preserves the foliation $\cG^{u, \iin}$ (Proposition <ref>, Item <ref>), we conclude that $\psi^\iin_\delta$ maps $C^u_\iin$ inside a $(\cG^{u, \iin}/\,\cG^{s, \iin})$-cone field of uniformly increased slope in $\delta$.

The second item is similar.
The field $C^s_\iin$ is contained in a $(K,\ \varphi_* \cG^{s,\out}/\, (\varphi \psi^\out)_* \cG^{u, \out})$-cone field
(Proposition <ref>, Item <ref>).
The  $\varphi^\inv$ is supported on a compact set so it maps a $(K,\ \varphi_* \cG^{s,\out}/\, (\varphi \psi^\out)_* \cG^{u, \out})$-cone field of $\Pin$ inside a $(K',\ \cG^{u, \out}/\, \psi^\out_* \cG^{u, \out})$-cone field where $K'$ is a uniform constant.
According to Lemma <ref>,
the differential of $(\psi^\out_\delta)^\inv$ is bounded on $\Pout$ uniformly in $\delta$.
Since $(\psi^\out_\delta)^\inv$ preserves the foliation $\cG^{s, \out}$ (Proposition <ref>, Item <ref>), we deduce that it maps a $(K',\ \cG^{u, \out}/\, \psi^\out_* \cG^{u, \out})$-cone field inside a $(\cG^{s, \out}/\, \cG^{u, \out})$-cone field of slope bounded below, uniformly in $\delta$.

We are now able to show Lemma <ref>.

Let us show the existence of the cone field $C^u_\Sigma$.
The effects of the differential of $\varphi \, \psi^\out \, f_{\out, \Sigma} \, f_\Sigma^n$ on the slope of a $(\zeta^u/\,\zeta^s)$-cone of $\Sigma$ are the following.

* $f_\Sigma^n$ maps a $(K,\ \zeta^u/\,\zeta^s)$-cone field on a $(K,\ \ \zeta^u/\, \zeta^s)$-cone field for $n \geq 0$ (Fact <ref>),

* $f_{\out, \Sigma}$ maps a $(K,\ \zeta^u/\, \zeta^s)$-cone field on $\Sigma$ inside a $(\cG^{u, \out}/\, \cG^{s, \out})$-cone field, of slope $K_1 \leq \cst K$ on $\Pout$ (Lemma <ref>),

* the differential of $\varphi \psi^\out$ is bounded on $\Pout$ uniformly in $\delta$ (Lemma <ref>), so it maps a $(K_1, \cG^{u, \iin}, \cG^{s, \iin})$-cone field inside a $(K_2,\ (\varphi \psi^\out)_* \cG^{u, \out}/\, \varphi_* \cG^{s, \out})$-cone field on , $K_2 \leq \cst K_1$ and the constant is uniform in $\delta$.

To summarize, a $(K,\ \zeta^u/\zeta^s)$-cone field on $\Sigma$ is mapped by $\varphi \, \psi^\out_\delta \, f_{\out, \Sigma} \, f_\Sigma^n$ inside a $(K',\ (\varphi \psi^\out)_* \cG^{u, \out}/\, \varphi_* \cG^{s, \out})$-cone field on $\Pout$, with $K' \leq \cst K$, where the constant is uniform in $\delta >0$.
Finally, according to Item <ref> of Proposition <ref>, it suffices to choose $C^u_\Sigma$ a $(K^u, \zeta^u/\,\zeta^s)$-cone field of small enough slope $K^u$ (in front of the constant $k'$ given by the proposition).

The existence of the $C^s_\iin$ field is shown symmetrically, and requires to choose a $(K^s,\ \zeta^s/\, \zeta^u)$-cone field of fairly large $K^s$ opening.

* $(\varphi \psi^\out)^\inv$ maps $C^s_\iin$ inside a $(\hat K,\cG^{s, \out}/\, \cG^{u, \out})$-cone field (Lemma <ref>),

* $f_{\out, \Sigma}^\inv$ maps a $(\hat K,\cG^{s, \out}/\,\cG^{u, \out})$-cone field on $\Pout$ inside a $(K_1,\ \zeta^u/\, \zeta^s)$-cone field on $\Sigma$, $K_1 \leq \cst \hat K$ (Lemma <ref>),

* $f_\Sigma^{-n}$ maps a $(K_1,\zeta^s/\,\zeta^u)$-cone field over a $(K_1,\zeta^s/\,\zeta^u)$-cone field for $n \geq 0$ (Fact <ref>)

To summarize, the field $C^s_\iin$ is mapped by $(\varphi, \psi^\out_\delta \, f_{\out, \Sigma} f_\Sigma^n)^\inv$ inside a $(K',\ \zeta^s/\,\zeta^u)$-cone field on $\Sigma$, where $K'$ is a constant uniform in $\delta >0$. So it suffices to choose $K^s = K'$ and $C^s_\Sigma$ is the $(K^s,\zeta^s/\,\zeta^u)$-cone field.

We fix a pair of cone fields $(C^u_\Sigma, C^s_\Sigma)$ on $\Sigma$ which satisfies Lemma <ref>.
The cone fields $C^u_\Sigma$ and $C^s_\Sigma$ have constant slope $K^u$ and $K^s$.
The following corollary (illustrated by Figure <ref>) follows directly from Fact <ref>.

Trajectory of a cone $C^u_\Sigma$ under $f_\Sigma^n$

There exists an integer $N_\Sigma \geq 0$ such that

* $(f_\Sigma)_* C^u_\Sigma \subset \intr C^u_\Sigma, \qq \mb { and } \qq \forall v \in C^u_\Sigma$, $\forall n \geq N_\Sigma$, $\Vert (f_\Sigma^n)_* v \Vert \geq 2 \Vert v \Vert$,

* $(f_\Sigma)^\inv_* C^s_\Sigma \subset \intr C^s_\Sigma, \qq \mb { and } \qq \forall v \in C^s_\Sigma$, $\forall n \geq N_\Sigma$, $\Vert (f_\Sigma^n)^\inv_* v \Vert \geq 2 \Vert v \Vert$

We fix an integer $N_\Sigma$ which satisfies Corollary <ref>.

§.§ Integer N_0

In this subsection, we show the existence of an integer $N_0$ which will allow us to obtain Items <ref> and <ref> of Proposition <ref> and this uniformly in $\delta$.
We recall that Item <ref> concerns the points $p \in \Sigma_\psi$ whose future orbit by $X_\psi$ intersects once $\Pin_\psi$ and then $N_0$ consecutive times $\Sigma_\psi$,
and Item <ref> concerns points $p \in (\Pin \ssm \hat \cV_\delta)_\psi$ whose future orbit through $X_\psi$ intersects a single time $(\Pin \cap \hat \cV_\delta )_\psi$ then $N_0$ consecutive times $\Sigma_\psi$.
Let $p \in \Sigma$ and $i, j \geq 0$ such that the following map is well defined in the neighborhood of $p$ :

$$f_\Sigma^j \, f_{\Sigma, \iin} \, \psi_\delta \, f_{\out, \Sigma} \, f_\Sigma^i(p) \colon \Sigma \to \Sigma$$

We look for an integer $N_0$, such that if $j \geq N_0$, the composition map above satisfies the cone fields condition on $\Sigma$ for the pair $(C^u_\Sigma, C^s_\Sigma)$.
We know that the iterates of $f_\Sigma$ contribute to close $C^u_\Sigma$ and expand its vectors exponentially (Fact <ref>).
We will show a result (Lemma <ref>) which gives a lower bound for the contraction of the composition map $f_{\Sigma, \iin} \, \psi_\delta \, f_{\out, \Sigma} \colon \Sigma \to \Sigma$ on the vectors of $C^u_\Sigma$ and on the slope of a $(\zeta^u/\,\zeta^s)$-cone, and this uniformly in $\delta$.
Let us start by estimating the effect of the crossing maps $f_{\out, \Sigma} \colon \Sigma \to \Pout$ and $f_{\Sigma, \iin} \colon \Pin \to \Sigma$.

There exists a constant $\cst >0$ such that:

* if $p \in \Sigma$ such that $q = f_{\out, \Sigma}(p) \in \Pout$ is well defined, then for any $v \in T_p \Sigma$

$$ \cst^\inv \dist(q, \cO_*)^\inv \leq \frac{\Vert (f_{\out, \Sigma})_* v\Vert}{\Vert v \Vert} \leq \cst \dist(q, \cO_*)^\inv   $$

* if $p \in \Pin$ such that $q = f_{\Sigma, \iin}(p) \in \Sigma$ is well defined, then for any $v \in T_p \Pin$

$$ \cst^\inv \dist(p, \cO_*)
\leq \frac{\Vert (f_{\Sigma, \iin})_* v\Vert}{\Vert v \Vert} \leq
\cst \dist(p, \cO_*) $$

Let $p \in \Sigma$ and $v \in T_p \Sigma$. Let $\tau = \tau(p)$ be the time such that
$X^\tau(p) = f_{\out, \Sigma}(p) = q \in \Pout$.
Since the segment of $X$-orbit $]p, q[$ is disjoint from $\Sigma$, the time $\tau$ is uniformly bounded (Fact <ref> with $\cU \subset \cup_{t=0}^1 \Sigma$).
Let $\pi \colon TP \to T\Pout$ be the projection on $T\Pout$ parallel to $\R.X$.
We have $\l( f_{\out, \Sigma} \r)_* v = \pi(X^\tau_*v)$.
Since the time $\tau$ is bounded, the norm $\Vert X^\tau_*v \Vert$ and the angle $\measuredangle(X^\tau_*v, X)_q$ are proportional to the norm $\Vert v \Vert$ and the angle $\measuredangle(X, v)_p$, respectively, and the proportionality constant is uniform in $p$.
From this observation and from the general Fact <ref>, there exists a uniform constant $\cst >0$ such that

\cst^\inv \l\vert \frac{\sin \measuredangle (v, X)_p}{\sin \measuredangle (f_{\out, \Sigma})_*v , X)_q} \r\vert
\leq \frac{\Vert \l( f_{\out, \Sigma} \r)_* v \Vert}{\Vert v \Vert} \leq
\cst \l\vert \frac{\sin \measuredangle (v, X)_p}{\sin \measuredangle (f_{\out, \Sigma})_*v , X)_q} \r\vert

On the other hand, by uniform transversality we know that the angle between a vector $v \in T\Sigma$ and $X$ is uniformly bounded,
and that the angle between a vector $v \in T_q\Pout$ and $X$ is bounded below by a factor uniformly proportional to the distance $\dist(\cdot, \cO_*)$ (Lemma <ref>).
We deduce the existence of a constant such that

\frac{\cst^\inv }{\dist(q, \cO_*)}
\leq \frac{\Vert \l( f_{\out, \Sigma} \r)_* v \Vert}{\Vert v \Vert} \leq
\frac{\cst }{\dist(q, \cO_*)}

The same arguments applied to $f_{\Sigma, \iin}^\inv$ allow us to prove the second item.

The following lemma bounds uniformly in $\delta$ the differential of the composition map $f_{\Sigma, \iin} \, \psi_\delta \, f_{\out, \Sigma}$ (when well defined) and its action on the slope of the cone field $C^u_\Sigma$ and $C^s_\Sigma$.

