2.1 Reduction of Theorem A to Theorem 2.1

After applying Proposition 2.1, we are thus left with a family $(f_{\varepsilon ,k})$ of $C^k$ - symplectic diffeomorphisms, each $f_{\varepsilon ,k}$ being conjugated to $f_{\varepsilon }$ and admitting a separatrix $\Sigma _{\varepsilon ,k}$ . Because the conclusions of Theorem A are clearly invariant by conjugation, to prove Theorem A, we just need to prove that if $k\geq r$ and $\varepsilon $ is small enough, each separatrix $\Sigma _{\varepsilon ,k}$ is accumulated by a set of positive measure of KAM circles for $f_{\varepsilon ,k}$ . This is the content of Theorem 2.1 below that we shall apply to the family of $C^k$ -diffeomorphisms $f_{\varepsilon ,k}$ defined by (1.2), (1.3), and (2.4), but that holds for any family (that we still denote in what follows by $(f_{\varepsilon })_{\varepsilon \in I}$ to alleviate the notations) of symplectic $C^k$ -diffeomorphisms satisfying the following hypothesis.

Let $(f_{\varepsilon })_{\varepsilon \in I}$ , ( $I\ni 0$ open interval of ${\mathbb R}$ ) be a family of $C^k$ -symplectic diffeomorphisms of ${\mathbb R}^2$ that satisfies:

1. (H1) each $f_{\varepsilon }$ has a (non-split) separatrix $\Sigma _{\varepsilon }$ associated to the hyperbolic point o;

2. (H2) the map $I\ni \varepsilon \to f_{\varepsilon }-\text {id}\in C^k({\mathbb R}^2,{\mathbb R}^2)$ is continuous (the norm on $C^k$ is the usual $C^k$ -norm);

3. (H3) on some neighborhood V of o, each $f_{\varepsilon }$ coincides with the time-1 map of a symplectic vector field $J\nabla Q_{\varepsilon }(x,y)$ where $Q_{\varepsilon }(x,y)=q_{\varepsilon }(xy)$ , $q_{\varepsilon }\in C^{k+1}({\mathbb R}^2)$

   $$ \begin{align*}q_{\varepsilon}(t)=\lambda t+O^2(t),\quad \lambda>0;\end{align*} $$

4. (H4) on ${\mathbb R}^2$ , $f_{0}=\phi ^1_{X_{0}}$ , where $X_{0}=J\nabla H_{0}$ is a Hamiltonian vector field that coincides with $J\nabla Q_{0}$ on V.

Remark 2.2. The intersection $\Sigma _{\varepsilon }\cap V_{}$ is the union

$$ \begin{align*}\Sigma_{\varepsilon}\cap V_{}=(W^s_{f_{\varepsilon}}(o)\cap V_{})\cup (W^{u}_{f_{\varepsilon}}(o)\cap V_{})\end{align*} $$

and

$$ \begin{align*}W^{s}_{f_{\varepsilon}}(o)\cap V_{}=({\mathbb R}\times\{0\}) \cap V_{},\quad W^{u}_{f_{\varepsilon}}(o)\cap V_{}=(\{0\}\times {\mathbb R}) \cap V_{}.\end{align*} $$

One then has the following.

We shall give the proof of Theorem 2.1 in §8.

The proof of Proposition 2.1 occupies the rest of this section. It will be based on a first reduction obtained by performing some steps of Birkhoff normal forms (Proposition 2.3) and then the application of various Sternberg-like normal forms (Corollary 2.4 and Proposition 2.5).

2.2 Birkhoff normal form for the time-periodic vector field $X_{\varepsilon }^t$

A preliminary step in Sternberg’s classical linearization theorem is to first conjugate the considered system (diffeomorphism or vector field) defined in the neighborhood of the hyperbolic fixed point o to a system which is tangent to an integrable model to some high enough order. This is what we do in this subsection and in a symplectic framework (see Proposition 2.3) by using Birkhoff normal form techniques.

2.2.1 Periodically forced vector fields

Let $X:{\mathbb R}\times {\mathbb R}^2\ni (t,x)\mapsto X^{t}(z):= X(t,z)\in {\mathbb R}^2$ be a smooth time-dependent symplectic vector field: for each t, the 1-form $i_{X_{t}}\omega $ is closed (and hence locally exact). For $t,s\in {\mathbb R}$ , we denote by $\phi ^{t,s}_{X}$ the flow of X between times s and t when it is defined (see page 3 for the definition of $\phi ^{t,s}_{X}$ ). If $t\mapsto g^{t}(\cdot )$ is a one-parameter family of symplectic diffeomorphisms, one has

(2.7) $$ \begin{align} g^{t}\circ \phi_{X}^{t,s}\circ (g^{s})^{-1}=\phi^{t,s}_{\tilde X},\end{align} $$

where $\tilde X:(t,z)\mapsto \tilde X^{t}(z):=\tilde X(t,z)$ is the smooth time-dependent symplectic vector field

(2.8) $$ \begin{align}\tilde X^{t}=\partial_{t}g^{t}\circ (g^{t})^{-1}+(g^{t})_{*}X^{t}. \end{align} $$

Conversely, if (2.8) is satisfied, then so is (2.7). Note that if $g^t$ depends 1-periodically on t, then (2.7) yields the more classical conjugation equation

$$ \begin{align*}g\circ \phi^{1,0}_{X}\circ g^{-1}=\phi_{\tilde X}^{1,0}, \end{align*} $$

where $g=g^0=g^1$ ( $g^t$ is 1-periodic in t).

Assume now that $X^{t}$ depends 1-periodically in t and, in a smooth way, on a small parameter $\varepsilon \in {\mathbb R}$ ; we furthermore assume that it is of the form

(2.9) $$ \begin{align} X_{\varepsilon}^t(z)=J\nabla H_{\varepsilon}^t(z),\end{align} $$

where ( $z=(z_{1},z_{2})\in {\mathbb R}^2$ )

(2.10) $$ \begin{align} H_{\varepsilon}^{t}(z)=\lambda_{\varepsilon}(t)z_{1}z_{2}+O^3(z),\quad \int_{{\mathbb T}}\lambda_{\varepsilon}(t)\,dt>0,\quad \lambda_{0}(t)=\lambda\in{\mathbb R}^*_{+},\end{align} $$

$H_{\varepsilon }:{\mathbb R}/{\mathbb Z}\times {\mathbb R}^2\to {\mathbb R}$ , $H_{\varepsilon }:(t,z)\mapsto H_{\varepsilon }(t,z):=H_{\varepsilon }^{t}(z)$ being a smooth function. Assume also that, for some $j\in {\mathbb N}^*$ ,

$$ \begin{align*}g^t_{\varepsilon}(z)=\phi^{1}_{J\nabla G_{\varepsilon}^t}(z)=\text{id}+O^j(z),\quad G_{\varepsilon}^t(z)=O^{j+1}(z),\end{align*} $$

where $G:I\times {\mathbb R}/{\mathbb Z}\times ({\mathbb R}^2,o)\ni (\varepsilon ,t,z)\mapsto G_{\varepsilon }(t,z):=G_{\varepsilon }^{t}(z)\in {\mathbb R}$ is a smooth function. Then, one has

$$ \begin{align*}\partial_{t}g_{\varepsilon}^{t}\circ (g_{\varepsilon}^{t})^{-1}=J\nabla \partial_{t}G_{\varepsilon}^t+O^{j+1}(z), \end{align*} $$

$$ \begin{align*}(g_{\varepsilon}^{t})_{*}X_{\varepsilon}^{t}=J\nabla H_{\varepsilon}^t\circ (g_{\varepsilon}^{t})^{-1}=J\nabla H_{\varepsilon}^{t}+J\nabla\{G_{\varepsilon}^{t},H_{\varepsilon}^t\}+O^{j+1}(z), \end{align*} $$

(here $\{A,B\}$ denotes the Poisson bracket $\{A,B\}={\langle }\nabla A,J\nabla B{\rangle } $ ) so that $\tilde X_{\varepsilon }^t$ defined by (2.8) is of the form

(2.11) $$ \begin{align} \tilde X_{\varepsilon}^t= J\nabla \tilde H_{\varepsilon}^t,\end{align} $$

with

(2.12) $$ \begin{align}\tilde H_{\varepsilon}^t&=H_{\varepsilon}^t+\partial_{t}G_{\varepsilon}^{t}+\{G_{\varepsilon}^{t},H_{\varepsilon}^{t}\}+O^{j+2}(z) \end{align} $$

(2.13) $$ \begin{align} & \kern1.65pc =H_{\varepsilon}^t+\partial_{t}G_{\varepsilon}^t+\{G_{\varepsilon}^t,H_{2,\varepsilon}^t\}+O^{j+2}(z), \end{align} $$

where we have denoted $H_{2,\varepsilon }^t(z_{1},z_{2})=\lambda _{\varepsilon }(t)z_{1}z_{2}$ .

If in the preceding equation, one chooses $G^{t}_{\varepsilon }=G_{\varepsilon ,2}^t$ with $G_{\varepsilon ,2}^t(z)=a_{\varepsilon ,0}(t)z_{1}z_{2}$ , where $a_{\varepsilon ,0}$ is the 1-periodic function defined by

$$ \begin{align*}a_{\varepsilon,0}(t)=-\int_{0}^t\bigg(\lambda_{\varepsilon}(s)-\int_{{\mathbb T}}\lambda_{\varepsilon}(u)\,du\bigg)\,ds, \end{align*} $$

one has

$$ \begin{align*} \tilde H_{\varepsilon}^t(z)=\bar\lambda_{\varepsilon}z_{1}z_{2}+O^3(z),\end{align*} $$

where $\bar \lambda _{\varepsilon }=\int _{{\mathbb T}}\lambda _{\varepsilon }(t)\,dt$ . In other words, performing a change of coordinates (2.8) on $X_{\varepsilon }^t$ with $g^t_{\varepsilon }=g_{\varepsilon ,2}^t=\phi ^{1}_{J\nabla G^t_{\varepsilon ,2}}$ , we can assume that in (2.10), $\lambda _{\varepsilon }(t)$ does not depend on t:

(2.14) $$ \begin{align} H_{\varepsilon}^t(z)=\lambda_{\varepsilon}z_{1}z_{2}+O^3(z),\quad \lambda_{\varepsilon}\in{\mathbb R}^*_{+}\end{align} $$

(we write $\lambda _{\varepsilon }$ in place of $\bar \lambda _{\varepsilon }$ ).

2.2.2 Birkhoff normal form

Having put $H_{\varepsilon }^t$ under the form (2.14), we now eliminate by successive conjugations (2.8) non-diagonal higher-order terms in z from $H_{\varepsilon }^t$ (note that they depend on t).

The following lemma describes this elimination procedure.

Lemma 2.2. Let $j\in {\mathbb N}$ , $j\geq 2$ . Assume that, for some polynomials $q_{\varepsilon }(s)=\lambda s +O(s^2)\in {\mathbb R}[s]$ of degree $\leq [ j/2]$ depending smoothly on $\varepsilon $ ,

$$ \begin{align*}H_{\varepsilon}^t(z)=q_{\varepsilon}(z_{1}z_{2})+O^{j+1}(z).\end{align*} $$

Then, there exist a smooth family $(\tilde q_{\varepsilon })_{\varepsilon }$ of polynomials $\tilde q_{\varepsilon }(s)=\lambda s+O(s^2)\in {\mathbb R}[s]$ of degree $\leq [(j+1)/2]$ and a smooth family of smooth maps $G_{\varepsilon }:{\mathbb R}/{\mathbb Z}\times ({\mathbb R}^2,o)\ni (t,z)\to G_{\varepsilon }(t,z)=G^t_{\varepsilon }(z)\in {\mathbb R}^2$ such that on a neighborhood of o,

(2.15) $$ \begin{align} \begin{cases}G_{\varepsilon}^t(z)=O^{j+1}(z),\\ H_{\varepsilon}^t(z)+\partial_{t}G_{\varepsilon}^t(z)+\{G_{\varepsilon}^t,H_{\varepsilon}^t\}(z)=\tilde q_{\varepsilon}(z_{1}z_{2})+O^{j+2}(z).\end{cases}\end{align} $$

Moreover, if for $\varepsilon =0$ , $H_{0}^t$ does not depend on t, one can choose $G_{0}^t$ to be independent of t.

Proof. See Appendix A.

Let now $X_{\varepsilon }^{t}$ be the family of vector fields of (1.2).