* the differentials of
$f_{\Sigma, \iin} \, \psi_\delta \, f_{\out, \Sigma} \colon \Sigma \to \Sigma$
are bounded uniformly in $\delta$;

* there exists a constant $K>0$ uniform in $\delta$ such that

* $(f_{\Sigma, \iin} \, \psi_\delta \, f_{\out, \Sigma})_* C^u_\Sigma$
is inside a $(K, \zeta^u/\,\zeta^s)$-cone field,

* $(f_{\Sigma, \iin} \, \psi_\delta \, f_{\out, \Sigma})^\inv_* C^s_\Sigma$
is inside a $(K, \zeta^s/\,\zeta^u)$-cone field.

Let $\psi = \psi_\delta$.
Let $p \in \Sigma$ such that $q = f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \in \Sigma$ is well defined.
Let us write

\frac{\Vert (f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma})_* v \Vert}{\Vert v \Vert } =
\underset{(a)}{\underbrace{\frac{\Vert (f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma})_* v \Vert}{\Vert ( \psi \, f_{\out, \Sigma})_* v \Vert }}} \
\underset{(b)}{\underbrace{\frac{\Vert (\psi \, f_{\out, \Sigma})_* v \Vert }{\Vert (f_{\out, \Sigma})_* v \Vert }}} \
\underset{(c)}{\underbrace{\frac{\Vert (f_{\out, \Sigma})_* v \Vert }{\Vert v \Vert }}}

Let us note $p_1 = f_{\out, \Sigma}(p) \in \Pout$ and $p_2 = \psi(p_1) \in \Pin$.
There exists a constant $\cst>0$ such that

* from Lemma <ref>: $ \cst^\inv \dist(p_2, \cO_*)
\leq (a) \leq \cst \dist(p_2, \cO_*) $;

* from Lemma <ref>: $\cst^\inv \leq (b) \leq \cst $;

* from Lemma <ref>: $ \cst^\inv \dist(p_1, \cO_*)^\inv
\leq (c) \leq
\cst \dist(p_1, \cO_*)^\inv $.

Up to change the constant, we have

$$ \cst^\inv \frac{\dist(p_2, \cO_*)}{\dist(p_1, \cO_*)} \leq
\frac{\Vert (f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma})_* v \Vert}{\Vert v \Vert }
\leq \cst \frac{\dist(p_2, \cO_*)}{\dist(p_1, \cO_*)}

Since $p_2 = \psi(p_1)$, according to Lemma <ref>, up to change the constant, we have a constant $\cst$ uniform in $\delta$ such that

\leq \frac{\vert (f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma})_* v \vert }{\vert v \vert } \leq
\cst. $$

Let us show the second item.
We prove only the statement concerning the image of the cone field $C^u_\Sigma$, the one for $C^s_\Sigma$ being symmetric.

* We know from Lemma <ref> that the composition map $\varphi \psi^\out f_{\out, \Sigma} \colon \Sigma \to \Pin$ maps $C^u_\Sigma$ inside $C^u_\iin$, and this for any $\delta>0$.

* The differential of $\psi^\iin_\delta$ is bounded uniformly in $\delta$ (Lemma <ref>) and
$C^u_\iin$ is contained in a $(K,\cG^{u, \iin}/,(\psi^\iin)_*^\inv(\cG^{s, \iin}))$-cone field (Proposition <ref>, Item <ref>).
Since $\psi^\iin_\delta$ preserves $\cG^{u, \iin}$, we deduce that $\psi^\iin_\delta$ maps $C^u_\iin$ inside a $(\cG^{u, \iin}/\,\cG^{s, \iin})$-cone field with a slope bounded uniformly in $\delta$.

* Using Lemma <ref>, we deduce that the crossing map $f_{\Sigma, \iin}$ maps the cone field $\psi^\iin_* C^u_\iin$ inside a $(\zeta^u/\,\zeta^s)$-cone field on $\Sigma$ with a slope bounded uniformly in $\delta$.

Recall that we study the composition
$f_\Sigma^j \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^i$.
We can now prove the existence of the uniform integer $N_0$ in $\delta$ such that this composition satisfies the cone fields condition for the pair $(C^u_\Sigma, C^s_\Sigma)$ on $\Sigma$ for $j \geq N_0$.
This is the following corollary.

There exists $N_0$ such that for any $\delta>0$, if
$f_\Sigma^j \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^i$ is well defined in $p \in \Sigma$ for some $i \geq 0$ and $j \geq N_0$, then, denoting $q$ the image of $p$,

* $\l(f_\Sigma^j \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^i\r)_*$ maps $C^u_\Sigma(p)$ inside $C^u_\Sigma(q)$ and expands the norm of the vectors of $C^u_\Sigma(p)$ by a factor 2,

* $\l( f_\Sigma^j \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^i \r)_*^\inv$ maps $C^s_\Sigma(q)$ inside $C^s_\Sigma(p)$ and expands the norm of the vectors of $C^s_\Sigma(q)$ by a factor 2

Trajectory of a cone $C^u_\Sigma$ under the map $f_\Sigma^{N_0} \, f_{\Sigma, \iin} \, f_{\out, \Sigma} \, f_\Sigma^i$

We decompose $f_\Sigma^j \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^i$ in three pieces:

* $f_\Sigma^i$ leaves the cones $C^u_\Sigma$ invariant and does not contract the vectors of $C^u_\Sigma$ for $i \geq 0$ (Corollary <ref>),

* $ f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma}$ maps $C^u_\Sigma$ to a $(\zeta^u/\,\zeta^s)$-cone on $C^u_\Sigma$, of uniformly bounded slope, and contracts the vectors of $C^u_\Sigma$ by a factor uniformly bounded below, and the constants are uniform in $\delta$ (Lemma <ref>),

* $f_\Sigma^j$ maps a $(\zeta^u/\,\zeta^s)$-cone field of bounded slope inside a $(\zeta^u/\,\zeta^s)$-cone field of arbitrarily small slope for $j$ large enough, and expands its vectors arbitrarily strongly (Fact <ref>).

We deduce the existence of an integer $N_0$ which satisfies the first item if $j \geq N_0$.
The proof of the second item is symmetric.

Let us show that the composition $f_\Sigma^n, f_{\Sigma, \iin} \, \psi^\iin$ restricted to $\Pin \ssm \cV_\delta$ at the domain and $\Sigma \cap \cV_\delta$ at the destination has a contraction bounded below on the norm of the vectors of $C^u_\iin$, and the bound is uniform in $\delta$, moreover the map contributes to close a $(\zeta^u/\,\zeta^s)$-cone field.

There exists a constant $\cst>0$ such that for any $\delta>0$ small enough, if $\psi^\iin = \psi^\iin_\delta$,
for any $p \in \Pin \ssm \cV_\delta$ such that there exists $n \geq 1$ with the image
$q = f_\Sigma^n \, f_{\Sigma, \iin} \, \psi^\iin(p) \in \Sigma \cap \cV_\delta$ well defined, then

* $(f_\Sigma^n \, f_{\Sigma, \iin} \, \psi^\iin)_* C^u_\iin(p)\, \subset \, \intr C^u_\Sigma(q), \qq \mb{and} \qq \forall \, v \in C^u_\iin(p)$, $\Vert (f_\Sigma^n \, f_{\Sigma, \iin} \, \psi^\iin)_* v \Vert \geq \cst \Vert v \Vert$

* $(f_\Sigma^n \, f_{\Sigma, \iin} \, \psi^\iin)^\inv_* C^s_\Sigma(q) _, \subset \, \intr C^s_\iin(p), \qq \mb{and} \qq \forall v \in C^s_\Sigma(q)$, $\ \Vert (f_\Sigma^n \, f_{\Sigma, \iin} \, \psi^\iin)^\inv_* v \Vert \geq \cst \Vert v \Vert$

Let us show the first item, the proof of the second being similar.
In order not to introduce new constants, we can assume that each normalized neighborhood $\cV_i$ of $\cO_i$ contains a $1$-neighborhood of $\cO_i$.
We denote $\cV_1$ the union of $1$-neighborhoods of $\cO_i$ in $\cV_i$.
The proof relies on the following key points.

* The differential of $f_{\Sigma, \iin}$ is bounded below by a factor uniformly proportional to $\dist(\cdot, \cO_*)$ from Lemma <ref>, so greater than $\delta$ on $\Pin \ssm \cV_\delta$.

* The orbit of any point $p \in \Pin$ exits a $1$-neighborhood of $\cO_*$ before exiting $P$.

* The number $n$ of crossings through $\Sigma$ of the orbit of a point crossing from $\cV_1(\cO_i)$ to $\cV_\delta(\cO_i)$ is of the order of $\log_2(\delta^\inv)$.
The corresponding iteration $f_\Sigma^{n}$ of the return map on $\Sigma$ expands the norm of the vectors $v \in E^\uu$ by a factor greater than $\delta^\inv$ (up to a uniform multiplicative constant), which will compensate for the contraction of $f_{\Sigma, \iin}$.

Let us insist on the fact that we will have an exact compensation of a contracting effect of order $\delta$ and a expanding effect of order $\delta^\inv$.
The contraction of the composition map $f_\Sigma^n , f_{\Sigma, \iin} \, \psi^\iin \colon \Pin \ssm \cV_\delta \to \Sigma \cap \cV_\delta$ will thus be bounded below by a constant independent of $\delta$.
We formally state the last two items.
Let $l$ be the minimum diameter of the disks of section $\Sigma$ and $0< \delta< l$.

Let $p \in \Pin \ssm \cV_\delta$. Then there exists $T \geq 0$ such that for all $t \leq T$, $X^t(p) \in P \ssm \cV_\delta$ and $X^T(p) \in P \ssm \cV_1$.

If $p \in \Pin \ssm \cV_1$, the fact is true for $T=0$.
Otherwise, $p \in \Pin \cap \cV_i$, where $\cV_i$ is the normalized neighborhood of the orbit $\cO_i \in \cO_*$ for some $i$, given coordinates $\xi_i = (x,y,\theta)$.
We have $p = (x,x,\theta)$, with $\vert x \vert > \delta$ and $X^t(x,y,\theta) = (2^{-t} x, 2^t x, \theta +t)$.
The distance of $X^t(p)$ to the orbit $\cO_i$ is therefore greater than $2^t \delta$.
We deduce that the orbit of $p$ leaves $\cV_1$ in time $t=\log_2(\delta^\inv)$ without intersecting $\cV_\delta$.

Let $q \in \Sigma \cap \cV_\delta$ be such that there exists $T \geq 0$ with $X^{-T}(q) \in P \ssm \cV_1$.
Then the orbit segment $[X^{-T}(q), q]$ intersects the section $\Sigma$ a number $n_0 \geq \log_2( l \delta^\inv) -1$ of times in a normalized neighborhood $\cV_j$ of $\cO_j \in \cO_*$.