Proposition 2.3. For any $N\geq 1$ there exist an open neighborhood $V_{N}$ of o, a smooth two-parameters family $(b_{\varepsilon }^t)_{\varepsilon \in I,t\in {\mathbb R}/{\mathbb Z}}$ (I some open interval containing 0) of smooth symplectic diffeomorphisms satisfying $b_{\varepsilon }^{t}(o)=o$ , $Db_{\varepsilon }^t(o)=\mathrm {id}$ and a smooth family of polynomials $q_{\varepsilon ,N}(s)=\lambda s+O(s^2)$ of degree $\leq [(N+1)/2]$ , such that, for any $\varepsilon \in I$ , $t\in {\mathbb R}/{\mathbb Z}$ , $(x,y)\in V_{N}$ one has

$$ \begin{align*}X_{\varepsilon}^{(1),t }&\mathop{=}_{\mathrm{defin.}}\ (\partial_{t}b^t_{\varepsilon})\circ (b^t_{\varepsilon})^{-1}+(b^t_{\varepsilon})_{*}X_{\varepsilon}^t\\ &\ \, =J\nabla Q_{\varepsilon,N}+O^{N+1}(x,y) \quad \text{with }Q_{\varepsilon,N}(x,y)=q_{\varepsilon,N}(xy),\end{align*} $$

and for $\varepsilon =0$ , $b^t_{0}$ is independent of t.

Proof. Applying the preceding Lemma 2.2 and relations (2.8)–(2.12) inductively (starting from (2.14)), we thus construct polynomials $q_{\varepsilon ,j}$ of degree $\leq [ j/2]$ ( $j\geq 2$ ) and functions $G_{\varepsilon ,j}^t=O^{j+1}(z)$ such that if one defines

$$ \begin{align*}b^t_{\varepsilon}=g^t_{\varepsilon,N}\circ\cdots\circ g^t_{\varepsilon,2}=\text{id}+O^2(z),\quad g^t_{\varepsilon,j}=\phi^1_{J\nabla G^t_{\varepsilon,j}}=\text{id}+O^{j}(z),\end{align*} $$

one has

$$ \begin{align*}\tilde X_{\varepsilon}^{t}:&=\partial_{t}b_{\varepsilon}^{t}\circ (b_{\varepsilon}^{t})^{-1}+(b_{\varepsilon}^{t})_{*}X_{\varepsilon}^{t}\\ &=J\nabla Q_{\varepsilon,N}+O^{N+1}(z)\quad \text{with } Q_{\varepsilon,N}(z)=q_{\varepsilon,N}(z_{1}z_{2}), \end{align*} $$

all depending on $\varepsilon $ being smooth. Moreover, if $X_{0}^t$ is independent of t, the diffeomorphism $b_{0}^t$ is independent of t.

Remark 2.3. Note that because $b_{0}^t\equiv b_{0}$ is independent of t, the vector field

$$ \begin{align*}X_{0}^{(1)}=(b_{0})_{*}X_{0}\end{align*} $$

is autonomous.

2.3 Symplectic Sternberg theorem for the autonomous vector field $X_{0}^{(1)}$

We shall need a symplectic version of the famous theorem by S. Sternberg (on smooth linearization of hyperbolic germs of smooth vector fields, see [Reference Sternberg19]), as proved in [Reference Banyaga, de la Llave and Wayne7] or [Reference Chaperon8] (see also [Reference Sternberg20]). We follow here the exposition of [Reference Banyaga, de la Llave and Wayne7].

Let $Z_{i}$ , $i=1,2$ , be two symplectic smooth autonomous vector fields such that, for some $\lambda \in {\mathbb R}^*$ and $N\in {\mathbb N}$ , one has

(2.16) $$ \begin{align}\begin{cases} Z_{i}(x,y)=-\lambda x\dfrac{\partial}{\partial x}+\lambda y\dfrac{\partial}{\partial y}+O^2(x,y)\quad (i=1,2),\\[9pt] Z_{1}(x,y)-Z_{2}(x,y)=O^{N+1}(x,y). \end{cases} \end{align} $$

Theorem 2.2. [Reference Banyaga, de la Llave and Wayne7, Theorem 1.2]

There exist positive constants $A,B$ for which the following holds. Let $m\in {\mathbb N}^*$ large enough and $N=[(m+B)/A]+1\geq 1$ . If (2.16) is satisfied, then there exists a $C^{m}$ symplectic change of coordinates such that on a neighborhood of o,

(2.17) $$ \begin{align} \begin{cases}(S_{0})_{*}Z_{1}=Z_{2},\\ S_{0}(o)=0,\qquad \ \ DS_{0}(o)=\mathrm{id}.\end{cases} \end{align} $$

We apply the preceding theorem to the case $Z_{1}=X_{0}^{(1)}$ and $Z_{2}=J\nabla Q_{0,N}$ ( $X_{0}^{(1)}$ , $Q_{0,N}$ given by Proposition 2.3 when $\varepsilon =0$ ). In view of Proposition 2.3, the condition (2.16) is satisfied and we hence get a symplectic diffeomorphism $S_{0}$ satisfying $S_{0}(o)=0$ , $DS_{0}(o)=\text {id}$ , and such that on a neighborhood of o,

$$ \begin{align*}(S_{0})_{*}X_{0}^{(1)}=J\nabla Q_{0,N}.\end{align*} $$

For each value of $t\in {\mathbb R}/{\mathbb Z}$ and $\varepsilon \in I$ , the diffeomorphism $(S_{0}\circ b^t_{\varepsilon })$ fixes the origin and its derivative at the origin is the identity. It can thus be extended as a symplectic $C^{m}$ -diffeomorphism $R^t_{\varepsilon }$ of ${\mathbb R}^2$ (cf. Lemma B.1). Note that the dependence of $R_{\varepsilon }^t$ with respect to t is smooth and 1-periodic ( $t\in {\mathbb R}/{\mathbb Z}$ ). We now define on ${\mathbb R}/{\mathbb Z}\times {\mathbb R}^2$ the time-periodic vector field $X_{\varepsilon }^{(2)}:(t,(x,y))\in ({\mathbb R}/{\mathbb Z})\times {\mathbb R}^2\to {\mathbb R}^2$ by

(2.18) $$ \begin{align} X_{\varepsilon}^{(2),t}\mathop{=}_{\mathrm{defin.}}(\partial_{t}R^t_{\varepsilon})\circ (R^t_{\varepsilon})^{-1}+(R^t_{\varepsilon})_{*}X_{\varepsilon}^t,\end{align} $$

and we observe that on a neighborhood of o,

$$ \begin{align*}(R^t_{0})_{*}X_{0}=X_{0}^{(2)}=J\nabla Q_{0,N}.\end{align*} $$

Because the conjugacy relation (2.18) is equivalent to (see §2.2.1)

$$ \begin{align*}\text{for all } t,s,\quad R^t_{\varepsilon}\circ \phi^{t,s}_{X_{\varepsilon}}\circ (R_{\varepsilon}^s)^{-1}=\phi^{t,s}_{X^{(2)}_{\varepsilon}},\end{align*} $$

we get by taking $t=1$ , $s=0$ , and setting $R_{\varepsilon }:=R^1_{\varepsilon }=R^0_{\varepsilon }$ , the following corollary.

Corollary 2.4. If $m\in {\mathbb N}^*$ is large enough and $N=[(m+B)/A]+1$ , there exists a smooth family $(R_{\varepsilon })$ of $C^m$ symplectic diffeomorphisms of ${\mathbb R}^2$ such that $R_{\varepsilon }(o)=o$ , $DR_{\varepsilon }(o)=\mathrm {id}$ , and on a neighborhood of o,

(2.19) $$ \begin{align}f_{\varepsilon}^{(1)}:&\mathop{=}_{\mathrm{defin.}}R_{\varepsilon}\circ f_{\varepsilon}\circ (R_{\varepsilon})^{-1} \end{align} $$

(2.20) $$ \begin{align} &\quad \ \, \kern3pc =\phi^1_{J\nabla Q_{\varepsilon,N}}+O^{N+1}(x,y).\end{align} $$

Moreover,

(2.21) $$ \begin{align} (R_{0})_{*}X_{0}=J\nabla Q_{0,N}.\end{align} $$

Note that the last equation shows that

(2.22) $$ \begin{align} f_{0}^{(1)}=\phi^1_{J\nabla Q_{0,N}}.\end{align} $$

2.4 Symplectic Sternberg normal form for the diffeomorphism $f_{\varepsilon }^{(1)}$

Theorem 2.2 has a version for smooth germs of symplectic diffeomorphisms which are hyperbolic at the origin. This is theorem 1.1 of [Reference Banyaga, de la Llave and Wayne7]. In our paper, we shall need a parametric version of that result, which is not explicitly stated in [Reference Banyaga, de la Llave and Wayne7] but that can be checked after close examination of the proof.

Proposition 2.5. There exist constants $A_{1},B_{1}$ depending on $\lambda \in {\mathbb R}^*$ such that the following holds. Let $m\in {\mathbb N}^*$ large enough and $N=[m/2]-3$ . If $(g_{1,\varepsilon })_{\varepsilon \in I}$ and $(g_{2,\varepsilon })_{\varepsilon \in I}$ ( $I\ni 0$ some open interval of ${\mathbb R}$ ) are two continuous (with respect to $\varepsilon \in I$ ) families of $C^m$ symplectic diffeomorphisms such that

(2.23) $$ \begin{align} \begin{cases}\text{for all } \varepsilon,\quad g_{1,\varepsilon}(o)=g_{2,\varepsilon}(o)=o,\\[-2pt] Dg_{1,0}(o)=\mathrm{diag}(\lambda,\lambda^{-1}) \quad \mbox{(is hyperbolic)},\\[-2pt] g_{1,\varepsilon}(x,y)=g_{2,\varepsilon}(x,y)+O^{N+1}(x,y),\\[-2pt] g_{1,0}=g_{2,0}, \end{cases}\end{align} $$

then, there exists a continuous family $(S^{(1)}_{\varepsilon })_{\varepsilon }$ (with respect to $\varepsilon \in I$ small enough) of $C^k$ symplectic diffeomorphisms such that $S^{(1)}_{\varepsilon }(o)=o$ , $DS^{(1)}_{\varepsilon }(o)=\mathrm {id}$ with $k=[NA_{1}-B_{1}]-1$ , and

$$ \begin{align*}\begin{cases}S^{(1)}_{\varepsilon}\circ g_{1,\varepsilon}\circ (S^{(1)}_{\varepsilon})^{-1}=g_{2,\varepsilon},\\[-2pt] S^{(1)}_{0}=\mathrm{id}.\end{cases}\end{align*} $$

2.5 Proof of Proposition 2.1

It will be a consequence of Corollary 2.4 and Proposition 2.5.

We first choose N so that $k=[NA_{1}-B_{1}]-1$ and we define m by $N=[(m+B)/A]+1$ . If k is large enough, m will satisfy the assumption of Corollary 2.4. We then apply Proposition 2.5 to $g_{1,\varepsilon }=f_{\varepsilon }^{(1)}$ , $g_{2,\varepsilon }=\phi ^1_{J\nabla Q_{\varepsilon ,N}}$ , which satisfies (2.23) (note that (2.20) is satisfied). This provides us with a continuous family $(S_{\varepsilon }^{(1)})_{\varepsilon }$ of $C^k$ symplectic diffeomorphisms defined in a fixed neighborhood of o such that $S^{(1)}_{\varepsilon }(o)=o$ , $DS^{(1)}_{\varepsilon }(o)=\text {id}$ , and on a neighborhood of o,

(2.24) $$ \begin{align} \begin{cases}S^{(1)}_{\varepsilon}\circ f_{\varepsilon}^{(1)}\circ (S^{(1)}_{\varepsilon})^{-1}=\phi^1_{J\nabla Q_{\varepsilon,N}},\\[-2pt] S^{(1)}_{0}=\text{id}.\end{cases}\end{align} $$

We can extend these $S^{(1)}_{\varepsilon }$ as symplectic $C^k$ diffeomorphisms $S^{(2)}_{\varepsilon }$ of ${\mathbb R}^2$ which depend continuously on $\varepsilon $ (cf. Lemma B.1). We then define

$$ \begin{align*}\Theta_{\varepsilon,k}=S_{\varepsilon}^{(2)}\circ R_{\varepsilon}\end{align*} $$

and we observe that on a neighborhood of o,

$$ \begin{align*}\begin{cases} \Theta_{\varepsilon,k}\circ f_{\varepsilon}\circ \Theta_{\varepsilon,k}^{-1}=\phi^1_{J\nabla Q_{\varepsilon,N}},\\ (\Theta_{{0},k})_{*}X_{0}=J\nabla Q_{0,N}; \end{cases}\end{align*} $$

indeed, the first equality comes from (2.19 and the first equation of (2.24), while the second is a consequence of (2.21) and the second equation of (2.24).

To conclude the proof, we rename $q_{\varepsilon ,N}$ , $Q_{\varepsilon ,N}$ as $q_{\varepsilon ,k}$ , $Q_{\varepsilon ,k}$ .

Note: From now on, and until the end of §8, we shall work in the setting of Theorem 2.1 with a family of $C^k$ symplectic diffeomorphisms satisfying conditions (H1)–(H4).