Let $(\cV_j, \xi_j)$ be an  of $\cO_j \in \cO_*$ such that $q \in \cV_j$.
Let $\cV_\delta \subset \cV_j$ be the $\delta$-neighborhood of $\cO_j$.
By definition of $l$, and by Definition <ref>, Item <ref>, the section $\Sigma$ contains the set $R_l := \{\theta=0, \, \vert x \leq l, \, \vert y \vert \leq l\}$.
Let $q = (x, y, \theta)$ with $x^2 + y^2 \leq \delta^2$ and we have $X^{-t} (p) = (2^{t} x, 2^{-t} y, \theta -t)$ for $t \geq 0$.
Then the negative orbit $X^{-t}(p)$ intersects the set $R_l \subset \Sigma$ a number greater than $n = \log_2 (l \delta^\inv) -1$ times before exiting $\cV_1$.

Let $p \in \Pin \ssm \cV_\delta$ such that $q = f^n_\Sigma f_{\Sigma, \iin} \psi^\iin (p)$ is well defined in $\Sigma \cap \cV_\delta$.
According to Fact <ref>
the orbit of $p$ exits a $1$-neighborhood at a point $p' = X^T(p)$.
According to Fact <ref>,
the orbit segment from $p'$ to $q$ intersects $n_0$ times the $\Sigma$ section in a normalized $\cV_j$ neighborhood of a $\cO_j \in \cO_*$ orbit.
In the neighborhood of $p$ we have the following equality

$$ f^n_\Sigma f_{\Sigma, \iin} \psi^\iin = f^{n_0}_\Sigma f^{n-n_0}_\Sigma f_{\Sigma, \iin} \psi^\iin$$

with $n-n_0 \geq 0$ and the map $f^{n_0}_\Sigma$ is the iterate $n_0$ of the restriction of $f_\Sigma$ to $\Sigma \cap \cV_j$.
We can study the effect of each map of the composition $f^n_\Sigma f_{\Sigma, \iin} \psi^\iin$ in the neighborhood of a point $p \in \Pin \ssm \cV_\delta$.

The  $\psi^\iin = \psi^\iin_\delta$ maps $C^u_\iin$ inside a $(\hat K,\cG^{u, \iin}/\, \cG^{s, \iin})$-cone field, where $\hat K$ is a constant uniform in $\delta$ (Lemma <ref>).
Its differential is bounded (below) by a constant $\cst$ uniform in $\delta$ (Lemma <ref>).

*  The crossing map $f_{\Sigma, \iin}$ maps a $(\hat K,\cG^{u, \iin}/\cG^{s, \iin})$-cone field inside a $(\zeta^u /\,\zeta^s)$-cone field on $\Sigma$ of slope $K_1 \leq \cst \hat K$ (Lemma <ref>).
Its differential at a point $\psi^\iin(p)$ is bounded below by $\cst \dist(\psi^\iin(p),\cO_*)$
(Lemma <ref>).
According to <ref>, this factor is bounded below by $\cst \dist(p,\cO_*) > \cst \delta$.

The iteration $f_\Sigma^{n-n_0} \colon \Sigma \to \Sigma$ maps a $(K_1,\zeta^u /\,\zeta^s)$-cone field to a $(K_1, \zeta^u /\,\zeta^s)$-cone field (Fact <ref>) and contracts the vectors of a $(K_1,\zeta^u /\,\zeta^s)$-cone by a factor bounded below by a $\cst>0$ which depends only on $K_1$.

*  Finally, we can use Fact <ref> with an expansion factor $\mu=2$ and an iterate $n_0 \geq \log_2(l \delta^\inv) -1$ to show that $f_\Sigma^{n_0}$ maps a $(K_1,\zeta^u/\, \zeta^s)$-cone inside a $(\zeta^u/\,\zeta^s)$-cone field of slope $K_2 \leq \cst \, \delta^2 \, K_1$, where the constant depends only on $l$.
It expands the vectors of a $(K_1,\ \zeta^u/\, \zeta^s)$-cone by a factor greater than $\cst \delta$, where the constant depends only on $K_1$ and $l$.

We deduce that one can choose $\delta$ small enough so that the image $(f^n_\Sigma f_{\Sigma, \iin} \psi^\iin)_* C^u_\iin$ is contained inside $C^u_\Sigma$.
Moreover, up to change the constant $\cst$ which is uniform in $\delta$, we have in the neighborhood of $p$ :

\begin{align*}
\l\Vert \l( f_\Sigma^n f_{\Sigma, \iin} \psi^\iin \r)_* v \r\Vert = \l\Vert \l( f_\Sigma^{n_0} f_\Sigma^{n-n_0} f_{\Sigma, \iin} \psi^\iin \r)_* v \r\Vert
& \geq \cst\, \delta^\inv \l\Vert \l(f_{\Sigma, \iin} \psi^\iin \r)_* v \r\Vert & \mb{(Item~\ref{it: composition; f_sigma^n-n0} and~\ref{it: composition; f_sigma^n0})} \\
& \geq \cst\, \delta^\inv  \delta \Vert (\psi^\iin)_*v \Vert
& \mb{(Item~\ref{it: composition; f_sigma,in})} \\
& \geq \cst\, \Vert v \Vert
& \mb{(Item~\ref{it: composition; psi^in})}
\end{align*}

It is easy to check that Lemma <ref> is still true (up to change the constant) for any orbit trajectory starting from $p \in \Pin \ssm \cV_\delta$ and arriving at
$q = f_\Sigma^n \, f_{\Sigma, \iin} \, \psi^\iin(p) \in \Sigma \cap \cV_{r.\delta}$, where $r>0$ is a uniform constant previously fixed.

We can now show the following corollary which states that the composition

\begin{equation} \label{eq: map from Pin minus V_delta to Sigma through Pin}
f_\Sigma^j \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^i \, f_{\Sigma, \iin} \, \psi^\iin \colon \Pin \ssm \cV_\delta \to \Sigma
\end{equation}

satisfies the cone fields condition for the pair $(C^u_\iin \cup C^u_\Sigma, C^s_\iin \cup C^s_\Sigma)$
on the set of points $p \in \Pin \ssm \cV_\delta$ whose image by $\varphi \, \psi^\out  \, f_{\out, \Sigma} \, f_\Sigma^i \, f_{\Sigma, \iin} \, \psi^\iin$ belongs to $\Pin \cap \cV_\delta$.
We refer to Figure <ref>.

Trajectory of a cone $C^u_\iin$ under the map $f_\Sigma^j \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^i \, f_{\Sigma, \iin} \, \psi^\iin \colon \Pin \ssm \cV_\delta \to \Sigma$ crossing $\Pin \cap \cV_\delta$

There exists $N_0$ such that
for all $\delta>0$ small enough,
the following condition is satisfied.
Let $p \in \Pin \ssm \cV_\delta$ such that the following are well defined for $i\geq 0$ and $j \geq N_0$

* $p' = \varphi \, \psi^\out  \, f_{\out, \Sigma} \, f_\Sigma^i \, f_{\Sigma, \iin} \, \psi^\iin(p) \in \Pin \cap \cV_\delta $, and

* $q =  f_\Sigma^j \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^i \, f_{\Sigma, \iin} \, \psi^\iin(p) \in \Sigma$.

* $f_\Sigma^j \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^i \, f_{\Sigma, \iin} \, \psi^\iin$ maps $C^u_\iin(p)$ inside $C^u_{\Sigma}(q)$ and expands the norm of the vectors of $C^u_\iin(p)$ by a factor $2$,

* $(f_\Sigma^j \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^i \, f_{\Sigma, \iin} \, \psi^\iin)^\inv$ maps $C^s_\Sigma(q)$ inside $C^s_\iin(p)$ and expands the norm of the vectors of $C^s_\Sigma(q)$ by a factor $2$.

Let us show the first item concerning unstable cones, the proof of the second is similar.
The fact that the point $p' = \varphi \, \psi^\out  \, f_{\out, \Sigma} \, f_\Sigma^i \, f_{\Sigma, \iin} \, \psi^\iin(p)$ belongs to $\Pin \cap \cV_\delta$ (by assumption of the corollary) implies that the point $f_\Sigma^i \, f_{\Sigma, \iin} \, \psi^\iin (p)$ belongs to $\Sigma \cap \cV_{r.\delta}$, in other words the positive orbit of $p$ intersects the $\Sigma$ section one last time in the neighborhood $\cV_{r.\delta}$ before exiting through $\Pout$, where $r>0$ is a (uniform) constant which satisfies Lemma <ref>.

We decompose the map
$f_\Sigma^j \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^i \, f_{\Sigma, \iin} \, \psi^\iin$
in the neighborhood of $p$ in three pieces:

* The map $f_\Sigma^i \, f_{\Sigma, \iin} \, \psi^\iin \colon \Pin \ssm \cV_\delta \to \Sigma \cap \cV_{r .\delta}$ is addressed in Lemma <ref> and Remark <ref>.
For $\delta$ small enough, it maps the cone $C^u_\iin(p)$ inside $C^u_\Sigma$ and contracts the vectors of $C^u_\iin$ by a factor bounded below by a constant $c_0>0$ uniform in $\delta$.

* The map $f_{\Sigma, \iin} \psi f_{\out, \Sigma} \colon \Sigma \to \Sigma$ is addressed in Lemma <ref>. Its differential is bounded by a constant $c_1>0$ and it maps the cone field $C^u_\Sigma$ inside a $(K_1,\ \zeta^u/\, \zeta^s)$-cone field, with $c_1$ and $K_1$ uniform in $\delta$.

* From Fact <ref>, there exists an integer $N_0$, such that if $j \geq N_0$, then $f_\Sigma^j$ maps a $(K_1,\ \zeta^u/\,\zeta^s)$-cone field inside $C^u_\Sigma$ and expands the norm of the vectors of a $(K_1,\ \zeta^u/\,\zeta^s)$-cone by a factor greater than $2 (c_1 c_0)^\inv$.

Putting together, we obtain the existence of an integer $N_0$ that the composition (<ref>) with $n \geq N_0$ maps the cone $C^u_\iin(p)$ inside $C^u_\Sigma(q)$ and expands the vectors of $C^u_\iin$ by a factor $2$.

We fix the integer $N_0$ which satisfies both Corollaries <ref> and <ref>.

§.§ Parameter δ_0

In this section, we show the existence of a parameter $\delta = \delta_0$ which will allow us to obtain Item <ref> of Proposition <ref>.
The integer $N_0$ is fixed in the previous section.
We define $\delta_0$ satisfying the following

There exists $\delta_0$ such that the maps $f_\Sigma^{N_0} \, f_{\Sigma, \iin} \, \psi^\iin$ and $f_\Sigma^{-N_0} \, f_{\out, \Sigma}^\inv \, (\varphi \psi^\out)^\inv$ are well defined from $ \Pin \cap \cV_{\delta_0}$ to $\Sigma$.

A point $p \in \cap \cV_\delta$ is in a $\delta$-neighborhood of a periodic orbit $\cO_i \in \cO_*$.
For arbitrarily small $\delta$, the orbit by the flow of $X$ of a point $p \in \Pin \cap \cV_\delta$ spends an arbitrarily long time in the neighborhood of $\Lambda$ so intersects an arbitrarily large number of times the section $\Sigma$ before intersecting $\Pout$.
It suffices to recall that the diffeomorphisms $\psi^\iin$ and $\varphi \psi^\out$ have a bounded effect on the distance $\dist(\cdot, \cO_*)$, uniformly in $\delta$ (Lemma <ref>).
We deduce the existence of $\delta_0$.

We fix the parameter $\delta_0$ which satisfies Proposition <ref>.

§.§ Integer N_1

The parameters $\lambda = \lambda_0$, $\epsilon = \epsilon_0$ and $\delta = \delta_0$ are fixed, as well as the cones $(C^u_\iin, C^s_\iin)$ on $\Pin$ and $(C^u_\Sigma, C^s_\Sigma)$ on $\Sigma$.
From now on we call $\psi^\iin = \psi^\iin_{\lambda_0, \epsilon_0, \delta_0}$, $\psi^\out = \psi^\out_{\lambda_0, \epsilon_0, \delta_0}$ the  of $\pP$ given by Proposition <ref> and <ref>.
Let $\psi := \psi^\iin \varphi \psi^\out$ be the composition map.
Let $\cV_0 := \psi^\iin(\cV_{\delta_0})$.
In this section, we show the existence of an integer $N_1$ which will allow us to have Items <ref> and <ref> of Proposition <ref>.
Recall that Item <ref> concerns the points $p \in (\Pin \ssm \cV_0)_\psi$ whose future orbit by $X_\psi$ intersects $\Sigma_\psi$ a number $N_1$ of consecutive times,
and Item <ref> concerns the points $p \in \Sigma_\psi$ whose future orbit by $X_\psi$ intersects $(\Pin \ssm \cV_0)_\psi$ a number $N_1$ of times.

Let us start by showing that the map $f^j_\Sigma \, f_{\Sigma, \iin} \, \psi^\iin \colon \Pin \to \Sigma$
restricted to $\Pin \ssm \cV_{\delta_0}$ satisfies the cone fields condition for the pair $(C^u_\iin \cup C^u_\Sigma, C^s_\iin \cup C^s_\Sigma)$ for a sufficiently large iterate $j$.
We refer to Figure <ref>.

Cone $C^u_\iin$ mapped by $f^j_\Sigma \, f_{\Sigma, \iin} \si^\iin$ restricted to $\Pin \ssm \cV_{\delta_0}$

There exists $N_1$ such that for any $p \in \Pin \ssm \cV_{\delta_0}$ if the image $q = f^n_\Sigma \, f_{\Sigma, \iin} \, \psi^\iin(p) \in \Sigma$ is well defined for a $n \geq N_1$,

* $(f^n_\Sigma \, f_{\Sigma, \iin} \, \psi^\iin)_* C^u_\iin(p) \subset \intr C^u_\Sigma (q)$ and
$\Vert (f^n_\Sigma \, f_{\Sigma, \iin} \, \psi^\iin)_* v \Vert \geq 2 \Vert v \Vert$ for all $v \in C^u_\iin(p)$

* $(f^n_\Sigma \, f_{\Sigma, \iin} \, \psi^\iin)^\inv_* C^s_\Sigma(q) \subset \intr C^s_\Sigma (p)$ and
$\Vert (f^n_\Sigma \, f_{\Sigma, \iin} \, \psi^\iin)^\inv_* v \Vert \geq 2 \Vert v \Vert$ for all $v \in C^s_\Sigma(p)$

We show the existence of an integer $N_1$ such that the first item is true.
The second item is shown in a similar way.
Let us study each map of the composition.

* The cone field $\psi^\iin(C^u_\iin)$ is inside a $(\hat K,\cG^{u,\iin}/, \cG^{s, \iin})$-cone field (Lemma <ref>) and $\psi^\iin$ has a differential uniformly bounded by a constant $c_0>0$.

* The crossing map $f_{\Sigma, \iin} \colon \Pin \to \Sigma$ maps a $(\hat K,\cG^{u, \iin}/\, \cG^{s, \iin})$-cone field on $\Pin$ inside a $(\zeta^u /\,\zeta^s)$-cone field on $\Sigma$ of uniform slope $K_1$ (Lemma <ref>).
Moreover, the differential of $f_{\Sigma, \iin}$ restricted to $\Pin \ssm \cV_{\delta_0}$ is uniformly bounded by a constant $c_1>0$ because it is a crossing map of the flow $X^t$ between a compact surface $\Pin \ssm \cV_{\delta_0}$ transverse to the  $X$ and a compact surface $\Sigma$  transverse to the  $X$ and the time of the crossing is uniformly bounded.

* By hyperbolicity of $f_\Sigma$ (Fact <ref>), there exists an integer $N_1$ such that if $n \geq N_1$, then $f_\Sigma^n$ maps a $(K_1,\zeta^u/\,\zeta^s)$-cone field inside $C^u_\Sigma$ and expands the norm of the vectors of a $(K_1,\zeta^u/\,\zeta^s)$-cone by a factor greater than $2 (c_1 c_0)^\inv$

Putting together we obtain the existence of an integer $N_1$ such that the composition $f^j_\Sigma \, f_{\Sigma, \iin} \, \psi^\iin \colon \Pin \to \Sigma$ with $n \geq N_1$ maps the cone $C^u_\iin(p)$ inside $C^u_\Sigma(q)$ and expands the vectors of $C^u_\iin$ by a factor $2$.

We recall the definition in Subsection <ref> of the following composition
$f_{\iin, \psi} = \varphi \psi^\out \foutin \psi^\iin \colon \Pin \to \Pin$.
Denote $\hat f_{\iin, \psi} \colon \Pin \ssm \cV_{\delta_0} \to \Pin \ssm \cV_{\delta_0}$
the restriction of $f_{\iin, \psi}$ to the set $\Pin \ssm \cV_{\delta_0}$ at the domain and destination.
Let us show that the composition map

$$ \hat f_{\iin, \psi}^j \varphi \psi^\out f_{\out, \Sigma} f_\Sigma^i \colon \Sigma \to \Pin \ssm \cV_{\delta_0} $$

satisfies the cone fields condition for the pair $(C^u_\Sigma \cup C^u_\iin, C^s_\Sigma \cup C^s_\iin)$ if $j$ is large enough.

Trajectory of a cone $C^u_\Sigma$ under the map $\hat f_{\iin, \psi}^{j} \varphi \psi^\out f_{\out, \Sigma} f_\Sigma^i$

There exists an integer $N_1 \geq 0$ such that for all
$p \in \Sigma$, if the image $\hat f_{\iin, \psi}^j \varphi \psi^\out f_{\out, \Sigma} f_\Sigma^i(p) \in \Pin \ssm \cV_{\delta_0}$ is well defined for $j \geq N_1$ and $i \geq 0$, in other words $f_{\iin, \psi}^k \varphi \psi^\out f_{\out, \Sigma} f_\Sigma^i(p) \in \Pin \ssm \cV_\delta$ for all $k=0, \dots, j$,

* $\hat f_{\iin, \psi}^j \, \varphi \psi^\out \, f_{\out, \Sigma} \, f_\Sigma^i$ maps $C^u_\Sigma(p)$ inside $C^u_\iin(q)$ and expands the norm of the vectors of $C^u_\Sigma(p)$ by a factor $2$,

* $(\hat f_{\iin, \psi}^j \, \varphi \psi^\out \, f_{\out, \Sigma} \, f_\Sigma^i)^\inv$ maps $C^s_\iin(p)$ inside $C^s_\Sigma(q)$ and expands the norm of the vectors of $C^s_\iin(p)$ by a factor $2$,

Let us insist on the fact that each crossing of the orbit of such a point $p$ on the surface $\Pin$ takes place on the complementary of $\cV_{\delta_0}$ by assumption.

The composition is decomposed into three pieces.

* $f_\Sigma^i$ maps $C^u_\Sigma$ on $C^u_\Sigma$ and the contraction of the norm of the vectors is greater than a uniform constant $c_0>0$ for $i \geq 0$ (Corollary <ref>).

* We know that $\l( \varphi \psi^\out \, f_{\out, \Sigma} \r)_* C^u_\Sigma \subset \intr C^u_\iin$ (Lemma <ref>).
Moreover, by hypotheses the codomain of $f_{\out, \Sigma}$ for our restriction is contained in $\Pout \ssm (\varphi \psi^\out)^\inv (\cV_{\delta_0})$, so uniformly far from the set $\cO_*$.
Using Lemma <ref>, we deduce that the differential of $f_{\out, \Sigma}$ is bounded, uniformly in $p$.
The diffeomorphisms $\varphi$, $\psi^\out$ are compactly supported so the differential of $\varphi \, \psi^\out \, f_{\out, \Sigma}$ is bounded by a constant $c_1>0$ (uniform in $p$).

* The map $\hat f_{\iin, \psi} \colon \ssm \cV_{\delta_0} \to \Pin \ssm \cV_{\delta_0}$, which coincides with the restriction of $f_{\iin, \psi}$ to the set $(\Pin \ssm \cV_{\delta}) \cap f_{\iin, \psi}^\inv( \Pin \ssm \cV_\delta)$ maps the field $C^u_\iin$ inside $C^u_\iin$ and expands the norm of the vectors of $C^u_\iin$ by a factor $2$ (Proposition <ref>).
It is enough to choose $N_1 \geq -\log_2(c_0 c_1)$ so that the expansion of $\hat f_{\iin, \psi}$ compensates the contraction of the previous maps.
This integer is uniform in $p$.

The second item is shown in a similar way.

We fix an integer $N_1$ which satisfies Lemmas <ref> and <ref>.

§.§ Proof of Proposition <ref>

We are now able to prove Proposition <ref>.
We have fixed in the previous sections the parameters $\lambda= \lambda_0$, $\epsilon = \epsilon_0$ (Subsection <ref>), $\delta=\delta_0$ (Subsection <ref>),
a pair of cone fields $(C^u_\iin, C^s_\iin)$ on $\Pin$ (Subsection <ref>) and a pair of cone fields $(C^u_\Sigma, C^s_\Sigma)$ on $\Sigma$ (Subsection <ref>),
and we defined integers $N_\Sigma$ (Subsection <ref>), $N_0$ (Subsection <ref>), $N_1$ (Subsection <ref>).
These choices have been made in the right order, which means that all the results from Subsections <ref> to <ref> are true for the parameters $\lambda_0, \epsilon_0, \delta_0$.
Denote $\psi^\iin = \psi^\iin_{\lambda_0, \epsilon_0, \delta_0}$ and $\psi^\out = \psi^\out_{\lambda_0, \epsilon_0, \delta_0}$.
We use the notations introduced in Section <ref>.
Let $\psi = \psi^\iin \varphi \psi^\out$ be the modified gluing map, $P_\psi = P/\,\psi$ the closed manifold and $X_\psi$ the vector field induced by $X$ on $P_\psi$. The projection is $\pi_\psi \colon P \to P_\psi$. Define the section $S_{0, \psi} = \pi_\psi( \Pin \cup \Sigma )$, $f_{0, \psi}$ the first-return map of the flow of $X_\psi$ on $S_{0, \psi}$ and $\cV_0 = \psi^\iin (\cV_{\delta_0})$.