4.1 Fundamental domains

Let $V_{}$ be the domain of Theorem 2.1. One can choose $x_{*}>0$ such that, for any $\varepsilon $ small enough,

$$ \begin{align*}(x_{*},0)\in V_{}\quad \text{and}\quad f_{\varepsilon}^{-1}(x_{*},0)\notin V_{}.\end{align*} $$

For $y_{*}>0$ small enough, we define the vertical segment

$$ \begin{align*}L_{x_{*},y_{*}}:=\{(x_{*},ty_{*}),\ 0<t<1\}\end{align*} $$

and the domain

$$ \begin{align*}\mathcal{F}_{\varepsilon,x_{*},y_{*}}\end{align*} $$

as the interior of the contour defined by (see Figure 2)

1. (a) the segment $[f_{\varepsilon }(x_{*},0),(x_{*},0)]$ ;

2. (b) the transversal $\overline {L_{x_{*},y_{*}}}$ ;

3. (c) the piece of hyperbola joining $(x_{*},y_{*})$ to $f_{\varepsilon }(x_{*},y_{*})$ (cf. Remark 2.1);

4. (d) the curve $f_{\varepsilon }(\overline {L_{x_{*},y_{*}}})$ .

We shall often drop the index $x_{*}$ in the notations of $L_{x_{*},y_{*}}$ , $\mathcal {F}_{\varepsilon ,x_{*},y_{*}}$ and simply set

$$ \begin{align*}L_{y_{*}}:=L_{x_{*},y_{*}}\quad \text{and}\quad \mathcal{F}_{\varepsilon,y_{*}}=\mathcal{F}_{\varepsilon,x_{*},y_{*}}.\end{align*} $$

If $y_{*}$ is small enough, one has $\mathcal {F}_{\varepsilon ,y_{*}},\overline {L_{y_{*}}}\subset V_{}$ . We set

$$ \begin{align*}\tilde{\mathcal{F}}_{\varepsilon,y_{*}}=\mathcal{F}_{\varepsilon,y_{*}}\cup L_{y_{*}}.\end{align*} $$

Figure 2 Fundamental domain $\mathcal {F}_{\varepsilon ,y_{*}}$ for $f_{\varepsilon }$ and the first return map $\hat f_{\varepsilon }$ .

4.2 First return maps

Our aim in this subsection is to define the first return map of $f_{\varepsilon }$ in $\tilde {\mathcal {F}}_{\varepsilon ,y_{*}}$ .

Because $\Sigma _{\varepsilon }$ is a separatrix for $f_{\varepsilon }$ , we can define (see Remark 2.2)

$$ \begin{align*}N_{}(\varepsilon)\mathop{=}_{\mathrm{defin.}}\min\{n\in{\mathbb N}^*,f_{\varepsilon}^{-n}(]f_{\varepsilon}(x_{*}),x_{*}])\subset V_{}\}.\end{align*} $$

We note that if $\varepsilon $ is small enough, $N_{}(\varepsilon )$ is independent of $\varepsilon $ , so we shall denote it by $N_{}$ . Moreover, if $\varepsilon $ and $y_{*}$ are small enough,

(4.29) $$ \begin{align} N_{}\mathop{=}_{\mathrm{defin.}}\min\{n\in{\mathbb N}^*,f_{\varepsilon}^{-n}(\tilde{\mathcal{F}}_{\varepsilon,y_{*}})\subset V_{}\}.\end{align} $$

Lemma 4.1. There exists a constant $0<c_{*}<1$ such that, for $(x,y)\in \mathcal {F}_{\varepsilon ,c_{*}y_{*}}$ ,

(4.30) $$ \begin{align} \tilde n_{\varepsilon}(x,y):\mathop{=}_{\mathrm{defin.}}\min\{j\in{\mathbb N}^*, f_{\varepsilon}^j(x,y)\in f_{\varepsilon}^{-N_{}}(\tilde{\mathcal{F}}_{\varepsilon,y_{*}})\}<\infty. \end{align} $$

One has

(4.31) $$ \begin{align} \tilde n_{\varepsilon}(x,y)\asymp \ln(xy)/\lambda.\end{align} $$

Proof. Note that the domain $f_{\varepsilon }^{-N_{}}(\mathcal {F}_{\varepsilon ,y_{*}})\subset \tilde V_{}$ is the interior of the contour defined by:

1. (a) the segment $[f_{\varepsilon }^{-(N_{}-1)}(x_{*},0),f_{\varepsilon }^{-N_{}}(x_{*},0)]\subset W^{u}_{f_{\varepsilon }}(o)\cap V_{}\subset \{0\}\times {\mathbb R}$ ;

2. (b) the curve $f_{\varepsilon }^{-N_{}}(\overline {L_{y_{*}}})$ ;

3. (c) a curve joining $f_{\varepsilon }^{-N_{}}(x_{*},y_{*})$ to $f_{\varepsilon }^{-(N_{}-1)}(x_{*},y_{*})$ ;

4. (d) the curve $f_{\varepsilon }^{-(N_{}-1)}(\overline {L_{y_{*}}})$ ,

and

$$ \begin{align*}f_{\varepsilon}^{-N_{}}(\tilde{\mathcal{F}}_{\varepsilon,y_{*}})=f_{\varepsilon}^{-N_{}}(\mathcal{F}_{\varepsilon,y_{*}})\cup f_{\varepsilon}^{-N_{}}(L_{y_{*}}). \end{align*} $$

Note that the lines $f_{\varepsilon }^{-N_{}}(\overline {L_{y_{*}}})$ , $f_{\varepsilon }^{-(N_{}-1)}(\overline {L_{y_{*}}})$ are transversal to the segment $[f_{\varepsilon }^{-(N_{}-1)} (x_{*},0), f_{\varepsilon }^{-N_{}}(x_{*},0)]$ .

Now let $(x,y)\in \tilde {\mathcal {F}}_{\varepsilon ,y_{*}}$ . We denote by $\mathcal {H}_{x,y}$ the hyperbola

$$ \begin{align*}\mathcal{H}_{x,y}:=\{ (x',y'),\ x'y'=xy\},\end{align*} $$

and if $z,z'\in \mathcal {H}_{x,y}$ , by $\mathcal {H}_{x,y}(z,z')$ , the arc of hyperbola of $\mathcal {H}_{x,y}$ between z and $z'$ which is open in z and closed in $z'$ . If $y>0$ is small enough, $\mathcal {H}_{x,y}$ intersects $f_{\varepsilon }^{-N_{}}(\tilde {\mathcal {F}}_{\varepsilon ,y_{*}})\subset V_{}$ in an arc of hyperbola of the form $\mathcal {H}_{x,y}(p,f_{\varepsilon }^{-1}(p))$ with $p\in f_{\varepsilon }^{-(N_{}-1)}({L_{y_{*}}})$ and $f_{\varepsilon }^{-1}(p)\in f_{\varepsilon }^{-N_{}}({L_{y_{*}}})$ . The sets $f^{-j}_{\varepsilon }(\mathcal {H}_{x,y}(p,f_{\varepsilon }^{-1}(p))$ , $j\geq 0$ form a partition of the semi-arc of parabola $\bigcup _{n\geq 0}\mathcal {H}_{x,y}(p,f_{\varepsilon ,k}^{-n}(p))$ which contains $(x,y)$ . In particular, there exists $j\geq 0$ (in fact $j\geq 1$ ) such that $(x,y)\in f^{-j}_{\varepsilon }(\mathcal {H}_{x,y}(p,f_{\varepsilon }^{-1}(p))$ or equivalently

$$ \begin{align*}f_{\varepsilon}^j(x,y)\in \mathcal{H}_{x,y}(p,f_{\varepsilon}^{-1}(p))\subset f^{-N_{}}_{\varepsilon}(\tilde{\mathcal{F}}_{\varepsilon,y_{*}}).\end{align*} $$

This proves (4.30).

To prove (4.31), we note that there exists an interval $I_{2}$ not containing 0 and depending only on $x_{*},y_{*}$ such that $f_{\varepsilon }^{-N_{}}(\tilde {\mathcal {F}}_{\varepsilon ,y_{*}})\subset {\mathbb R}\times I_{2}$ . We then use Lemma 3.1 and the fact that $|\bar t_{I_{2}}(x,y)-\tilde n_{\varepsilon }(x,y)|\leq 1$ .

We now define

(4.32) $$ \begin{align} n_{\varepsilon}=N_{}+\tilde n_{\varepsilon}.\end{align} $$

By (4.31), one has

(4.33) $$ \begin{align} n_{\varepsilon}(x,y)\asymp \ln(xy)/\lambda.\end{align} $$

The map $\hat f_{\varepsilon }:\tilde {\mathcal {F}}_{\varepsilon ,c_{*}y_{*}}\to \tilde {\mathcal {F}}_{\varepsilon ,y_{*}}$ , defined by

(4.34) $$ \begin{align} \hat f_{\varepsilon}=f_{\varepsilon}^{n_{\varepsilon}},\end{align} $$

is the first return map of $f_{\varepsilon }$ in $\tilde {\mathcal {F}}_{\varepsilon ,y_{*}}$ (for points starting in $ \tilde {\mathcal {F}}_{\varepsilon ,c_{*}y_{*}}$ ). Note that $\hat f_{\varepsilon }$ is not $C^k$ on the whole domain $\tilde {\mathcal {F}}_{\varepsilon ,c_{*}y_{*}}$ .

4.3 Estimates on first return maps

We denote for $a\in {\mathbb R}$

$$ \begin{align*}T_{a}:(u,v)\mapsto (u+a,v)\end{align*} $$

and we recall the definition (3.25) of the symplectic diffeomorphisms $\Xi _{1},\Xi _{2}$ .

We observe that there exist open sets $W_{1}\subset {\mathbb R}^*_{+}\cap {\mathbb R}$ , $W_{2}\subset {\mathbb R}\times {\mathbb R}_{+}^*$ such that, for any $\varepsilon $ and $y_{*}>0$ small enough,

$$ \begin{align*}\tilde{\mathcal{F}}_{\varepsilon,y_{*}}\subset W_{1}\subset V_{},\quad f_{0}^{-N_{}}(\tilde{\mathcal{F}}_{\varepsilon,y_{*}})\subset W_{2}\subset V_{}.\end{align*} $$

Lemma 4.2. There exists a $C^k$ function $\sigma _{0,N_{}}\in C^k({\mathbb R}^*_{+},{\mathbb R})$ such that on $\Xi _{2}(W_{2})$ , one has

(4.35) $$ \begin{align} \Xi_{1}\circ f_{0}^{N_{}}\circ \Xi_{2}^{-1}=T_{\sigma_{0,N_{}}}. \end{align} $$

Proof. From condition (H4), one can write on ${\mathbb R}^2$

$$ \begin{align*}f_{0}=\phi^1_{J\nabla H_{0}}\end{align*} $$

and hence

$$ \begin{align*}f_{0}^{N_{}}=\phi^{N_{}}_{J\nabla H_{0}} \quad\text{where}\ H_{0}\ |_{V_{}}=Q_{0}.\end{align*} $$

If $(u,v)\in \Sigma _{2}(W_{2})$ and $(\tilde u,\tilde v)=\Xi _{1}(f_{0}^{N_{}}(\Xi _{2}^{-1}(u,v)))$ , one then has

$$ \begin{align*}Q_{0}(\Xi_{1}^{-1}(\tilde u,\tilde v))=Q_{0}(f_{0}^{N_{}}(\Xi_{2}^{-1}(u,v)))=Q_{0}(\Xi_{2}^{-1}(u,v))\end{align*} $$

and hence

$$ \begin{align*}q_{0}(\tilde v)=q_{0}(v)\end{align*} $$

and thus $\tilde v=v$ . Because the map $(u,v)\mapsto (\tilde u, \tilde v)$ is symplectic, this forces $\tilde u=u+\tilde \sigma _{0,N_{}}(v)$ for some $C^k$ function $\tilde \sigma _{0,N_{}}$ ; this function can be extended as a $C^k$ function $\sigma _{0,N_{}}: {\mathbb R}\to {\mathbb R}$ .

Recall the definition (4.34) of $\hat f_{\varepsilon }$ .