Define the following pair of cone fields on $S_{0, \psi}$:

$$(C^u_\psi, C^s_\psi) = \l( \pi_\psi( \psi^\iin_* (C^u_\iin) \sqcup C^u_\Sigma),\  \pi_\psi(\psi^\iin_* (C^s_\iin) \sqcup C^s_\Sigma) \r) $$

and the metric
$g_\psi = \pi_\psi(\psi^\iin_* g^\iin) \sqcup \pi_\psi(g^\Sigma)$ on $P_\psi$,
where $g^\Sigma$ and $g^\iin$ are restrictions of the metric fixed on $P$.
Recall that the metric $g$ is the metric fixed at the beginning of Section <ref>, which coincides with $dx^2 + dy^2 + d\theta^2$ in each $(\cV_i, \xi_i = (x,y,\theta))$, and which is adapted to the hyperbolic splitting of the flow.
Let us note $\Vert \cdot \Vert_\psi, \Vert \cdot \Vert_\Sigma, \Vert \cdot \Vert_\iin$ the associated norms.
Recall that for practical reasons, we have studied lifts in $P$ conjugated by $\psi^\iin$ of the map $f^n_{0, \psi}$.
This explains that we compose the projection $\pi_\psi$ by $\psi^\iin$ on $\Pin$ to fit.
Recall that :

* $S_{0, \psi}$ has points of self-intersections $p$, at these points it has two or three tangent planes $T^* = T^*_p \subset T_p S_{0, \psi}$.
The differential of $f_{0, \psi}$ in $p$ is well defined for a choice of tangent plane on $p$ and on $f_{0, \psi}(p)$.
A cone field $C$ on $S_{0, \psi}$ is a collection of cones $C(p, T^i)$ on each of the tangent planes $T^i \subset T_p S_{0, \psi}$.

* If $p \in \Pin_\psi \subset S_{0, \psi}$, we denote $T^\iin_p \subset T_p S_{0, \psi}$ the tangent plane induced by $T_{\tilde p} P^\iin$, where $\tilde p \in \Pin$ is the lift of $p$ in $\Pin$, and we denote $(C^u_\psi(p, T^\iin_p), C^s_\psi(p, T^\iin_p))$ the pair $(C^u_\psi, C^s_\psi)$ on $T^\iin_p$ in $p$.

If $q \in \Sigma_\psi \subset S_{0, \psi}$, we denote $T^\Sigma_\psi \subset T_p S_{0, \psi}$ the (or one of the two) tangent plane(s) induced by $T_{\tilde q} P^\Sigma$, where $\tilde q \in \Sigma$ is a lift of $q$ in $\Sigma$, and we denote $(C^u_\psi(q, T^\Sigma_q), C^s_\psi(q, T^\Sigma_q)$ the pair $(C^u_\psi, C^s_\psi)$ on $T^\Sigma_q$ in $q$.

§.§.§ Lifts in P

We summarize the lifts in $P$ of the points, cones, vectors and norms that we will use.

* (maps)

* If $p \in \Sigma_\psi$ and $f_{0, \psi}(p) \in \Sigma_\psi$ then a lift of $f_{0, \psi}$ is $f_\Sigma \colon \Sigma \to \Sigma$.

* If $p \in \Sigma_\psi$ and $f_{0, \psi}(p) \in \Pin_\psi$ then a lift of $f_{0, \psi}$ is $\psi \, f_{\out, \Sigma}: \Sigma \to \Pin$.

* If $p \in \Pin_\psi$ and $f_{0, \psi}(p) \in \Sigma_\psi$ then a lift of $f_{0, \psi}$ is $f_{\Sigma, \iin}: \Pin \to \Sigma$.

* If $p \in \Pin_\psi$ and $f_{0, \psi}(p) \in \Pin_\psi$ then a lift of $f_{0, \psi}$ is $\psi \, \foutin: \Pin \to \Pin$

* (points)

* If $p \in \Pin_\psi$, there exists a unique lift $\tilde p$ of $p$ in $\Pin$, then denote $\check p := (\psi^\iin)^\inv (\tilde p)$, in other words we have $\pi_\psi (\psi^\iin (\check p)) =p$.
Moreover if $p \in (\Pin \ssm \cV_0)_\psi$ then $\check p \in \Pin \ssm \cV_{\delta_0}$ by definition of $\cV_0 = \psi^\iin (\cV_{\delta_0})$.

* If $q \in \Sigma_\psi$, there exists $\tilde q \in \Sigma$ a lift of $q$ in $P$.

* (cones)

* If $p \in \Pin_\psi$, a lift of the cone $C^u_\psi(p, T^\iin_p)$ is the cone $\psi^\iin_* C^u_\iin ( \check p )$.

* If $q \in \Sigma$, a lift of the cone $C^u_\psi(q, T^\Sigma_q)$ is the cone $C^u_\Sigma (\tilde q)$.

* (vectors)

* If $v \in T \Pin_\psi$, there exists a unique lift of $v$ in $TP$ of the form $\psi^\iin_* \check v$ with $\check v \in T \Pin$.
Moreover, if $v \in C^u_\psi(p, T^\iin_p)$, then $\check v \in C^u_\iin(\check p)$.

* If $w \in T \Sigma_\psi$, there exists $\tilde w \in T \Sigma$ a lift of $w$ in $P$.
Moreover, if $w \in C^u_\psi(q, T^\Sigma_q)$, then $\tilde w \in C^u_\Sigma(\tilde q)$.

* (norm)

* If $v \in T \Pin_\psi$, then $\Vert v \Vert_\psi = \Vert \check v \Vert_\iin = \Vert \check v \Vert$ where $\check v$ is given by the previous item and $\Vert \cdot \Vert$ is the norm from the Riemannian metric on $P$.

* If $w \in T \Sigma_\psi$, then $\Vert w \Vert_\psi = \Vert \tilde w \Sigma = \Vert \tilde w \Vert$ where $\tilde w$ is given by the previous item and $\Vert \cdot \Vert$ is the norm from the Riemannian metric on $P$.

All the above items are direct consequences of the definitions of the cones and the metric on $P_\psi$.
Let us show Proposition <ref>.

Let $p \in S_{0, \psi}$, $n \geq 1$ and $q=f_{0, \psi}^n(p)$.
Let us show each item of the proposition.
We only show the strict invariance and expansion of unstable cones $C^u_\psi$.
The proof for the stable cones $C^s_\psi$ is symmetric.

Suppose that $p \in (\Pin \cap \cV_0)_\psi$.

Let $\tilde p := \psi^\iin \check p$ be a lift of $p$ in $P$ with $\check p \in \Pin \cap \cV_{\delta_0}$ (Fact <ref>, Item <ref>).

According to Proposition <ref>, the maps
$f_\Sigma^{N_0} , f_{\Sigma, \iin} \psi^\iin \colon \Pin \to \Sigma$ and
$f_\Sigma^{-N_0} f_{\out, \Sigma}^\inv \, (\varphi \psi^\out)^\inv \colon \to \Sigma$ are well defined in $\check p$.
From Fact <ref>, Item <ref>, the lifts of the iterates $f^{k}_{0, \psi}(p)$ in $P$ are the iterates :

* $ f_\Sigma^{k-1} \, f_{\Sigma, \iin} (\tilde p) = f_\Sigma^{k-1} \, f_{\Sigma, \iin} \psi^\iin (\check p) $ for $k = 1, \dots, N_0$, and

* $f_\Sigma^{k-1} \, f_{\out, \Sigma}^\inv \, \psi^\inv (\tilde p) =f_\Sigma^{k-1} \, f_{\out, \Sigma}^\inv \, (\varphi \psi^\out)^\inv (\check p) $ for $k=-N_0, \dots, -1$.

We deduce that the iterates $f^k_{0, \psi}(p)$ belong to $\Sigma_\psi$ for $k= -N_0, \dots, -1, 1, \dots, N_0$.

Suppose $p \in (\Pin \ssm \cV_0)_\psi$, $q \in \Sigma_\psi$, $n \geq N_1$, and $f^k_{0,\psi}(p) \in \Sigma_\psi$ for $k=1, \dots, n-1$.

Let $\tilde p := \psi^\iin \check p$ be a lift of $p$ in $P$ with $\check p \in \Pin \cap \cV_{\delta_0}$ (Fact <ref>, Item <ref>).
According to Fact <ref>, Item <ref>, $f^{n-1}_\Sigma, f_{\Sigma, \iin}$ is a lift of $f^n_{0, \psi}$ in the neighborhood of $p$.
Let $\tilde q := f^{n-1}_\Sigma \, f_{\Sigma, \iin} \psi^\iin (\check p) \in \Sigma $.
This is a lift of $q = f^n_{0, \psi}(p)$ in $P$.

According to Item <ref>, Fact <ref>, a lift of $C^u_\psi(p, T^\iin_p)$ is $\psi^\iin_* C^u_\iin (\check p)$ and a lift of $C^u_\psi(q, T^\Sigma_q)$ is $C^u_\Sigma(\tilde q)$
We use Lemma <ref>.
We have

\begin{align*}
(f_{0, \psi}^n)_* C^u_\psi(p, T^\iin_p)
& = (\pi_\psi)_* (f^{n-1}_\Sigma, f_{\Sigma, \iin} \, \psi^\iin)_* C^u_\iin (\check p) \\
& \subset (\pi_\psi)_* \, \intr C^u_\Sigma(\tilde q)
& \mb{(Lemma~\ref{lem: N1 for Pin minus v0 to Sigma})} \\
& = \intr C^u_\psi(q, T^\Sigma_q)
\end{align*}

Let $v \in C^u_\psi(p, T^\iin_p)$.
Let $\psi^\iin_* \check v$ be a lift of $v$ in $T \Pin$, with $v \in C^u_\iin(\check p)$ (Fact <ref> Item <ref>).
Then $(f^{n-1}_\Sigma f_{\Sigma, \iin} \psi^\iin)_* \check v$ is a lift of $(f_{0, \psi}^n)_* v$ in $T \Sigma$, and we have

\begin{align*}
\dfrac{\Vert (f_{0, \psi}^n)_* v \Vert_\psi}{\Vert v \Vert_\psi} & = \dfrac{\Vert (f^{n-1}_\Sigma \, f_{\Sigma, \iin} \,\psi^\iin)_* \check v \Vert}{\Vert \check v \Vert}
& \mb{(Fact~\ref{fact: lifts in P of points}, Item~\ref{fact: lifts; it: norm})}\\
& \geq  2 &  \mb{(Lemma~\ref{lem: N1 for Pin minus v0 to Sigma})} \\
\end{align*}

Suppose $p \in (\Pin \ssm \cV_0)_\psi$, $q \in (\Pin \ssm \cV_0)_\psi$ and $f^k_{0, \psi}(p) \in \Sigma_\psi$ for $k=1, \dots, n-1$.

Let $\tilde p := \psi^\iin \check p$ be a lift of $p$ in $P$ with $\check p \in \Pin \cap \cV_{\delta_0}$ (Fact <ref>, Item <ref>).
According to Fact <ref>, Item <ref>, $\psi \foutin$ is a lift of $f^n_{0, \psi}$ in the neighborhood of $p$.
Let $\tilde q := \psi \foutin (\psi^\iin (\check p)) = \psi^\iin \, f_{\iin, \psi} (\check p) \in \Pin $ (Equation (<ref>)).
This is a lift of $q = f^n_{0, \psi}(p)$ in $P$.
Let $\check q := (\psi^\iin)^\inv \tilde q = f_{\iin, \psi} (\check p)$.