Lemma 4.3. There exists a continuous family $(\hat \eta _{\varepsilon })_{\varepsilon }$ of $C^k$ symplectic diffeomorphisms defined on ${\mathbb R}^2$ and a neighborhood W of $ f_{\varepsilon }^{-1}(\tilde {\mathcal {F}}_{\varepsilon ,y_{*}})\cup \tilde {\mathcal {F}}_{\varepsilon ,y_{*}}\cup f_{\varepsilon }(\tilde {\mathcal {F}}_{\varepsilon ,y_{*}})$ such that

(4.36) $$ \begin{align}\begin{cases}\lim_{\varepsilon\to 0} \|\hat \eta_{\varepsilon}-\mathrm{id}\|_{k}=0,\\ \hat\eta_{\varepsilon}(W\cap({\mathbb R}\times\{0\})\subset {\mathbb R}\times\{0\} \end{cases} \end{align} $$

and on a neighborhood of $\tilde {\mathcal {F}}_{\varepsilon ,c_{*}y_{*}}$ , one has

(4.37) $$ \begin{align} \Xi_{1}\circ \hat f_{\varepsilon}\circ \Xi_{1}^{-1}=\hat \eta_{\varepsilon}\circ T_{\hat l_{\varepsilon}},\end{align} $$

where

(4.38) $$ \begin{align}\hat l_{\varepsilon}(v)=\sigma_{0,N_{}}(v)+\hat n_{\varepsilon}(u,v)q_{\varepsilon}'(v)-\ln v\quad \text{with }\hat n_{\varepsilon}=\tilde n_{\varepsilon}\circ \Xi_{1}^{-1}.\end{align} $$

Proof. We write (we use (4.34), (4.32), (H3)):

$$ \begin{align*} \hat f_{\varepsilon}&=f_{\varepsilon}^{N_{}+\tilde n_{\varepsilon}}\\ &=f_{\varepsilon}^{N_{}}\circ \phi_{J\nabla Q_{\varepsilon}}^{\tilde n_{\varepsilon}}\\ &=\eta_{\varepsilon}\circ f_{0}^{N_{}}\circ \phi_{J\nabla Q_{\varepsilon}}^{\tilde n_{\varepsilon}},\end{align*} $$

with

(4.39) $$ \begin{align} \eta_{\varepsilon}=f_{\varepsilon}^{N_{}}\circ f_{0}^{-N_{}}.\end{align} $$

As a consequence, if we set

(4.40) $$ \begin{align} \hat \eta_{\varepsilon}=\Xi_{1}\circ \eta_{\varepsilon}\circ \Xi_{1}^{-1}\quad \text{and}\quad \hat n_{\varepsilon}=\tilde n_{\varepsilon}\circ \Xi_{1}^{-1}, \end{align} $$

we have, using (4.35),

$$ \begin{align*}\Xi_{1}\circ \hat f_{\varepsilon}\circ \Xi_{1}^{-1}&=\hat\eta_{\varepsilon}\circ (\Xi_{1}\circ f_{0}^{N_{}}\circ\Xi_{2}^{-1}) \circ (\Xi_{2}\circ \phi^{\tilde n_{\varepsilon}}_{J\nabla Q_{\varepsilon}} \circ \Xi_{2}^{-1}) \circ (\Xi_{2}\circ\Xi_{1}^{-1}) \\ &=\hat \eta_{\varepsilon}\circ T_{\sigma_{0,N_{}}}\circ \phi^{\tilde n_{\varepsilon}\circ \Xi_{1}^{-1}}_{J\nabla (Q_{\varepsilon}\circ \Xi_{2}^{-1})} \circ T_{-\ln v}\\ &=\hat \eta_{\varepsilon}\circ T_{\sigma_{0,N_{}}}\circ T^{\hat n_{\varepsilon}}_{q^{\prime}_{\varepsilon}}\circ T_{-\ln v}\\ &=\hat \eta_{\varepsilon}\circ T_{\sigma_{0,N_{}}+\hat n_{\varepsilon}q^{\prime}_{\varepsilon}-\ln v}, \end{align*} $$

which is (4.37) together with (4.38).

Note that by (4.39), (4.40), Remark 2.2, and the fact that ${\mathbb R}\ni \varepsilon \mapsto f_{\varepsilon }\in C^k(V_{k},{\mathbb R}^2)$ is continuous, one has

$$ \begin{align*}\begin{cases}\lim_{\varepsilon\to 0}\|\hat \eta_{\varepsilon}-\text{id}\|_{C^k}=0,\\ \hat\eta_{\varepsilon}(W\cap ({\mathbb R}\times\{0\}))\subset {\mathbb R}\times\{0\}. \end{cases}\\[-4pc] \end{align*} $$

5.1 Gluing

Let $\mathcal {F}$ be an open set of ${\mathbb R}^2$ , L a one-dimensional submanifold of ${\mathbb R}^2$ and f an orientation preserving smooth diffeomorphism from a neighborhood of $\mathcal {F}\cup L$ to a neighborhood of $f(\mathcal {F}\cup L)$ . We assume that:

1. (1) $f(\mathcal {F}\cup L)\cap (\mathcal {F}\cup L)=\emptyset $ ;

2. (2) $\mathcal {F}\cup L$ is a two-dimensional submanifold of ${\mathbb R}^2$ with boundary and this boundary is $\partial (\mathcal {F}\cup L)=L$ ; in particular, for any point $p\in L$ , there exists an open set $U_{p}$ , $p\in U_{p}\subset {\mathbb R}^2$ , and a smooth diffeomorphism $\varphi _{p}:U_{p}\to \varphi _{p}(U_{p})\subset {\mathbb R}^2$ such that $\varphi _{p}(U_{p}\cap L)=\varphi _{p}(U_{p})\cap ({\mathbb R}\times \{0\})$ and $\varphi _{p}(U_{p}\cap \mathcal {F})=\varphi _{p}(U_{p})\cap ({\mathbb R}\times {\mathbb R}^*_{+})$ ;

3. (3) for any $p\in \mathcal {F}\cup L$ and $U_{p}$ , one has $U_{p}\cap f(\mathcal {F}\cup L)=\emptyset $ ;

4. (4) for any $p\in L$ one has $f^{-1}(f(U_{p})\cap \mathcal {F})=\varphi _{p}^{-1}(\varphi _{p}(U_{p})\cap ({\mathbb R}\times {\mathbb R}^*_{-}))$ , for any of the previous chart $(U_{p},\varphi _{p})$ at p.

We define the topological space $(\mathcal {F}\cup L,\mathcal {T})$ as being the set $\mathcal {F}\cup L$ endowed with the following topology $\mathcal {T}$ : a subset S of $\mathcal {F}\cup L$ is an element of $\mathcal {T}$ (that is an open set) if for every $p\in S$ , there exists an open set $V\subset {\mathbb R}^2$ (contained in a neighborhood of $\mathcal {F}\cup L$ where f is defined) such that $V\cap f(\mathcal {F}\cup L)=\emptyset $ and $p\in (V\cup f(V))\cap (\mathcal {F}\cup L)\subset S$ .

We can then define the following differentiable structure on $(\mathcal {F}\cup L,\mathcal {T})$ as follows: (a) if $p\in \mathcal {F}$ , we define the local chart $C_{p}:=(W_{p},\text {id})$ , where $W_{p}$ is an open set of ${\mathbb R}^2$ such that $p\in W_{p}\subset \mathcal {F}$ ; and (b) if $p\in L$ , we define the local chart $C_{p}:=(W_{p},\psi _{p})$ where $W_{p}$ is the open set of $\mathcal {F}\cup L$ (see condition (3)), $W_{p}=(\mathcal {F}\cup L)\cap (U_{p}\cup f(U_{p}))$ (here $(U_{p},\varphi _{p})$ is the local chart for $p\in L$ as defined in (2)), and where $\psi _{p}$ is defined by (we use condition (4))

$$ \begin{align*}\begin{cases}\psi_{p}=\varphi_{p}\quad \text{on } U_{p}\cap (\mathcal{F}\cup L)=\varphi_{p}^{-1}((\varphi_{p}(U_{p})\cap ({\mathbb R}\times{\mathbb R}_{+})),\\ \psi_{p}=\varphi_{p}\circ f^{-1}\quad \text{on } f(U_{p})\cap \mathcal{F}=f\circ\varphi_{p}^{-1}((\varphi_{p}(U_{p})\cap ({\mathbb R}\times{\mathbb R}_{-}^*)). \end{cases} \end{align*} $$

We denote by $\mathcal {A}$ the collection of all these local charts $C_{p}$ and we set $(\mathcal {F}\cup L)/f= (\mathcal {F}\cup L,\mathcal {T},\mathcal {A})$ .

5.2 The space $(\mathcal {F}_{\varepsilon ,y_{*}}\cup L_{y_{*}})/f_{\varepsilon }$

If $\varepsilon $ and $y_{*}$ are small enough, item (1) is satisfied and we can find charts $(p,U_{p})$ such that items (2)–(4) are satisfied. See Figure 3. We can then define the manifold $(\mathcal {F}_{\varepsilon ,y_{*}}\cup L_{y_{*}})/f_{\varepsilon }$ . We shall see that it is an annulus without boundary, cf. Lemma 5.3.

Figure 3 Gluing: $(\mathcal {F}_{\varepsilon ,y_{*}}\cup L_{y_{*}})/f_{\varepsilon }$ .

Note that if $0<c_{*}<1$ , the smaller set $\tilde {\mathcal {F}}_{\varepsilon ,c_{*}y_{*}}=\mathcal {F}_{\varepsilon ,c_{*}y_{*}}\cup L_{c_{*}y_{*}}$ is an open subset of $(\mathcal {F}_{\varepsilon ,y_{*}}\cup L_{y_{*}})/f_{\varepsilon }$ (which means that it belongs to $\mathcal {T}$ ) and it can be endowed with the topology and differentiable structure induced by the inclusion. We denote $(\mathcal {F}_{\varepsilon ,c_{*}y_{*}}\cup L_{c_{*}y_{*}})/f_{\varepsilon }$ the thus obtained submanifold of $(\mathcal {F}_{\varepsilon ,y_{*}}\cup L_{y_{*}})/f_{\varepsilon }$ . The following lemma is then tautological.

We shall need in §6 the following lemma.

Proof. We shall in fact construct this measure $\pi _{\varepsilon ,y_{*}}$ on the bigger set $\hat {\mathcal {F}}_{\varepsilon ,y_{*}}/f_{\varepsilon }$ ,

$$ \begin{align*}\hat{\mathcal{F}}_{\varepsilon,y_{*}}=\tilde{\mathcal{F}}_{\varepsilon,y_{*}}\cup \sigma(\tilde{\mathcal{F}}_{\varepsilon,y_{*}}),\end{align*} $$

where $\sigma :{\mathbb R}^2\to {\mathbb R}^2$ is the reflection $(x,y)\mapsto (x,-y)$ (it commutes with $f_{\varepsilon }$ in $V_{}$ , see condition (H3)). From Remark 5.1, there exists a symplectic form $\omega _{\varepsilon }$ on $\hat {\mathcal {F}}_{\varepsilon ,y_{*}}/f_{\varepsilon }$ . Note that the first return map $\hat f_{\varepsilon }$ is not defined on the whole set $\hat {\mathcal {F}}_{\varepsilon ,y_{*}}/f_{\varepsilon }$ but nevertheless

$$ \begin{align*}(\hat f_{\varepsilon})^*\omega_{\varepsilon}=\omega_{\varepsilon}\end{align*} $$

whenever this formula makes sense. The probability measure $\pi _{\varepsilon ,y_{*}}$ defined by

$$ \begin{align*}\pi_{\varepsilon,y_{*}}(A)=\int_{A}|\omega_{\varepsilon}|\Bigg/\int_{\mathcal{F}_{\varepsilon,y_{*}}}|\omega_{\varepsilon}|\end{align*} $$

is $\hat f_{\varepsilon }$ invariant.

5.3 Normalization of $f_{\varepsilon }$

We now uniformize the abstract annulus $\tilde {\mathcal {F}}_{\varepsilon ,y_{*}}/f_{\varepsilon }$ . To do that, it is enough to normalize $f_{\varepsilon }$ in the sense of item 2 of the following lemma.