From Item <ref>, Fact <ref>, $\psi^\iin_* C^u_\iin (\check p)$ is a lift of $C^u_\psi(p, T^\iin_p)$ and $\psi^\iin_* C^u_\iin(\check q)$ is a lift of $C^u_\psi(q, T^\iin_q)$.
We use Proposition <ref>.
We have

\begin{align*}
(f_{0, \psi}^n)_* C^u_\psi(p, T^\iin_p) & = (\pi_\psi)_* (\psi \, \foutin \, \psi^\iin)_* C^u_\iin (\check p) \\
& =  (\pi_\psi \psi^\iin)_* (f_{\iin, \psi})_* C^u_\iin (\check p) \\
& \subset (\pi_\psi \psi^\iin)_* (\intr C^u_\iin(\check q))
& \mb{(Proposition~\ref{prop: lambda epsilon cones Pin})} \\
& = (\pi_\psi)_* (\intr \psi^\iin_* C^u_\iin(\check q))) \\
& = \intr C^u_\psi( q, T^\iin_q) \\
\end{align*}

Let $v \in C^u_\psi(p, T^\iin_p)$.
Let $\psi^\iin_* \check v$ be a lift of $v$ in $T \Pin$, with $v \in C^u_\iin(\check p)$ (Fact <ref> Item <ref>).
Then $\psi^\iin_* (f_{\iin, \psi})_* \check v$ is a lift of $(f_{0, \psi}^n)_* v$ in $T \Pin$, and we have

\begin{align*}
\dfrac{\Vert (f_{0, \psi}^n)_* v \Vert_\psi}{\Vert v \Vert_\psi} & = \dfrac{\Vert (f_{\iin, \psi})_* \check v \Vert}{\Vert \check v \Vert}
& \mb{(Fact~\ref{fact: lifts in P of points}, Item~\ref{fact: lifts; it: norm})}\\
& \geq  2 & \mb{(Proposition~\ref{prop: lambda epsilon cones Pin})} \\
\end{align*}

Suppose that $p \in (\Pin \ssm \cV_0)_\psi$, $q \in \Sigma_\psi$, $n \geq N_0 +1$,
there exists $1 \leq m \leq n-N_0$ such that $f^m_{0, \psi}(p) \in (\Pin \cap \cV_0)_\psi$ and $f^k_{0, \psi}(p) \in \Sigma_\psi$ for $k = 1, \dots, m-1, m+1, \dots, n-1$.

Let $\tilde p := \psi^\iin \check p$ be a lift of $p$ in $P$ with $\check p \in \Pin \cap \cV_{\delta_0}$ (Fact <ref>, Item <ref>).
From Fact <ref>, Item <ref>, $f_\Sigma^{n-m} \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^{m-1} \, f_{\Sigma, \iin}$ is a lift of $f^n_{0, \psi}$ in the neighborhood of $p$.
Let $\tilde q := f_\Sigma^{n-m} \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^{m-1} \, f_{\Sigma, \iin} \psi^\iin (\check p) \in \Sigma $.
This is a lift of $q = f^n_{0, \psi}(p)$ in $P$.

According to Item <ref>, Fact <ref>, a lift of $C^u_\psi(p, T^\iin_p)$ is $\psi^\iin_* C^u_\iin (\check p)$ and a lift of $C^u_\psi(q, T^\Sigma_q)$ is $C^u_\Sigma(\tilde q)$.
We have $n-m \geq N_0$.
By Corollary <ref> :

\begin{align*}
(f_{0, \psi}^n)_* C^u_\psi(p, T^\iin_p)
& = (\pi_\psi)_* (f_\Sigma^{n-m} \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^{m-1} \, f_{\Sigma, \iin} \psi^\iin)_* C^u_\iin (\check p) \\
& \subset (\pi_\psi)_* \, \intr C^u_\Sigma(\tilde q) \\
& = \intr C^u_\psi(q, T^\Sigma_q) \\
\end{align*}

Let $v \in C^u_\psi(p, T^\iin_p)$.
Let $\psi^\iin_* \check v$ be a lift of $v$ in $T \Pin$, with $v \in C^u_\iin(\check p)$ (Fact <ref> Item <ref>).
Then $f_\Sigma^{n-m} \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^{m-1} \, f_{\Sigma, \iin} \psi^\iin)_* \check v$ is a lift of $(f_{0, \psi}^n)_* v$ in $T \Sigma$, and we have

\begin{align*}
\dfrac{\Vert (f_{0, \psi}^n)_* v \Vert_\psi}{\Vert v \Vert_\psi} & = \dfrac{\Vert f_\Sigma^{n-m} \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^{m-1} \, f_{\Sigma, \iin} \psi^\iin)_* \check v \Vert}{\Vert \check v \Vert}
& \mb{(Item~\ref{fact: lifts; it: norm})}\\
& \geq  2 & \mb{(Corollary~\ref{coro: N0 for Pin minus V0 to Sigma through Pin inter VO})} \\
\end{align*}

Suppose $p \in \Sigma_\psi$, $q \in \Sigma_\psi$, and $f^k_{0, \psi}(p) \in \Sigma$ for $k=1, \dots, n-1$.

Let $\tilde p \in \Sigma$ be a lift of $p$ in $P$.
According to Fact <ref>, Item <ref>, $f_\Sigma^n$ is a lift of $f^n_{0, \psi}$ in the neighborhood of $p$.
Let $\tilde q := f_\Sigma^n (\tilde p)$.
This is a lift of $q = f^n_{0, \psi}(p)$ in $P$.

According to Item <ref>, Fact <ref>, a lift of $C^u_\psi(p, T^\Sigma_p)$ is $C^u_\Sigma (\tilde p)$ and a lift of $C^u_\psi(q, T^\Sigma_q)$ is $C^u_\Sigma(\tilde q)$.
We use Corollary <ref>.
We have

\begin{align*}
(f_{0, \psi}^n)_* C^u_\psi(p, T^\Sigma_p) & = (\pi_\psi)_* (f_\Sigma^n)_* C^u_\Sigma (\tilde p) \\
& \subset (\pi_\psi)_* (\intr C^u_\Sigma(\tilde q))
& \mb{(Corollary~\ref{coro: N_Sigma})} \\
& = \intr C^u_\psi( q, T^\Sigma_q) \\
\end{align*}

Moreover, if $n \geq N_\Sigma$:
Let $v \in C^u_\psi(p, T^\Sigma_p)$ and $\tilde v \in C^u_\Sigma(\tilde p)$ be a lift of $v$.
We have

\begin{align*}
\dfrac{\Vert (f_{0, \psi}^n)_* v \Vert_\psi}{\Vert v \Vert_\psi} & = \dfrac{\Vert (f_\Sigma^n)_* \check v \Vert}{\Vert \check v \Vert}
& \mb{(Fact~\ref{fact: lifts in P of points}, Item~\ref{fact: lifts; it: norm})}\\
& \geq  2 & \mb{(Corollary~\ref{coro: N_Sigma})} \\
\end{align*}

Suppose $p \in \Sigma_\psi$, $q \in \Pin_\psi$ and $f^k_{0, \psi}(p) \in \Sigma_\psi$ for $k=1, \dots, n-1$.

Let $\tilde p \in \Sigma$ be a lift of $p$ in $P$.
From Fact <ref>, Item <ref>, $ \psi \, f_{\out, \Sigma} \, f_\Sigma^n$ is a lift of $f^n_{0, \psi}$ in the neighborhood of $p$.
Let $\tilde q := \psi \, f_{\out, \Sigma} \, f_\Sigma^n \, (\tilde p) \in \Pin $.
This is a lift of $q = f^n_{0, \psi}(p)$ in $P$.
Denote $\check q := (\psi^\iin)^\inv \tilde q = \varphi \, \psi^\out \, f_{\out, \Sigma} \, f_\Sigma^n (\tilde p)$.

According to Item <ref>, Fact <ref>, a lift of $C^u_\psi(p, T^\Sigma_p)$ is $C^u_\Sigma (\tilde p)$ and a lift of $C^u_\psi(q, T^\iin_q)$ is $\psi^\iin_* C^u_\iin(\check q)$.
We use Lemma <ref>.

\begin{align*}
(f_{0, \psi}^n)_* C^u_\psi(p, T^\Sigma_p) & = (\pi_\psi)_* (\psi \, f_{\out, \Sigma} \, f_\Sigma^n)_* C^u_\iin (\tilde p) \\
& =  (\pi_\psi \psi^\iin)_* ( \varphi \, \psi^\out \, f_{\out, \Sigma} \, f_\Sigma^n )_* C^u_\Sigma (\tilde p) \\
& \subset (\pi_\psi \psi^\iin)_* (\intr C^u_\iin(\check q))
& \mb{(Lemma~\ref{lem: cones Sigma})} \\
& = (\pi_\psi)_* \intr \psi^\iin_* C^u_\iin(\check q)) \\
& = \intr C^u_\psi(q, T^\iin_q) \\
\end{align*}

Suppose that $p \in \Sigma_\psi$, $q \in \Sigma_\psi$, $n \geq N_0+1$, and there exists $1 \leq m \leq n-N_0$ such that $f_{0, \psi}^m(p) \in \Pin_\psi$, and $f^k(p) \in \Sigma_\psi$ for $k=1, \dots, m-1, m+1, \dots, n-1$.

Let $\tilde p \in \Sigma$ be a lift of $p$ in $P$.
From Fact <ref>, Item <ref>, $f_\Sigma^{n-m} \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^{m-1}$ is a lift of $f^n_{0, \psi}$ in the neighborhood of $p$.
We set $\tilde q := f_\Sigma^{n-m} \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^{m-1} (\tilde p)$.
This is a lift of $q = f^n_{0, \psi}(p)$ in $P$.

According to Item <ref>, Fact <ref>, a lift of $C^u_\psi(p, T^\Sigma_p)$ is $C^u_\Sigma (\tilde p)$ and a lift of $C^u_\psi(q, T^\Sigma_q)$ is $C^u_\Sigma(\tilde q)$.
Since $n-m \geq N_0$, we can use Corollary <ref>.
We have

\begin{align*}
(f_{0, \psi}^n)_* C^u_\psi(p, T^\Sigma_p) & = (\pi_\psi)_* (f_\Sigma^{n-m} \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^{m-1})_* C^u_\Sigma (\tilde p) \\
& \subset (\pi_\psi)_* (\intr C^u_\Sigma(\tilde q))
& \mb{(Corollary~\ref{coro: N0 for sigma to sigma through Pin})} \\
& = \intr C^u_\psi( q, T^\Sigma_q) \\
\end{align*}

Let $v \in C^u_\psi(p, T^\Sigma_p)$ and $\tilde v \in C^u_\Sigma(\tilde p)$ be a lift of $v$.
We have

\begin{align*}
\dfrac{\Vert (f_{0, \psi}^n)_* v \Vert_\psi}{\Vert v \Vert_\psi} & = \dfrac{\Vert (f_\Sigma^{n-m} \, f_{\Sigma, \iin} \, \psi \, f_{\out, \Sigma} \, f_\Sigma^{m-1})_* \check v \Vert}{\Vert \check v \Vert}
& \mb{(Fact~\ref{fact: lifts in P of points}, Item~\ref{fact: lifts; it: norm})}\\
& \geq  2 & \mb{(Corollary~\ref{coro: N0 for sigma to sigma through Pin})} \\
\end{align*}

$p \in \Sigma_\psi$, $q \in (\Pin \ssm \cV_0)_\psi$, and there exist $m \geq N_1$, and integers $1 \leq k_1 < \dots <k_m = n$ such that
and $f_{0, \psi}^{k_1}(p), \dots, f_{0, \psi}^{k_m}(p) \in (\Pin \ssm \cV_0)_\psi$
and $f_{0, \psi}^k(p) \in \Sigma$ for $k \neq k_1, \dots, k_m$, $1 \leq k \leq n-1$.

Let $\tilde p \in \Sigma$ be a lift of $p$ in $P$.
According to Fact <ref>, Item <ref>, $(\foutin \psi)^{m-1} f_{\out, \Sigma} f_\Sigma^{k_1-1} $ is a lift of $f^n_{0, \psi}$ in the  of $p$, and we write in the following decomposition :

$$(\foutin \psi)^{m-1} f_{\out, \Sigma} f_\Sigma^{k_1 -1} = \psi^\iin \hat f_{\iin, \psi}^{m-1} \varphi \psi^\out f_{\out, \Sigma} f_\Sigma^{k_1 -1}  $$

with $\hat f_{\iin, \psi} \colon \Pin \ssm \cV_{\delta_0} \to \Pin \ssm \cV_{\delta}$ is the restriction of $f_{\iin, \psi}$ to the corresponding sets.

Let $\tilde q := \psi^\iin \hat f_{\iin, \psi}^{m-1} \varphi \psi^\out f_{\out, \Sigma} f_\Sigma^{k_1 -1} \in \Pin$.
This is a lift of $q = f^n_{0, \psi}(p)$ in $P$.
We set $\check q := (\psi^\iin)^\inv \tilde q = (\hat f_{\iin, \psi})^{m-1} \varphi \psi^\out f_{\out, \Sigma} f_\Sigma^{k_1 -1} (\tilde p)$.

According to Item <ref>, Fact <ref>, a lift of $C^u_\psi(p, T^\Sigma_p)$ is $C^u_\Sigma (\tilde p)$ and a lift of $C^u_\psi(q, T^\iin_q)$ is $\psi^\iin_* C^u_\iin(\check q)$.
We use Lemma <ref>.
We have

\begin{align*}
(f_{0, \psi}^n)_* C^u_\psi(p, T^\Sigma_p) & = (\pi_\psi)_* (\psi^\iin \hat f_{\iin, \psi}^{m-1} \varphi \psi^\out f_{\out, \Sigma} f_\Sigma^{k_1 -1})_* C^u_\iin (\tilde p) \\
& =  (\pi_\psi \psi^\iin)_* (\hat f_{\iin, \psi}^{m-1} \varphi \psi^\out f_{\out, \Sigma} f_\Sigma^{k_1 -1})_* C^u_\Sigma (\tilde p) \\
& \subset (\pi_\psi \psi^\iin)_* (\intr C^u_\iin(\check q))
& \mb{(Lemma~\ref{lem: N1 for Sigma to Pin minus V0})} \\
& = (\pi_\psi)_* \intr \psi^\iin_* C^u_\iin(\check q)) \\
& = \intr C^u_\psi(q, T^\iin_q) \\
\end{align*}

Let $v \in C^u_\psi(p, T^\Sigma_p)$, and $\tilde v \in C^u_\Sigma(\tilde p)$ be a lift of $v$ in $T \Sigma$.
Then $(\psi^\iin)_* (\hat f_{\iin, \psi}^{m-1} \varphi \psi^\out f_{\out, \Sigma} f_\Sigma^{k_1 -1})_* \tilde v$ is a lift of $(f_{0, \psi}^n)_* v$ in $T \Pin$, and we have

\begin{align*}
\dfrac{\Vert (f_{0, \psi}^n)_* v \Vert_\psi}{\Vert v \Vert_\psi} & = \dfrac{\Vert (\hat f_{\iin, \psi}^{m-1} \varphi \psi^\out f_{\out, \Sigma} f_\Sigma^{k_1 -1})_* \tilde v \Vert}{\Vert \tilde v \Vert}
& \mb{(Fact~\ref{fact: lifts in P of points}, Item~\ref{fact: lifts; it: norm})}\\
& \geq  2 & \mb{(Lemma~\ref{lem: N1 for Sigma to Pin minus V0})} \\
\end{align*}

§.§ Additional result

We can improve Proposition <ref> by adding (for the same parameters) the existence of another pair of cone fields on $S_{0, \psi}$, denoted $(\hat C^u_\psi, \hat C^s_\psi)$, and which will have a complementary role to the one of the pair $(C^u_\psi, C^s_\psi)$ in the next section. Let us explain.

In the next section, we will show (Proposition <ref>) that we can remove a neighborhood of the boundary of $\Pin_\psi$ in $S_{0, \psi}$ to obtain a compact section $S_\psi$ with boundary, transverse to the vector field $X$, and intersecting any orbit of $X_\psi$.
The first return map $f_\psi$ for the flow of $X_\psi$ on $S_\psi$ then satisfies the following property.
For any $p \in S_\psi$, there exists an integer $n^\star = n^\star(p) \geq 0$, such that $f^n_\psi$ satisfies the cone fields condition in the neighborhood of $p$ for the pair $(C^u_\psi, C^s_\psi)$ (restricted on $S_{0, \psi}$).
Since the integer $n^\star = n^\star(p)$ depends on the point $p$, this result is not symmetric by inversion of the time direction, in other words it does not follow that there exists an integer $\hat n^\star = \hat n^\star (p) \leq 0$ such that $f^{\hat n^\star}_\psi$ satisfies the cone fields condition in the neighborhood of $p$ for the pair $(C^s_\psi, C^u_\psi)$.
Now this is necessary to prove the existence of a hyperbolic splitting on $S_\psi$ using the invariant cone fields criterion, which will then give the Anosov structure for $X_\psi$ on $P_\psi$ (Proposition <ref>).
In fact, this is not the case for the pair $(C^s_\psi, C^u_\psi)$ given by Proposition <ref>. Let us explain.

We chose the pair $(C^s_\psi, C^u_\psi)$ so as to satisfy Item <ref> of Proposition <ref>, which gives a compatibility of cones from $\Sigma_\psi$ to $\Pin_\psi$ at the first crossing, and allows us to keep the control of the image of a cone during the trajectory of an orbit crossing $\Pin_\psi$ several times.
On the other hand, there is no compatibility of cones from $\Pin_\psi$ to $\Sigma_\psi$ on the first crossing.
To get around this problem, we use a pair $(\hat C^u_\psi, \hat C^s_\psi)$ which satisfies this property, i.e. such that we have strict invariance of the cones by $f_\psi$ at the first crossing from $\Pin_\psi$ to $\Sigma_\psi$.
This pair will also continue to satisfy all the items of Proposition <ref>, except that of Item <ref>.

Let us state the result. Let the parameters $\lambda_0>1$, $\epsilon_0>0$, $\delta_0>0$ and the integers $N_0$, $N_1$ and $N_\Sigma$ be given by Proposition <ref>.
$\psi^\iin = \psi^\iin_{\lambda_0, \epsilon_0, \delta_0}$,
$\psi^\out = \psi^\out_{\lambda_0, \epsilon_0, \delta_0}$,
$\psi = \psi^\iin \varphi \psi^\out$,
the  of $\pP$ associated with this choice of parameters.

There exists a pair $(\hat C^u_\psi, \hat C^s_\psi)$ of cone fields on $S_{0, \psi}$ which satisfy the following properties.
Let $p \in S_{0, \psi}$, let $n \geq 1$ and let $q = f_{0, \psi}^n(p)$.

* All the items of Proposition <ref> except Item <ref> are satisfied.

* Suppose $p \in \Pin_\psi$, $q \in \Sigma_\psi$ and $f^k_{0, \psi}(p) \in \Sigma_\psi$ for $k=1, \dots, n-1$.

* $(f_{0, \psi}^n)_* \hat C^u_\psi(p, T_p^\iin) \subset \intr \hat C^u_\psi(q, T_q^\Sigma)$ and,

* $(f_{0, \psi}^n)_*^\inv \hat C^s_\psi(q, T_q^\Sigma) \subset \intr \hat C^s_\psi(p, T_p^\iin)$.

The proof is the same, except for Section <ref>, Lemma <ref> (this is the part which allows us to obtain Item <ref> of Proposition <ref> in the conclusion).
The analogous of Lemma <ref> is

There exists $\hat C^u_\Sigma$ a $(\hat K^u,\zeta^u/\, \zeta^s)$-cone field and $\hat C^s_\Sigma$ a $(\hat K^s,\zeta^s/\, \zeta^u)$-cone field on $\Sigma$ which satisfy the following property.
For any $\delta>0$, if $\psi^\out = \psi^\out_\delta$, and for any $p \in \Pin$ such that the point $q = f_\Sigma^n \, f_{\Sigma, \iin} \, \psi^\iin (p) \in \Sigma$ is well defined for some integer $n \geq 0$, then:

* $(f_\Sigma^n \, f_{\Sigma, \iin} \, \psi^\iin)_* \hat C^u_\iin(p) \subset \intr \hat C^u_\Sigma(q)$, and

* $(f_\Sigma^n, f_{\Sigma, \iin} \, \psi^\iin)^\inv_* \hat C^s_\Sigma(q) \subset \intr \hat C^s_\iin(p)$

The proof is based on the same arguments.
We have to choose a slope $\hat K^u$ of $\hat C^u_\Sigma$ big enough and a slope $\hat K^s$ of $\hat C^s_\Sigma$ small enough to satisfy this compatibility.
The cone fields $(\hat C^u_\psi, \hat C^s_\psi)$ will be the quotient by $\pi_\psi$ of cone fields on $\Pin$ and on $\Sigma$ in $P$.
The cone fields on $\Pin$ are the same as for the pair $(C^u_\psi, C^s_\psi)$, namely $(\psi^\iin_* C^u_\iin, \psi^\iin_* C^s_\iin)$ where $C^u_\iin$ and $C^s_\iin$ are the cone fields given in Proposition <ref>.
We replace the cones $(C^u_\Sigma, C^s_\Sigma)$ by the cones $(\hat C^u_\Sigma, \hat C^s_\Sigma)$ on $\Sigma$.
By reproducing point by point the previous proof and replacing the cone fields $(C^u_\Sigma, C^s_\Sigma)$ by the cone fields $(\hat C^u_\Sigma, \hat C^s_\Sigma)$ we obtain the result announced in Proposition <ref>.