Proof. Using condition (H3) and the change of coordinates (3.25) of §3, we see that on a neighborhood of $\tilde {\mathcal {F}}_{\varepsilon ,y_{*}}$ , one has (we use the notation $(x,y)$ for $(u,v)$ )

$$ \begin{align*}\Xi_{1}\circ f_{\varepsilon}\circ \Xi_{1}^{-1}=T_{q^{\prime}_{\varepsilon}} :(x,y)\mapsto (x+q^{\prime}_{\varepsilon}(y),y).\end{align*} $$

If $g_{\varepsilon }$ is the (not necessarily symplectic) smooth diffeomorphism

(5.41) $$ \begin{align} g_{\varepsilon}:(x,y)\mapsto \bigg(\frac{x}{q^{\prime}_{\varepsilon}(y)},y\bigg),\end{align} $$

one has

(5.42) $$ \begin{align} g_{\varepsilon}\circ \Xi_{1}\circ f_{\varepsilon}\circ \Xi_{1}^{-1}\circ g_{\varepsilon}^{-1}=T_{1}.\end{align} $$

The set $\overline {(g_{\varepsilon }\circ \Xi _{1})(\mathcal {F}_{\varepsilon ,y_{*}})}$ is of the form

$$ \begin{align*}\overline{(g_{\varepsilon}\circ\Xi_{1})(\mathcal{F}_{\varepsilon,y_{*}})}=\{(x,y),\ y\in [0,c_{}],\ {\gamma}_{\varepsilon}(y)\leq x\leq {\gamma}_{\varepsilon}(y)+1\},\end{align*} $$

where $c_{}>0$ , ${\gamma }_{\varepsilon }:[0,c_{}]\to {\mathbb R}_{+}$ is $C^k$ , ${\gamma }_{\varepsilon }(0)=0$ , and the map ${\mathbb R}\ni \varepsilon \mapsto {\gamma }_{\varepsilon }\in C^k([0,c_{}],{\mathbb R})$ is continuous. This indeed follows from the definition of $\tilde {\mathcal {F}}_{\varepsilon ,y_{*}}$ in §4.1, the definition of $\Xi _{1}$ (3.25), and (5.41), (5.42). As a consequence, if we denote

(5.43) $$ \begin{align} j_{\varepsilon}:(x,y)\mapsto (x-\gamma_{\varepsilon}(y),y),\end{align} $$

we have

(5.44) $$ \begin{align}\begin{cases}j_{\varepsilon}\circ T_{1}=T_{1}\circ j_{\varepsilon},\\ j_{\varepsilon}( (g_{\varepsilon}\circ\Sigma_{1})(\mathcal{F}_{\varepsilon,y_{*}}))=\,]0,1[{\kern2pt}{\times}{\kern2pt}]0,c_{}[,\\ j_{\varepsilon}((g_{\varepsilon}\circ\Sigma_{1})(L_{y_{*}}))=\{0\}{\kern2pt}{\times}{\kern2pt}]0,c_{}[. \end{cases} \end{align} $$

By Remarks 5.2 and 5.3, the map

(5.45) $$ \begin{align} h_{\varepsilon}=j_{\varepsilon}\circ g_{\varepsilon}\circ \Xi_{1}\end{align} $$

is a diffeomorphism that sends $\tilde {\mathcal {F}}_{\varepsilon ,y_{*}}/f_{\varepsilon }$ to the standard annulus $([0,1[{\kern2pt}{\times}{\kern2pt}]0,c_{}[)/T_{1}\simeq ({\mathbb R}/{\mathbb Z}){\kern2pt}{\times}{\kern2pt}]0,c_{}[$ and such that

$$ \begin{align*}h_{\varepsilon}\circ f_{\varepsilon}\circ h_{\varepsilon}^{-1}=T_{1}.\end{align*} $$

To conclude the proof, we notice (2) is an immediate consequence of the definition (5.45) of $h_{\varepsilon }$ .

Remark 5.4. Note that if $T_{a}(x,y)=(x+a(y),y)$ , one has

$$ \begin{align*}(h_{\varepsilon}\circ\Xi_{1}^{-1})\circ T_{a}\circ (h_{\varepsilon}\circ \Xi_{1}^{-1})^{-1}=T_{\tilde a},\quad \tilde a(y)=a(y)/q_{\varepsilon}'(y). \end{align*} $$

5.4 The renormalization $\bar f_{\varepsilon }$ of $f_{\varepsilon }$

There exists $\delta _{}\in{\kern2pt}]{\kern2pt}0,c_{}[$ such that the map

(5.46) $$ \begin{align} \bar f_{\varepsilon}\mathop{=}_{\mathrm{defin.}}h_{\varepsilon}\circ \hat{f}_{\varepsilon}\circ h_{\varepsilon}^{-1}:{\mathbb R}/{\mathbb Z}{\kern2pt}{\times}{\kern2pt}]0,\delta_{}[{\kern2pt}\to {\mathbb R}/{\mathbb Z}{\kern2pt}{\times}{\kern2pt}]0,c_{}[\end{align} $$

is well defined and is a $C^k$ diffeomorphism onto its image.

Proposition 5.4. One has

(5.47) $$ \begin{align} \bar f_{\varepsilon}=\bar\eta_{\varepsilon}\circ T_{l_{\varepsilon}},\end{align} $$

where $\bar \eta _{\varepsilon }$ is a $C^k$ diffeomorphism defined on ${\mathbb R}/{\mathbb Z}{\kern2pt}{\times}{\kern2pt}]0,\delta _{}[$ and $l_{\varepsilon }\in C^{k}(]0,c_{}[, {\mathbb R}/{\mathbb Z})$ ; they satisfy

(5.48) $$ \begin{align} l_{\varepsilon}(y)=\frac{\sigma_{0,N_{}}(y)}{q^{\prime}_{\varepsilon}(y)}-\frac{\ln y}{q^{\prime}_{\varepsilon}(y)}\!\!\mod {\mathbb Z},\end{align} $$

(5.49) $$ \begin{align}\lim_{\varepsilon\to0}\|\bar\eta_{\varepsilon}-\mathrm{id}\|_{C^{k}}=0,\end{align} $$

(5.50) $$ \begin{align} \bar\eta_{\varepsilon}:(x,y)\mapsto (x+a_{\varepsilon}(x,y),y+yb_{\varepsilon}(x,y)), \end{align} $$

where $a_{\varepsilon }\in C^k$ , $b_{\varepsilon }\in C^{k-1}$ are functions defined on ${\mathbb R}/{\mathbb Z}\times (0,\delta _{})$ .

Moreover, the map $\bar f_{\varepsilon }$ preserves a probability measure $\bar \pi _{\varepsilon ,y_{*}}$ with positive density defined on ${\mathbb R}/{\mathbb Z}{\kern2pt}{\times}{\kern2pt}]0,c_{}[$ .

Proof. By (4.37) and Remark 5.4 after Lemma 5.3,

$$ \begin{align*}\bar f_{\varepsilon}&=(h_{\varepsilon}\circ\Xi_{1}^{-1})\circ \hat \eta_{\varepsilon}\circ (h_{\varepsilon}\circ\Xi_{1}^{-1})^{-1}\circ (h_{\varepsilon}\circ \Xi_{1}^{-1})\circ T_{\hat l_{\varepsilon}}\circ (h_{\varepsilon}\circ \Xi_{1}^{-1})^{-1}\\ &=\bar\eta_{\varepsilon}\circ T_{l_{\varepsilon}}, \end{align*} $$

where

(5.51) $$ \begin{align}\bar\eta_{\varepsilon}=(h_{\varepsilon}\circ \Xi_{1}^{-1})\circ \hat \eta_{\varepsilon}\circ (h_{\varepsilon}\circ \Xi_{1}^{-1})^{-1}\quad\text{and}\quad l_{\varepsilon}(y)=(1/q^{\prime}_{\varepsilon}(y))\hat l_{\varepsilon}(y).\end{align} $$

Because $\bar \eta _{\varepsilon }:=(h_{\varepsilon }\circ \Xi _{1}^{-1})\circ \hat \eta _{\varepsilon }\circ (h_{\varepsilon }\circ \Xi _{1}^{-1})^{-1}$ and $\bar f_{\varepsilon }$ are $C^{k}$ , the function ${l_{\varepsilon }}:{\kern2pt}]0,c_{}[{\kern2pt}\to {\mathbb R}/{\mathbb Z}$ is also $C^{k}$ and

$$ \begin{align*}l_{\varepsilon}(y)=(1/q^{\prime}_{\varepsilon}(y))\hat l_{\varepsilon}(y).\end{align*} $$

By (4.38) (remember that $\hat n_{\varepsilon }$ takes its value in ${\mathbb Z}$ ),

$$ \begin{align*}l_{\varepsilon}(y)&= \frac{\sigma_{0,N_{}}(y)}{q^{\prime}_{\varepsilon}(y)}+\hat n_{\varepsilon}(x,y)-\frac{\ln y}{q^{\prime}_{\varepsilon}(y)}\\ &=\frac{\sigma_{0,N_{}}(y)}{q^{\prime}_{\varepsilon}(y)}-\frac{\ln y}{q^{\prime}_{\varepsilon}(y)}\!\!\mod {\mathbb Z},\end{align*} $$

which is (5.48).

Equation (5.49) is a consequence of the definition of $\bar \eta _{\varepsilon }$ , cf. (5.51), the first equation of (4.36), and of the fact that ${\mathbb R}\ni \varepsilon \mapsto h_{\varepsilon }\in C^k$ is continuous (Lemma 5.3).

We now claim that if $\bar \eta _{\varepsilon }(x,y)=(x+a_{\varepsilon }(x,y), y+\bar b_{\varepsilon }(x,y))$ , one has for any y,

(5.52) $$ \begin{align} \bar b_{\varepsilon}(x,0)=0.\end{align} $$

Indeed, because

$$ \begin{align*}\bar \eta_{\varepsilon}:=(h_{\varepsilon}\circ \Xi_{1}^{-1})\circ \hat \eta_{\varepsilon}\circ (h_{\varepsilon}\circ \Xi_{1}^{-1})^{-1},\end{align*} $$

equality (5.52) is a consequence of the second equation of (4.36), of item (3) of Lemma 5.3, and of the fact that $\Xi _{1}({\mathbb R}_{+}^*\times \{0\})={\mathbb R}_{+}^*\times \{0\}$ .

To prove (5.50), we thus notice that equality (5.52) gives us for $\bar b_{\varepsilon }$ a decomposition

$$ \begin{align*}\begin{cases}\bar b_{\varepsilon}(x,y)=yb_{\varepsilon}(x,y),\\ b_{\varepsilon}\in C^{k-1} .\end{cases}\end{align*} $$

Finally, to conclude the proof of the proposition, we observe that because the map $\hat f_{\varepsilon }:\tilde {\mathcal {F}}_{\varepsilon ,c_{*}y_{*}}/f_{\varepsilon }\to \tilde {\mathcal {F}}_{\varepsilon ,y_{*}}/f_{\varepsilon }$ preserves the probability measure $\pi _{\varepsilon ,y_{*}}$ , cf. Lemma 5.2, the diffeomorphism $\bar f_{\varepsilon }:{\mathbb R}/{\mathbb Z}{\kern2pt}{\times}{\kern2pt}]0,\delta _{}[{\kern2pt}\to {\mathbb R}/{\mathbb Z}{\kern2pt}{\times}{\kern2pt}]0,c_{}[$ preserves the probability measure $\bar \pi _{\varepsilon ,y_{*}}=(h_{\varepsilon })_{*}\pi _{\varepsilon ,y_{*}}$ defined on ${\mathbb R}/{\mathbb Z}{\kern2pt}{\times}{\kern2pt}]0,c_{}[$ (in the sense that if $A\subset {\mathbb R}/{\mathbb Z}{\kern2pt}{\times}{\kern2pt}]0,c_{}[{\kern2pt}$ is a Borelian set such that $\bar f_{\varepsilon }^{-1}(A)\subset {\mathbb R}/{\mathbb Z}{\kern2pt}{\times}{\kern2pt}]0,c_{}[$ , one has $\bar \pi _{\varepsilon ,y_{*}}(A)=\bar \pi _{\varepsilon ,y_{*}}(\bar f_{\varepsilon }^{-1}(A))$ ).

6.1 The translated curve theorem

Let $\psi : {\mathbb R}/{\mathbb Z}{\kern2pt}{\times}{\kern2pt}]e^{-1},1[{\kern2pt}\to {\mathbb R}/{\mathbb Z}\times {\mathbb R}$ ( $\ln e=1$ ) be a $C^k$ diffeomorphism defined on the annulus (or cylinder) ${\mathbb R}/{\mathbb Z}{\kern2pt}{\times}{\kern2pt}]e^{-1},1[$ . We say that the graph $\mathrm {Gr}_{{\gamma }}:=\{(x,\gamma (x)): x\in {\mathbb R}/{\mathbb Z}\}$ of a continuous map $\gamma :{\mathbb R}/{\mathbb Z}\to {\mathbb R}/{\mathbb Z}{\kern2pt}{\times}{\kern2pt}]e^{-1},1[{\kern2pt}$ is translated by $\psi $ if for some $t\in {\mathbb R}$ ,

(6.53) $$ \begin{align}\psi(\mathrm{Gr}_{\gamma})=\mathrm{Gr}_{t+\gamma} \end{align} $$

and invariant if $t=0$ . If $\mathrm {Gr}_{{\gamma }}$ satisfies (6.53), there exists an orientation preserving homeomorphism of the circle $g:{\mathbb R}/{\mathbb Z}\to {\mathbb R}/{\mathbb Z}$ such that $\psi (x,\gamma (x))=\psi (g(x),t+\gamma (g(x))$ . If $t=0$ (respectively $t\ne 0$ ), we define (respectively with a clear abuse of language) the rotation number of ( $\psi $ on) the invariant (respectively translated) graph $\mathrm {Gr}_{{\gamma }}$ as the rotation number of the circle diffeomorphism g. We say that $\psi $ has the intersection property if for any continuous ${\gamma }:{\mathbb R}/{\mathbb Z}\to {\mathbb R}/{\mathbb Z}{\kern2pt}{\times}{\kern2pt}]e^{-1},1[$ , the curve $\mathrm {Gr}_{{\gamma }}:=\{(x,{\gamma }(x)):x\in {\mathbb R}/{\mathbb Z}\}$ intersects its image $\psi (\mathrm {Gr}_{{\gamma }})$ . Note the following important fact: If $\psi $ has the intersection property, any translated graph by $\psi $ is invariant.

We state the translated curve theorem by Rüssmann [Reference Rüssmann17] (which implies the invariant curve theorem by Moser [Reference Moser15]):

Theorem 6.1. (Rüssmann, [Reference Rüssmann17])

There exists $k_{0}\in {\mathbb N}$ for which the following holds. Let $k\geq k_{0}$ , $C,\mu>0$ , and $l:{\mathbb R}/{\mathbb Z}\to {\mathbb R}$ a $C^k$ map satisfying the twist condition,

(6.54) $$ \begin{align} \min_{y} |\partial_{y}l(y)|>\mu>0 \quad \text{and}\quad \|l\|_{C^{k_{0}}}\leq C,\end{align} $$

and define

$$ \begin{align*}\psi_{0}:(x,y)\mapsto (x+l(y),y).\end{align*} $$

There exists $\varepsilon _{0}=\varepsilon _{0}(C,\mu )>0$ such that for any $C^k$ diffeomorphism

$$ \begin{align*}\psi:{\mathbb R}/{\mathbb Z}{\kern2pt}{\times}{\kern2pt}]e^{-1},1[{\kern2pt}\to{\mathbb R}/{\mathbb Z}\times{\mathbb R}\end{align*} $$

satisfying

(6.55) $$ \begin{align} \|\psi-\psi_{0}\|_{C^{k_{0}}}<\varepsilon_{0},\end{align} $$

the diffeomorphism $\psi $ admits a set of positive Lebesgue measure of $C^{k-k_{0}}$ translated graphs contained in $({\mathbb R}/{\mathbb Z}){\kern2pt}{\times}{\kern2pt}]e^{-3/4},e^{-1/4}[$ . Moreover, all these translated graphs have Diophantine rotation numbers (they are in a fixed Diophantine class $DC(\kappa ,\tau )$ (the exponent is $\tau $ and the constant $\kappa $ ) that can be prescribed in advance once $\mu $ is fixed ( $k_{0}$ then depends on $\tau $ and $\varepsilon _{0}$ on $\kappa $ and $\tau $ )).

6.2 The rescaled diffeomorphism $\mathring {f}_{\varepsilon ,n}$

Let $\bar f_{\varepsilon }$ be the renormalized map defined in §5.4 and define $u_{\varepsilon },v_{\varepsilon }$ by

$$ \begin{align*}\bar f_{\varepsilon}(x,y)=(x+u_{\varepsilon}(x,y),y+v_{\varepsilon}(x,y)).\end{align*} $$

Because $\bar f_{\varepsilon }=\bar \eta _{\varepsilon }\circ T_{l_{\varepsilon }}$ (cf. (5.47)), one has using (5.50):

$$ \begin{align*}u_{\varepsilon}(x,y)&=l_{\varepsilon}(y)+a_{\varepsilon}(x+l_{\varepsilon}(y),y),\\ v_{\varepsilon}(x,y)&=yb_{\varepsilon}(x+l_{\varepsilon}(y),y). \end{align*} $$

Figure 4 The diffeomorphism $\bar f_{\varepsilon }$ on ${\mathbb R}/{\mathbb Z}{\kern2pt}{\times}{\kern2pt}[e^{-(n+1)},e^{-n}]$ .

Now, let $n\in {\mathbb N}^*$ large enough so that

(6.56) $$ \begin{align} ]e^{-(n+1)},e^{-n}[{\kern2pt}{\subset}{\kern2pt}]0,\delta_{}[\end{align} $$

(the $\delta _{}$ of (5.46)) and introduce the rescaled $C^{k}$ diffeomorphism $\mathring {f}_{\varepsilon ,n}$ defined on the annulus ${\mathbb R}/{\mathbb Z}{\kern2pt}{\times}{\kern2pt}]e^{-1},1[$ by

(6.57) $$ \begin{align} \mathring{f}_{\varepsilon,n}\mathop{=}_{\mathrm{defin.}}\Lambda_{e^n}\circ \bar f_{\varepsilon}\circ \Lambda_{e^n}^{-1}, \end{align} $$

where $\Lambda _{e^n}:(x,y)\mapsto (x,e^n y)$ . Let us denote

$$ \begin{align*}\mathring{f}_{\varepsilon,n}(x,y)=(x+u_{\varepsilon,n}(x,y), y+v_{\varepsilon,n}(x,y)).\end{align*} $$

A computation shows that:

(6.58) $$ \begin{align} \begin{cases}u_{\varepsilon,n}(x,y)&=l_{\varepsilon,n}(y)+a_{\varepsilon}(x+l_{\varepsilon,n}(y),e^{-n}y),\\ v_{\varepsilon,n}(x,y)&=yb_{\varepsilon}(x+l_{\varepsilon,n}(y),e^{-n}y), \end{cases}\end{align} $$

where

(6.59) $$ \begin{align} l_{\varepsilon,n}(y)=l_{\varepsilon}(e^{-n}y). \end{align} $$

We can now state the following important proposition the proof of which occupies the next subsection.

6.3 Proof of Proposition 6.1

6.3.1 Twist condition for $l_{\varepsilon ,n}$

Lemma 6.2. There exist $C,\mu>0$ such that, for any $\varepsilon $ small enough and any $n\gg 1$ , the map $l_{\varepsilon ,n}$ satisfies the twist condition (6.54) provided $k\geq k_{0}+1$ .

Proof. Using (5.48), (6.59), we have

$$ \begin{align*}l_{\varepsilon,n}(y)&=l_{\varepsilon}(e^{-n}y)\\ &=\frac{\sigma_{0,N_{}}(e^{-n}y)}{q^{\prime}_{\varepsilon}(e^{-n}y)}+\frac{n}{q^{\prime}_{\varepsilon}(e^{-n}y)}-\frac{\ln y}{q^{\prime}_{\varepsilon}(e^{-n}y)}\!\!\mod {\mathbb Z}\\ &=\frac{\sigma_{0,N_{}}(0)+n}{\lambda}-\frac{\ln y}{\lambda}+\theta_{\varepsilon,n}(y)\!\!\mod {\mathbb Z}, \end{align*} $$

where

$$ \begin{align*} \|\theta_{\varepsilon,n}\|_{C^{k-1}([e^{-1},1])}=O(e^{-n});\end{align*} $$

this last inequality is a consequence of the fact that $q_{\varepsilon }(s)=\lambda s+O(s^2)$ is continuous with respect to $\varepsilon $ (cf. condition H2) and of the fact that $\sigma _{0,N_{}}$ is $C^k$ (cf. Lemma 4.2). In particular, for some $C_{k}>0$ (depending on $\lambda $ ),

$$ \begin{align*}\| l_{\varepsilon,n} \|_{C^{k-1}}\leq C_{k}\end{align*} $$

and because $\partial _{y} l_{\varepsilon ,n}(y)=-1/ (\lambda y)+\partial _{y}\theta _{\varepsilon ,n}(y)$ and $y\in{\kern2pt}]e^{-1},1[$ ,

$$ \begin{align*}|\partial_{y} l_{\varepsilon,n}(y)|\geq 1/(2\lambda) .\end{align*} $$

Hence (6.54) holds uniformly in $\varepsilon ,n$ with $C=C_{k_{0}+1}$ and $\mu =1/(2\lambda )$ as soon as n is large enough.

6.3.2 $\mathring {f}_{\varepsilon ,n}$ is close to a twist

We observe that from (5.49), (5.50), (6.58), and Lemma 6.2, one has uniformly in n,

(6.60) $$ \begin{align} \lim_{\varepsilon\to0}\max(\|u_{\varepsilon,n}-l_{\varepsilon,n}\|_{C^{k-2}},\|v_{\varepsilon,n}\|_{C^{k-2}})=0.\end{align} $$

In particular, if n is large enough, inequality (6.55) is satisfied if $k\geq k_{0}+2$ with $\psi =\mathring {f}_{\varepsilon ,n}$ and $\psi _{0}:(x,y)\mapsto (x+l_{\varepsilon ,n}(y),y)$ .

We see from §§6.3.1 and 6.3.2 that, if

$$ \begin{align*}|\varepsilon|\leq \varepsilon_{1}\mathop{=}_{\mathrm{defin.}} \varepsilon_{0}(C_{k_{0}+1},1/(2\lambda))\end{align*} $$

and $n\gg 1$ , the assumptions of Theorem 6.1 are then satisfied by $\mathring {f}_{\varepsilon ,n}$ with $k-2$ in place of k. Under these conditions, there thus exists a set $\mathring {\mathcal {G}}_{\varepsilon ,n}$ of $C^{k-k_{0}-2}\ \mathring {f}_{\varepsilon ,n}$ -translated graphs, the union of which covers a set of positive Lebesgue measure in $({\mathbb R}/{\mathbb T}){\kern2pt}{\times}{\kern2pt}]e^{-3/4},e^{-1/4}[$ . We just have to check that these translated graphs are indeed invariant.

6.3.3 $\mathring {f}_{\varepsilon ,n}$ -translated graphs are invariant

Let $\mathring {{\gamma }}\subset ({\mathbb R}/{\mathbb T}){\kern2pt}{\times}{\kern2pt}]e^{-3/4},e^{-1/4}[$ be a $\mathring {f}_{\varepsilon ,n}$ -translated graph: $\mathring {f}_{\varepsilon ,n}(\mathring {{\gamma }})=\mathring {{\gamma }}+(0,t)$ for some $t\in {\mathbb R}$ . We shall prove that $t=0$ . We can without loss of generality assume that $t\geq 0$ (the case $t\leq 0$ is treated in a similar way).

Formula (6.60) shows that if $n\gg 1$ , one has $\mathring {f}_{\varepsilon ,n}(\mathring {{\gamma }})\subset ({\mathbb R}/{\mathbb T}){\kern2pt}{\times}{\kern2pt}]e^{-1},1[$ . From the conjugation relation (6.57), we see that (cf. (6.56))

$$ \begin{align*}\bar {\gamma}:=\Lambda_{e^n}^{-1}(\mathring{{\gamma}})\subset ({\mathbb R}/{\mathbb Z}){\kern2pt}{\times}{\kern2pt}]e^{-n-3/4}, e^{-n-1/4}[\subset ({\mathbb R}/{\mathbb Z}){\kern2pt}{\times}{\kern2pt}]0,\delta[\end{align*} $$

is a $\bar f_{\varepsilon }$ -translated graph such that

$$ \begin{align*}\bar f_{\varepsilon}(\bar {\gamma})=\bar{\gamma}+(0,e^{-n}t)\subset ({\mathbb R}/{\mathbb Z}){\kern2pt}{\times}{\kern2pt}]e^{-(n+1)},e^{-n}[\subset ({\mathbb R}/{\mathbb Z}){\kern2pt}{\times}{\kern2pt}]0,\delta[.\end{align*} $$

Let A be the open domain of $({\mathbb R}/{\mathbb Z}){\kern2pt}{\times}{\kern2pt}]0,c[$ between $({\mathbb R}/{\mathbb Z})\times \{0\}$ and $\bar {\gamma }$ . Because $t\geq 0$ , one has $A\subset \bar f_{\varepsilon }(A)\subset ({\mathbb R}/{\mathbb Z}){\kern2pt}{\times}{\kern2pt}]0,c[$ .

Assume by contradiction that $t>0$ ; then the set $\bar f_{\varepsilon }(A)\smallsetminus A$ contains a non-empty open set. We have seen (cf. Proposition 5.4) that $\bar f_{\varepsilon }$ preserves a probability measure $\bar \pi _{\varepsilon ,y_{*}}$ with positive density defined on $({\mathbb R}/{\mathbb Z}){\kern2pt}{\times}{\kern2pt}]0,c[$ , so $\bar \pi _{\varepsilon ,y_{*}}(\bar f_{\varepsilon }(A)\smallsetminus A)>0$ . However, this contradicts the invariance of $\bar \pi _{\varepsilon ,y_{*}}$ by $\bar f_{\varepsilon }$ .

The proof of Proposition 6.1 is complete.□

6.4 Invariant curves for $\bar f_{\varepsilon }$

We can now state the following.