§ MODIFICATION OF THE GLUING MAP AND PROOF OF THEOREM <REF>

In Section <ref>, we showed that there exist parameters $\lambda_0, \epsilon_0, \delta_0$ which satisfy Proposition <ref>.
We refer to the notations introduced in Section <ref>.
Let $\psi = \psi^\iin \varphi \psi^\out$ be the modified gluing map for such parameters and $X_\psi$ be the vector field induced by $X$ on $P_\psi = P/\,\psi$.
Recall that $S_{0, \psi} = \pi_\psi( \Pin \cup \Sigma )$ and $f_{0, \psi}$ is the first-return map of the flow of $X_\psi$ on $S_{0, \psi}$.
According to Proposition <ref>, there exists a pair of cone fields
$(C^u_\psi, C^s_\psi)$ on $S_{0, \psi}$ for which a (non-uniform) iterate of $f_{0, \psi}$ satisfies the cone fields condition in the neighborhood of some points, outside a small neighborhood $(\cV_0)_\psi$ of the periodic orbits $(\cO_*)_\psi$ in $S_{0, \psi}$.
Indeed, the section is not uniformly transverse and the expansion of the vector of the cones is missing along some orbits when approaching the periodic orbits $(\cO_*)_\psi$ contained in $\Pin_\psi$ (Item <ref>).
The goal of this section is therefore to remove a well-chosen neighborhood of the orbits $(\cO_*)_\psi$ in $S_{0, \psi}$ in order to have the hyperbolicity of the return map everywhere.

§.§ Gluing procedure and invariant cone fields on a global section

§.§.§ Global section

We remove from $S_{0, \psi}$ the projection of the neighborhood $\cV_0 := \psi^\iin (\cV_{\delta_0})$ of the orbits $\cO_*$. We obtain the surface

\begin{equation} \label{eq: S_psi}
S_\psi := \pi_\psi \l( (\Pin \ssm \cV_0 ) \cup \Sigma \r) = (\Pin \ssm \cV_0)_\psi \cup \Sigma_\psi \subset S_{0, \psi}
\end{equation}

It is an immersed compact global section.
Indeed, according to Item <ref> of Proposition <ref>, an orbit of $X_\psi$ which intersects $(\Pin \cap \cV_0)_\psi$ intersects the section $\Sigma_\psi$ in the past and in the future in uniformly bounded time.

Section $S_\psi$

We recall (see Subsection <ref>) that as for the section $S_{0, \psi}$, the surface $S_\psi$ admits points $p$ of self-intersection on which it can have two or three tangent planes $T^* = T^*_p \subset T_p S_\psi$.
Denote $T^\Sigma_p$ the tangent plane(s) to $\Sigma_\psi$ at $p$ and $T^\iin_q$ the tangent plane to $\Pin_\psi$ at $q$.
The surface $S_\psi$ has a metric $g_\psi$, which is a collection of metrics $g_\psi (p, T^*)$ on each of the tangent planes $T^* \subset T_p S_\psi$.
A cone field $C$ on $S_\psi$ is the data for each $p \in S_\psi$ and for each tangent plane $T^*$ to $S_\psi$ at the point $p$ of a cone $C(p, T^*) \subset T^*$.

§.§.§ Statement and proof of the proposition

If $f_\psi$ is the first return map of the flow of $X_\psi$ on $S_\psi$,
then we show that it has a hyperbolic behavior in the future of the orbit of the points $p \in \Sigma_\psi$.
More precisely, the following proposition gives the existence of a positive integer $n^\star(p)\geq 0$ for any point $p \in S_\psi$, uniformly bounded from above, such that the iterate $f^{n^\star}_\psi$ satisfies the cone fields condition in the neighborhood of $p$ for the pair $(C^s_\psi, C^u_\psi)$ of Proposition <ref>.

There exists an integer $\bN \geq 0$ such that,
for all $p \in S_\psi$, if $T_p$ is (one of) the tangent plane(s) to $S_\psi$ at $p$,
there exists an integer $n^\star = n^\star(p, T_p)$ with $0 \leq n^\star \leq \bN$,
and a tangent plane $T_q = T_q(p, T_p)$ to $S_\psi$ in
$q = f_\psi^{n^\star}(p) \in S_\psi$
such that :

* $(f_\psi^{n^\star})_* C^u_\psi(p, T_p) \subset \intr C^u_\psi(q, T_q)$,
and for all $v \in C^u_\psi(p, T_p)$, $\Vert (f_\psi^{n^\star})_* v \Vert \geq 2 \Vert v \Vert$

* $(f_\psi^{n^\star})^\inv_* C^s_\psi(q, T_q) \subset \intr C^s_\psi(p, T_p)$,
and for all $v \in C^s_\psi(q, T_q)$, $\Vert (f_\psi^{n^\star})^\inv_* v \Vert \geq 2 \Vert v \Vert$

Dichotomy diagram of orbits in Case <ref>

Dichotomy diagram of $f_\psi$-orbits on $S_\psi$

Dichotomy diagram of orbits in Case <ref>

Dichotomy diagram of $f_\psi$-orbits on $S_\psi$

Note that $f_\psi$ is a restriction of $f_{0, \psi}$.
We can use exclusively the results of Proposition <ref> on the return map $f_{0, \psi}$ of the flow of $X_\psi$ on the section $S_{0, \psi}$.
If $p \in S_\psi$ and $q \in S_\psi$, then there exist $n$ and $m$ such that $f^n_\psi(p) = q = f^m_{\psi, 0}(p)$ and $n \leq m$.
Indeed $f_\psi$ does not see the crossings on $(\Pin \cap \cV_0)_\psi$.
We will make dichotomy according to the starting point $p$ on the section $S_\psi$, and the trajectory of its orbit.
We represent the dichotomy by the graphs of Figure <ref>.

* Let $p \in (\Pin \ssm \cV_0)_\psi$ and $T_p = T_p^\iin$.

Two cases are possible depending on whether the future orbit of $p$ by the flow of $X_\psi$ crosses the section $\Sigma_\psi$ enough times without crossing $\Pin_\psi$ to have strict invariance and expansion of the cones, or whether it crosses $\Pin_\psi$ before.
This return number is the integer $N_1$ which satisfies Item <ref> of Proposition <ref>.

We make the following dichotomy:

* Either the orbit of $p$ by the flow of $X_\psi$ crosses $N_1$ consecutive times $\Sigma_\psi$ without crossing $\Pin_\psi$.

In other words $f_{0, \psi}^k(p) \in \Sigma_\psi$ for $k=1, \dots, N_1$.
Then according to Item <ref> Proposition <ref>,
Proposition <ref> is satisfied for $n^\star(p)=N_1$ and $T_q = T_q^\Sigma$.

* Either the orbit of $p$ by the flow of $X_\psi$ crosses $\Pin_\psi$ before crossing $N_1$ times $\Sigma_\psi$.

In other words there exists $k_0 \leq N_1$ minimal such that $f_{0, \psi}^{k_0}(p) \in \Pin_\psi$.
Then two cases are possible.

* Either $f_{0, \psi}^{k_0}(p) \in (\Pin \ssm \cV_0)_\psi$.

According to Item <ref> of Proposition <ref>, Proposition <ref> is satisfied for $n^\star(p) = k_0 \leq N_1$ and $T_q = T_q^\iin$.

* Either $f_{0, \psi}^{k_0}(p) \in (\Pin \cap \cV_0)_\psi$.

Then according to Item <ref> of Proposition <ref>, the future orbit of $f_{0, \psi}^{k_0}(p)$ by the flow of $X_\psi$ crosses $N_0$ consecutive times the section $\Sigma_\psi$ without crossing $\Pin_\psi$.
In other words $f^k_{0, \psi}\l(f^{k_0}_{0, \psi}(p)\r) \in \Sigma_\psi$ for $k=1, \dots, N_0$.

Then according to Item <ref> of Proposition <ref>, Proposition <ref> is satisfied for $n^\star(p) = N_0 + k_0-1 \leq N_0 + N_1 -1$ and $T_q = T_q^\Sigma$.

Note that the endpoint
$q$ on $\Sigma_\psi$ corresponds to the iteration $N_0 + k_0$ of the orbit of $p$ by $f_{0, \psi}$, and the iteration $N_0 + k_0 -1$ of the orbit of $p$ by $f_\psi$ because in the second case we do not count the crossing of the orbit through $(\Pin \cap \cV_0)_\psi$.

* Let $p \in \Sigma_\psi$, and $T_p = T_p^\Sigma$.

We make a dichotomy depending on whether the orbit of $p$ by the flow of $X_\psi$ crosses $\Sigma_\psi$ enough times without crossing $\Pin_\psi$ to have strict invariance and expansion of the cones, or if it crosses $\Pin_\psi$ before.

This return number is the integer $N_\Sigma$ which satisfies Item <ref> of Proposition <ref>.

* Either the orbit of $p$ by the flow of $X_\psi$ crosses $N_\Sigma$ consecutive times without crossing $\Pin$.

In other words $f^k_{0, \psi}(p) \in \Sigma_\psi$ for $k=0, \dots, N_\Sigma$.
Then according to Item <ref> of Proposition <ref>, Proposition <ref> is satisfied for $n^\star(p) = N_\Sigma$ and $T_q = T_q^\Sigma$.

* Either the orbit of $p$ by the flow of $X_\psi$ crosses $\Pin_\psi$ before.

In other words there exists $k_1 \leq N_\Sigma$ minimal such that $f^{k_1}_{0, \psi}(p) \in \Pin_\psi$.
Then two cases are possible, depending on whether this intersection point is in the neighborhood $(\cV_0)_\psi$ or not.

Either $f^{k_1}_{0, \psi}(p) \in (\Pin \cap \cV_0)_\psi$.

Then according to Item <ref> of Proposition <ref>, the future orbit of $f^{k_1}(p)$ by the flow of $X_\psi$ crosses $N_0$ consecutive times the section $\Sigma_\psi$ without crossing $\Pin_\psi$.
In other words $f^k_{0, \psi} \l( f^{k_1}_{0, \psi}(p) \r) \in \Sigma_\psi$ for $k=1, \dots, N_0$.

Then according to Item <ref> of Proposition <ref>, Proposition <ref> is satisfied for $n^\star(p) = N_0 + {k_1}-1 \leq N_0 + N_\Sigma -1$ and $T_q = T_q^\Sigma$.

Note that the endpoint $q$ on $\Sigma_\psi$ corresponds to the iteration $N_0 + k_1$ of $p$ by $f_{0, \psi}$, and the iteration $N_0 + k_1 -1$ of $p$ by $f_\psi$, because in the second case we do not count the crossing of the orbit by $(\Pin \cap \cV_0)_\psi$.

Either $f^{k_1}_{0, \psi}(p) \in (\Pin \ssm \cV_0)_\psi$.

Let $q_1 = f^{k_1}_{0, \psi}(p) = f^{k_1}_\psi(p)$.
Either the orbit of $q_1$ crosses the $\Sigma$ section enough consecutive times without crossing $\Pin_\psi$.
This return number is the integer $N_0$ which satisfies Item <ref> of Proposition <ref>, and we will have strict invariance and expansion of the cones.
Either the orbit of $q_1$ crosses $\Pin_\psi$ at a point $q_2$ with less than $N_0$ crossing through $\Sigma_\psi$ between each.
According to Item <ref> of Proposition <ref>, the point of intersection $q_2$ is in $(\Pin \ssm \cV_0)_\psi$.
In this case, we can reiterate this dichotomy by replacing $q_2$ by $q_1$.
This recurrence is finite because if the orbit of $q_1$ crosses $(\Pin \ssm \cV_0)_\psi$ enough times without crossing $(\Pin \cap \cV_0)_\psi$, then we have the strict invariance and expansion of cones. This return number is the integer $N_1$ which satisfies Proposition <ref>, Item <ref>.
Formally the dichotomy is the following.