Theorem 6.2. Let $k\geq k_{0}+2$ and $|\varepsilon |\leq \varepsilon _{1}$ . There exists $\nu _{}\in{\kern2pt}]0,\delta _{}[$ such that, for any $\nu \in{\kern2pt}]0,\nu _{}[$ , there exists a set $\bar {\mathcal {G}}_{\varepsilon ,\nu }$ of $C^{k-k_{0}-2}$ , $\bar f_{\varepsilon }$ -invariant graphs contained in $({\mathbb R}/{\mathbb Z}){\kern2pt}{\times}{\kern2pt}]e^{-1}\nu ,\nu [$ such that

$$ \begin{align*}\mathrm{Leb}_{{\mathbb R}^2}\bigg(\bigcup_{\bar{\gamma}\in\bar{\mathcal{G}}_{\varepsilon,\nu}} \bar{\gamma}\bigg)>0.\end{align*} $$

Proof. We choose n so that

(6.61) $$ \begin{align} ]e^{-(n+1)},e^{-n}[{\kern2pt}{\subset}{\kern2pt}]0,\nu[\end{align} $$

and we observe that when $\nu \to 0$ , one has $n\to \infty $ . Define

$$ \begin{align*} \mathring{f}_{\varepsilon,n}=\Lambda_{e^n}\circ \bar f_{\varepsilon}\circ \Lambda_{e^n}^{-1}.\end{align*} $$

By Proposition 6.1, there exists $\nu _{1}>0$ such that if $\nu \in{\kern2pt}]0,\nu _{1}[$ (n satisfying (6.61) is then large enough), the diffeomorphism $\mathring {f}_{\varepsilon ,n}$ admits $C^{k-k_{0}-2}$ -invariant curves in ${\mathbb T}{\kern2pt}{\times}{\kern2pt}]e^{-1},1[$ covering a set of positive Lebesgue measure; hence, $\bar f_{\varepsilon ,k}$ has $C^{k-k_{0}-2}$ -invariant curves in ${\mathbb T}{\kern2pt}{\times}{\kern2pt}]e^{-1}\nu ,\nu [$ covering a set of positive Lebesgue measure.

We shall denote

$$ \begin{align*}\bar{\mathcal{G}}_{\varepsilon}=\bigcup_{\nu\in{\kern2pt}]0,\nu_{1}[}\bar{\mathcal{G}}_{\varepsilon,\nu}.\end{align*} $$

7.1 Preliminary results

We define the function $\hat Z:{\mathbb R}\to {\mathbb R}^2$

$$ \begin{align*}\text{for all } t\in{\mathbb R},\quad \hat Z(t)=f_{\varepsilon,k}^{[t]}(\hat z(t-[t]))\end{align*} $$

( $[t]$ denotes the integer part of t that is the unique integer such that $[t]\leq t<[t]+1$ ).

Let us set

$$ \begin{align*}\tau\mathop{=}_{\mathrm{defin.}}\inf\{t\geq 1,\ \hat Z(t)\in \tilde{\mathcal{F}}_{\varepsilon,y_{*}}\}.\end{align*} $$

Note that

(7.62) $$ \begin{align} 2\leq \tau<\infty.\end{align} $$

Indeed, the left-hand side inequality is a consequence of the fact that $\tilde {\mathcal {F}}_{\varepsilon ,y_{*}}\kern-1pt\cap\kern-1pt f_{\varepsilon }(\tilde {\mathcal {F}\kern-1pt}_{\varepsilon ,y_{*}})=\emptyset $ . For the right-hand side, we observe that because $\hat Z(0)=\hat z(0)\in \tilde {\mathcal {F}}_{\varepsilon ,c_{*}y_{*}}$ , one has (see (4.34)), $\hat Z(n_{\varepsilon }(\hat z(0)))=f_{\varepsilon }^{n_{\varepsilon }(\hat z(0))}(\hat z(0))\in \tilde {\mathcal {F}}_{\varepsilon ,y_{*}}$ , hence $\tau \leq n_{\varepsilon }(\hat z(0))<\infty $ .

Proof. Assume by contradiction that $\hat Z:[0,\tau [{\kern2pt}\to {\mathbb R}^2$ is not injective; then, there exists $m_{i}\in {\mathbb N}$ , $0\leq s_{i}<1$ ,

(7.63) $$ \begin{align} 0\leq m_{i}+s_{i}<\tau,\quad i=1,2,\quad \hat Z(s_{1}+m_{1})=\hat Z (s_{2}+m_{2}).\end{align} $$

Hence, $f_{\varepsilon }^{m_{1}}(\hat \gamma )\cap f_{\varepsilon }^{m_{2}}(\hat \gamma )\ne \emptyset $ and if $m:=m_{2}-m_{1}\geq 0$ , $f_{\varepsilon }^{m}(\hat {\gamma })\cap \hat {\gamma }\ne \emptyset $ . In particular, there exists $t\in [m,m+1[$ such that $\hat Z(t)\in \tilde {\mathcal {F}}_{\varepsilon ,y_{*}}$ and then $t\geq \tau $ . As a consequence, $m> \tau -1$ and because $0\leq m<\tau $ ( $m_{1},m_{2}$ are both in the interval $[0,\tau [$ ), one has $m=[\tau ]$ , and hence $m_{2}=m=[\tau ]$ and $m_{1}=0$ . We then have from (7.63), $\hat Z(s_{2}+[\tau ])=\hat Z(s_{1})\in \tilde {\mathcal {F}}_{\varepsilon ,y_{*}}$ (because $s_{1}\in [0,1[$ ) and hence by the definition of $\tau $ , $s_{2}+[\tau ]\geq \tau $ , which contradicts $m_{2}+s_{2}<\tau $ .

7.2 Proof of Proposition 7.1

We first observe that

(7.64) $$ \begin{align} \hat\Gamma=\bigcup_{n\in{\mathbb Z}}f_{\varepsilon}^n(\hat {\gamma})=\hat Z({\mathbb R})=\bigcup_{n\in{\mathbb Z}}f_{\varepsilon}^n(\hat Z([0,\tau+1[)).\end{align} $$

Next we note the following.

1. (1) One has $\hat Z ([0,\tau +1[)=\hat Z([0,\tau ])$ .

2. (2) The set $\hat Z([0,\tau +1[)$ is $f_{\varepsilon }$ -invariant.

Item (1) is a consequence of

$$ \begin{align*}\hat Z ([0,\tau+1[)&=\hat Z ([0,\tau])\cup \hat Z ([\tau,\tau+1[)\\ &=\hat Z([0,\tau])\cup \hat Z([0,1[) \quad(\text{Lemma}\ {7.7})\\ &=\hat Z([0,\tau])\quad (1\leq \tau). \end{align*} $$

Item (2) follows from item (1) and

$$ \begin{align*}f_{\varepsilon}(\hat Z([0,\tau+1[))&=f_{\varepsilon}(\hat Z ([0,\tau]) )\\ &=\hat Z([1,\tau+1])\\ &=\hat Z([1,\tau])\cup \hat Z([\tau,\tau+1])\\ &=\hat Z([1,\tau])\cup \hat Z([0,1])\quad(\text{Lemma}\ {7.7})\\ &=\hat Z([0,\tau])\quad (1\leq \tau)\\ &=\hat Z([0,\tau+1[). \end{align*} $$

Item (2) and (7.64) yield

$$ \begin{align*}\hat \Gamma=\hat Z([0,\tau+1[).\end{align*} $$

This last identity shows that $\hat \Gamma $ is a connected, compact (cf. item (1)) subset of ${\mathbb R}^2$ which is $f_{\varepsilon }$ -invariant.

Let us prove that $\hat \Gamma $ is a one-dimensional submanifold of ${\mathbb R}^2$ . Because $\hat Z(\tau )=\hat Z(0)$ (Lemma 7.5), one has

$$ \begin{align*}\hat Z([0,\tau+1[)=\hat Z(]0,\tau+1[)=\hat Z(]0,\tau[)\cup \hat Z(]\tau-1,\tau+1[).\end{align*} $$

From Lemmata 7.2 and 7.3, the set $\hat Z(]0,\tau [)$ is a one-dimensional submanifold of ${\mathbb R}^2$ as well as the set $\hat Z(]\tau -1,\tau +1[)$ (note that $\hat Z(]\tau -1,\tau +1[)=f_{\varepsilon }(\hat Z([\tau -2,\tau [)$ ). The intersection of these two sets is $\hat Z(]\tau ,\tau +1[)$ and from Lemma 7.7, it is equal to $\hat Z(]0,1[)$ which is a one-dimensional submanifold of ${\mathbb R}^2$ . As a consequence, the union $\hat Z(]0,\tau [)\cup \hat Z(]\tau -1,\tau +1[)$ is one-dimensional submanifold of ${\mathbb R}^2$ .

This concludes the proof of Proposition 7.1□

8.1 Proof of (8.66) (positive measure)

This is a consequence of the inclusion (cf. (8.65))

$$ \begin{align*}h_{\varepsilon}^{-1}\bigg(\bigcup_{\bar{\gamma}\in\bar{\mathcal{G}}_{\varepsilon,\nu}}\bar {\gamma} \bigg)\subset \bigcup_{\hat \Gamma\in\hat{\mathcal{G}}_{\varepsilon,\nu}}\hat \Gamma \end{align*} $$

and of the fact that $\mathrm {Leb}_{2}(\bigcup _{\bar {\gamma }\in \bar {\mathcal {G}}_{\varepsilon ,\nu }}\bar {\gamma })>0$ (this is the content of Theorem 6.2).

8.2 Proof of (8.67) (accumulation)

Let $\bar {\gamma }\in \bar {\mathcal {G}}_{\varepsilon ,\nu }$ , $\bar {\gamma }\subset ({\mathbb R}/{\mathbb Z}){\kern2pt}{\times}{\kern2pt}]0,\nu [$ . From the definition (5.45) of the diffeomorphism $h_{\varepsilon }$ , we see that, for some positive constant $C_{\lambda }$ depending on $\lambda $ (cf. condition (H3)),

$$ \begin{align*}\hat{\gamma}=h_{\varepsilon}^{-1}(\bar{\gamma})\subset\{(x,y)\in \tilde{\mathcal{F}}_{\varepsilon,y_{*}},\quad xy\in{\kern2pt}]0,C_{\lambda}\nu[ \}.\end{align*} $$

However,

$$ \begin{align*}\hat \Gamma=\bigcup_{n\in{\mathbb Z}}f_{\varepsilon}^n(\hat {\gamma})=\bigg(\bigcup_{n\in{\mathbb Z}}f_{\varepsilon}^n(\hat {\gamma})\cap V_{}\bigg)\cup \bigcup_{\substack{n\in{\mathbb Z}\\ f_{\varepsilon}^n(\hat {\gamma})\not\subset V_{}}}f_{\varepsilon}^n(\hat {\gamma}).\end{align*} $$

From condition (H3), one has

$$ \begin{align*}\bigcup_{n\in{\mathbb Z}}f_{\varepsilon}^n(\hat{\gamma})\cap V_{}\subset V_{}\cap \{(x,y),\quad xy\in{\kern2pt}]0,C_{\lambda}\nu[\},\end{align*} $$

and hence, using Remark 2.2,

(8.68) $$ \begin{align} \mathrm{dist}\bigg( \bigcup_{n\in{\mathbb Z}} f_{\varepsilon}^n(\hat {\gamma})\cap V_{} ,\Sigma_{\varepsilon}\cap V_{} \bigg)=o_{\nu}(1) \quad (\text{uniform in }\ \hat{\gamma}).\end{align} $$

Now, recalling the definition (4.29) of the integer $N_{}$ of §4.2, one has

$$ \begin{align*} \bigcup_{\substack{n\in{\mathbb Z}\\ f_{\varepsilon}^n(\hat {\gamma})\not\subset V_{}}}f_{\varepsilon}^n(\hat {\gamma})\subset \bigcup_{n=1}^{N_{}} f^{-n}_{\varepsilon}(\hat{\gamma}), \end{align*} $$

and using the fact that $\mathrm {dist}(\hat {\gamma },\Sigma _{\varepsilon }\cap [(x_{*},0),f_{\varepsilon }(x_{*},0)[)=o_{\nu }(1)$ , one can see that ( $N_{}$ is fixed)

(8.69) $$ \begin{align} \mathrm{dist}\bigg(\!\bigcup_{\substack{n\in{\mathbb Z} \\ f_{\varepsilon}^n(\hat{\gamma})\not\subset V_{}}} f_{\varepsilon}^n(\hat {\gamma}) ,\Sigma_{\varepsilon}\cap \bigcup_{j=1}^{N_{}} f_{\varepsilon}^{-1}([(x_{*},0),f_{\varepsilon}(x_{*},0)[)) \bigg)=o_{\nu}(1),\end{align} $$

where the previous limit is uniform in $\hat {\gamma }$ .

Equations (8.68) and (8.69) give

$$ \begin{align*}\mathrm{dist}(\hat \Gamma,\Sigma_{\varepsilon})=o_{\nu}(1).\end{align*} $$

8.3 KAM circles for $f_{\varepsilon }$

Let $\hat \Gamma $ be a $C^r$ invariant curve for $f_{\varepsilon }$ of the form (8.65) and $g_{\hat \Gamma }$ the restriction of $f_{\varepsilon }$ to $\hat \Gamma $ . The map $g_{\hat \Gamma }$ can be identified with a circle diffeomorphism. Similarly, the restriction of $\bar f_{\varepsilon }$ to the invariant curve $\bar {\gamma }$ yields a circle diffeomorphism $g_{\bar {\gamma }}$ .

Let $\hat \alpha $ and $\bar \alpha $ be the rotation numbers of $g_{\hat \Gamma }$ and $g_{\bar {\gamma }}$ .

Because $\bar \alpha $ can be chosen in a fixed Diophantine class $DC(\kappa ,\tau )$ (see Remark 6.1), the rotation number $\hat \alpha $ is Diophantine with the same exponent $\tau $ . By the Herman–Yoccoz theorem on linearization of $C^r$ -circle diffeomorphisms [Reference Herman10, Reference Yoccoz22], this implies that if r is large enough (depending on $\tau $ which is fixed), the diffeomorphism $g_{\hat \Gamma }$ is linearizable; in other words, $\hat \Gamma $ is a KAM curve. However, one has a priori no control on the Diophantine constant of $\hat \alpha $ .

This concludes the proof of Theorem 2.1, whence of Theorem A.□

9.1 Construction of the example

We start with a smooth autonomous symplectic vector field of the form $X_{0}=J\nabla H_{0}$ , where $H_{0}:{\mathbb R}^2\to {\mathbb R}$ satisfies on some neighborhood V of $o=(0,0)$

$$ \begin{align*}H_{0}(x,y)=x y\quad \text{on} \ V \end{align*} $$

and has the property that $\Sigma =H_{0}^{-1}(H_{0}(0,0))$ is compact and connected. The set $\Sigma $ is a separatrix of

$$ \begin{align*}f\mathop{=}_{\mathrm{defin.}}\phi^1_{J\nabla H_{0}}\end{align*} $$

associated to the hyperbolic fixed point o.

Fixing $x_{*}>0$ small enough, we can define like in §4, for $y_{*}>0$ small enough, a fundamental domain $\tilde {\mathcal {F}}_{y_{*}}=\mathcal {F}_{y_{*}}\cup L_{y_{*}}\subset V$ , where $\mathcal {F}_{y_{*}}$ is defined by $(a)-(d)$ (§4.1) with $\phi ^1_{J\nabla H_{0}}$ in place of $f_{\varepsilon }$ . We can even assume that $\phi _{J\nabla H_{0}}^{-j}(\tilde {\mathcal {F}}_{y_{*}})\subset V$ , $j=1,2$ . There exists $c_{*}>0$ such that the first return map,

$$ \begin{align*}\hat f:\tilde{\mathcal{F}}_{c_{*}y_{*}}\to \mathcal{F}_{y_{*}},\end{align*} $$

is well defined. We can renormalize $f=\phi ^1_{J\nabla H_{0}}$ like in §5 by first normalizing f (cf. Lemma 5.3):

(9.70) $$ \begin{align} h\circ f\circ h^{-1}=T_{1}, \end{align} $$

where

(9.71) $$ \begin{align} h:\mathcal{F}_{y_{*}}\to [0,1[{\kern2pt}{\times}{\kern2pt}]0,c[ \end{align} $$

is symplectic (see (5.45), (5.41), and the fact that we choose $q(s)=s$ ) and then setting (cf. (5.46)):

(9.72) $$ \begin{align} \bar f\mathop{=}_{\mathrm{defin.}}h\circ \hat f\circ h^{-1}:{\mathbb R}/{\mathbb Z}{\kern2pt}{\times}{\kern2pt}]0,\delta[{\kern2pt}\to {\mathbb R}/{\mathbb Z}{\kern2pt}{\times}{\kern2pt}]0,c[.\end{align} $$

By (5.48) of Proposition 5.4, we have

(9.73) $$ \begin{align} \bar f=T_{l},\quad l(y)=\sigma(y)-\ln y\end{align} $$

for some smooth function $\sigma $ .

We can assume that $h(\tilde {\mathcal {F}}_{y_{*}})=[0,1[{\kern2pt}{\times}{\kern2pt}]0,c[$ and that $T^{-j}([0,1[{\kern2pt}{\times}{\kern2pt}]0,c[)\subset h^{-1}(V)$ , $j=1,2$ .

We now construct a symplectic perturbation $f_{\mathrm {pert}}:{\mathbb R}^2\to {\mathbb R}^2$ of f which admits $\Sigma $ as a separatrix. We shall need first the following lemma.

Let $\chi :{\mathbb R}\to {\mathbb R}$ be a smooth function equal to 1 on $[-c/2,c/2]$ and to 0 on ${\mathbb R}\smallsetminus [-(3/4)c,(3/4)c]$ , and define

(9.74) $$ \begin{align} S_{M}(x,y)=(s_{M}(x)y)\chi (y)+xy(1-\chi(y)).\end{align} $$

The canonical (hence symplectic) mapping $g_{M}$ associated to $S_{M}$ :

(9.75) $$ \begin{align} g_{M}(x,y)=(\tilde x, \tilde y) \ \ \iff\ \ \begin{cases}x=\dfrac{\partial S_{M}}{\partial y}(\tilde x,y),\\[10pt] \tilde y=\dfrac{\partial S_{M}}{\partial \tilde x}(\tilde x,y), \end{cases} \end{align} $$

is equal to the identity on $({\mathbb R}\smallsetminus [0,1]){\kern2pt}{\times}{\kern2pt} [-c,c]$ and satisfies for $(x,y)\in [0,1[{\kern2pt}{\times}{\kern2pt}]0,c/2[$

(9.76) $$ \begin{align} \begin{cases}\tilde x=s_{M}^{-1}(x),\\ \tilde y=s^{\prime}_{M}\circ s_{M}^{-1}(x)y. \end{cases} \end{align} $$

The following symplectic perturbation of f:

$$ \begin{align*}f_{\mathrm{pert}}:&\mathop{=}_{\mathrm{defin.}}h^{-1}\circ (g_{M}\circ T_{1})\circ h\\[-2pt] & \, \, =(h^{-1}\circ g_{M}\circ h)\circ f \end{align*} $$

(recall h satisfies (9.70)) is thus defined on ${\mathbb R}^2$ and coincides with f outside $f^{-1}(\tilde {\mathcal {F}}_{y_{*}})$ . Moreover, because $g_{M}({\mathbb R}\times \{0\})={\mathbb R}\times \{0\}$ ,

$$ \begin{align*}\Sigma \ \ \text{is a separatrix for } f_{\mathrm{pert}}.\end{align*} $$

Now, because $\tilde {\mathcal {F}}_{y_{*}}$ is a fundamental domain for $f_{\mathrm {pert}}$ ( $f_{\mathrm {pert}}$ coincide with f on $\tilde {\mathcal {F}}_{y_{*}}$ ), for some $c_{\mathrm {pert}}>0$ small enough, the first return map

$$ \begin{align*} \hat f_{\mathrm{pert}}:f_{\mathrm{pert}}^{-1}(\tilde{\mathcal{F}}_{c_{\mathrm{pert}}y_{*}})\to f_{\mathrm{pert}}^{-1}(\tilde{\mathcal{F}}_{y_{*}}),\end{align*} $$

is well defined and satisfies

$$ \begin{align*}\hat f_{\mathrm{pert}}=(\hat f\circ f^{-1})\circ f_{\mathrm{pert}}.\end{align*} $$

In particular, on

$$ \begin{align*}[-1,0[{\kern2pt}{\times}{\kern2pt}]0,c/2[,\end{align*} $$

one has (cf. (9.72), (9.73))

(9.77) $$ \begin{align}\bar f_{\mathrm{pert}}\mathop{=}_{\mathrm{defin.}}h\circ \hat f_{\mathrm{pert}}\circ h^{-1}&=\bar f\circ T_{-1}\circ g_{M}\circ T_{1} \end{align} $$

(9.78) $$ \begin{align} & \hspace{7pc} =T_{l-1}\circ g_{M}\circ T_{1} \end{align} $$

and

$$ \begin{align*}\bar f_{\mathrm{pert}}:{\mathbb R}{\kern2pt}{\times}{\kern2pt}]0,c/2[{\kern2pt}\to {\mathbb R}{\kern2pt}{\times}{\kern2pt}]0,c[\quad \text{satisfies}\quad \bar f_{\mathrm{pert}}\circ T_{1}=T_{1}\circ \bar f_{\mathrm{pert}};\end{align*} $$

in particular, it defines a smooth map $({\mathbb R}/{\mathbb Z}){\kern2pt}{\times}{\kern2pt}]0,c/2[{\kern2pt}\to ({\mathbb R}/{\mathbb Z}){\kern2pt}{\times}{\kern2pt}]0,c[$ .

Note that because $g_{M}$ is the identity outside $[0,1]{\kern2pt}{\times}{\kern2pt}[-c,c]$ , it admits a $T_{1}$ -periodization $\tilde g_{M}:{\mathbb R}{\kern2pt}{\times}{\kern2pt}[-c,c]\to {\mathbb R}{\kern2pt}{\times}{\kern2pt}[-c,c]$ (which means that $\tilde g_{M}$ and $g_{M}$ coincide on $[0,1]{\kern2pt}{\times} [-c,c]$ and $\tilde g_{M}$ commutes with $T_{1}$ ). This $\tilde g_{M}$ is defined by the same formula (9.75) as $g_{M}$ , where now the new function $\tilde s_{M}$ involved in (9.74) is the ${\mathbb Z}$ -periodization of $s_{M}$ . To simplify the notation, we shall continue to denote $\tilde g_{M}$ and $\tilde s_{M}$ by $g_{M}$ and $s_{M}$ .

Let

(9.79) $$ \begin{align} t:=t(x):=s_{M}^{-1}(x+1). \end{align} $$

Lemma 9.2. For $(x,y)\in [-1,0[{\kern2pt}{\times}{\kern2pt}]0,c/2[$ , the point $(\bar x,\bar y):=\bar f_{\mathrm {pert}}(x,y)$ satisfies with the notation (9.79)

(9.80) $$ \begin{align} \begin{cases} \bar x=t-1+\sigma(s_{M}'(t)\times y)-\ln(s_{M}'(t))-\ln y,\\ \ln \bar y=\ln(s_{M}'(t))+\ln y. \end{cases}\end{align} $$

Proof. Let $(x,y)\in [-1,0[{\kern2pt}{\times}{\kern2pt}]0,c/2[ $ ; with the notation $(x_{1},y_{1})=(g_{M}\circ T_{1})(x,y)=g_{M}(x+1,y)$ , one has from (9.78), $(\bar x,\bar y)=T_{l-1}(x_{1},y_{1})$ and from (9.76), (9.73),

$$ \begin{align*}\begin{cases}x_{1}= s_{M}^{-1}(x+1),\\ y_{1}=s_{M}'\circ s_{M}^{-1}(x+1)\times y, \end{cases}\text{and}\quad \begin{cases}\bar x=x_{1}-1+\sigma(y_{1})-\ln y_{1},\\ \bar y=y_{1}, \end{cases}\end{align*} $$

and hence (9.80).

9.2 Image of a piece of graph by $\bar f_{\mathrm {pert}}$

We take $M>0$ (from Lemma 9.1) large enough and we define

(9.81) $$ \begin{align} J_{M}=s_{M}(I)-1\subset [-1,0[, \end{align} $$

where I is the interval introduced in Lemma 9.1.

If $y:J\to {\mathbb R}_{+}^*$ , $x\to y(x)$ is a differentiable function, we denote by $\gamma _{J,y}$ its graph:

$$ \begin{align*}\gamma_{J,y}=\{(x,y(x)),\ x\in J\} \subset [-1,0[{\kern2pt}{\times}{\kern2pt}]0,c/2[.\end{align*} $$

Proposition 9.3. There exists a constant $y_{\mathrm {pert}}>0$ for which the following holds. Assume that $y:J_{M}\to ]0,y_{\mathrm {pert}}[$ , $x\mapsto y(x)$ is a differentiable function such that

$$ \begin{align*}\text{for all } x\in J_{M},\quad \bigg|\frac{d\ln y}{dx}+1\bigg|\leq 1/2.\end{align*} $$

Then, $\bar f_{\mathrm {pert}}({\gamma }_{J_{M},y})+({\mathbb Z},0)$ contains the graph ${\gamma\kern1pt }_{[-1,0[,\bar y}$ of a differentiable function $\bar y:[-1,0[{\kern2pt}\to {\mathbb R}_{+}^*$ (see Figure 6)

$$ \begin{align*}{\gamma}_{[-1,0[,\bar y}=\{(\bar x,\bar y(\bar x)),\ \bar x\in [-1,0[\},\end{align*} $$

such that

(9.82) $$ \begin{align} &\text{for all } \bar x\in [-1,0[,\quad \bigg|\frac{d\ln \bar y}{d\bar x}+1\bigg|\leq 1/2; \end{align} $$

(9.83) $$ \begin{align} &\sup_{\bar x\in [{-}1,0[} \ln\bar y(\bar x) \leq \sup_{x\in J_{M}}\ln y(x)-bM. \end{align} $$

Moreover, for some interval $J_{M}^{1}\subset J_{M}$ , one has

(9.84) $$ \begin{align} \gamma_{[{-}1,0[,\bar y}=\bar f_{\mathrm{pert}}(\gamma_{J_{1},y}).\end{align} $$