def}}{{=}}\Phi_{k}^{-1}(\ell_{(s,a,\bar{\upsilon}_{k}(\mu))})\,\,\colon\,a,s\in[-a_{P},a_{P}]\,\big{\\}}.$

####  9.1.1. Comparison of the homoclinic and heteroclinic settings

Our heterodimensional analysis is inspired by the one in [23, Chapter 6.4] for
homoclinic tangencies. Let us highlight some key differences and similarities.
For that recall that [23] considers a surface diffeomorphism with saddle $R$
having a homoclinic tangency $Z$. There are associated auxiliary local stable
and unstable foliations $\mathcal{W}^{\ast}_{\mu}$,
$\ast={\mathrm{s}},{\mathrm{u}}$, defined on a neighbourhood of $Z$, here
$\mu$ refers to a “renormalisation” parameter unfolding the tangency. In [23]
the renormalisation scheme converges to the quadratic family
$\varphi_{\mu}(x,y)=(y,y^{2}+\mu)$.

The construction in [23] implicitly uses the fact that (for suitable
parameters) the family $\varphi_{\mu}$ has a fixed point whose unstable
manifold has an “infinite” separatrix. Here we have a property with the same
flavour stated in Lemma 5.5.

The foliations $\mathcal{W}^{{\mathrm{s}}}_{\mu}$ and
$\mathcal{W}^{{\mathrm{u}}}_{\mu}$ converge (in the charts of the
corresponding renormalisation scheme) to foliations whose leaves are
horizontal lines and parabolas of the form $(x,x^{2}+\mu)$, respectively.
Here, the foliation $\mathcal{D}_{\bar{\upsilon}_{k}(\mu)}$ plays the role of
the unstable foliation $\mathcal{W}^{{\mathrm{u}}}_{\mu}$. Lemma 9.1 states a
convergence property of the foliation $\mathcal{D}_{\bar{\upsilon}_{k}(\mu)}$
(involving some projections). The stable foliation
$\mathcal{F}^{{\mathrm{s}}}_{Q,\bar{\upsilon}_{k}(\mu)}$ defined above is
similar to $\mathcal{W}^{{\mathrm{s}}}_{\mu}$.

Both foliations $\mathcal{W}^{{\mathrm{s}}}_{\mu}$ and
$\mathcal{W}^{{\mathrm{u}}}_{\mu}$ foliate a neighbourhood of the tangency
$Z$. Here, we have that the foliation $\mathcal{D}_{\bar{\upsilon}_{k}(\mu)}$
covers a neighbourhood of the heterodimensional tangency $\widetilde{Y}$ and
therefore of the reference domain $\Delta_{k}(\mu)$ of the blender
$\Upsilon_{\bar{\upsilon}_{k}(\mu)}$, see (9.2). There is a similar assertion
for $\mathcal{F}^{{\mathrm{s}}}_{Q,\bar{\upsilon}_{k}(\mu)}$.

In [23], the unstable leaves of the limit thick horseshoes approach to
parabola of the limit unstable foliation and its projection along stable
leaves “covers” several fundamental domains of the local unstable manifold of
the saddle $R$. Here, we have a similar property: the leaves of the strong
unstable foliation of the blenders $\Upsilon_{\bar{\upsilon}_{k}(\mu)}$ are
close to the leaves of $\mathcal{D}_{\bar{\upsilon}_{k}(\mu)}$. Due to the
lack of domination of our setting, it is not possible to get a similar
covering property. Instead, we prove that “projections” of the leaves of
$\mathcal{D}_{\bar{\upsilon}_{k}(\mu)}$ covers a fixed proportion of a fixed
fundamental domain of $W^{{\mathrm{u}}}_{\mathrm{loc}}(Q)$. This will be
enough to see that the blenders are involved in the robust cycles with the
saddle $Q$ and are homoclinically related to the saddle $P$.

\begin{overpic}[scale={0.1}]{diagonal-2} \put(78.0,92.0){\Large{$Q$}}
\put(98.0,75.0){{$\bullet$}} \put(177.0,140.0){\Large{$P$}}
\put(195.0,132.0){{$\bullet$}}
\put(210.0,60.0){\Large{$f^{N_{2}}_{\bar{\upsilon}_{k}(\mu)}$}}
\put(185.0,210.0){{$\\{\ell_{(0,a)}:a\in\mathbb{R}\\}$}} \end{overpic} Figure
5. The diagonal foliation on the leaf $\mathcal{F}^{{\mathrm{u}}}(P)$.

####  9.1.2. Convergence to parabolas

We now go to the details of our construction. Consider:

  * •

the projection $\pi_{1,2}\colon\mathbb{R}^{3}\to\mathbb{R}^{2}$ given by
$\pi_{1,2}(x,y,z)=(x,y)$;

  * •

the maps $\alpha,\,\beta:\mathbb{R}\to\mathbb{R}$ defined by

(9.6) $\begin{split}\alpha(a)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\sqrt{2}\,\beta_{2}\,(b_{2}-b_{3})\,a,\\\
\beta(\mu,s,a)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mu+b_{1}\,\varsigma_{2}\,s+(b_{2}+b_{3}-b_{4})\,\varsigma_{2}\,a^{2},\end{split}$

where $b_{1},b_{2},b_{3},b_{4}$ are parameters associated to the
heterodimensional tangency in (2.6) and $\varsigma_{2}$ is as in the
definition of the Hénon-like maps in (5.2);

  * •

the family of curves

(9.7)
$\overline{\ell}_{(s,a,\bar{\upsilon}_{k}(\mu))}(t)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\pi_{1,2}\Big{(}\widetilde{\ell}_{(s,a,{\bar{\upsilon}_{k}(\mu)})}(t)\Big{)}=\pi_{1,2}\Big{(}\Phi_{k}^{-1}\big{(}\ell_{(s,a,{\bar{\upsilon}_{k}(\mu)})}(t)\big{)}\Big{)};$

  * •

the re-scaling maps
$\widehat{s}_{k},\widehat{a}_{k},\widehat{t}_{k}\colon[-a_{P},a_{P}]\to\mathbb{R}$,
given by

(9.8)
$\begin{split}\widehat{s}_{k}(s)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\sigma_{Q}^{-2n_{k}}\,\sigma_{P}^{-2m_{k}}s,\\\
\widehat{a}_{k}(a)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\sigma_{Q}^{-n_{k}}\,\sigma_{P}^{-m_{k}}a,\\\
\widehat{t}_{k}(t)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\frac{\sqrt{2}}{\beta_{2}(b_{2}+b_{3}+b_{4})}\,\sigma_{Q}^{-2n_{k}}\,\sigma_{P}^{-2m_{k}}t.\end{split}$

Noting that $\widehat{s}_{k}(s)\in J_{k}$ for every $s\in[-a_{P},a_{P}]$, we
can define the curves

$\widehat{\ell}_{(s,a,{\bar{\upsilon}_{k}(\mu)})}(t)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\overline{\ell}_{(\,\widehat{s}_{k}(s)\,,\,\widehat{a}_{k}(a)\,,\,\bar{\upsilon}_{k}(\mu)\,)}(\,\widehat{t}_{k}(t)\,),\quad{(s,a)\in[-a_{P},a_{P}]}$

and the sets

$\widehat{L}_{(s,a,{\bar{\upsilon}_{k}(\mu)})}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{\\{}\,\widehat{\ell}_{(s,a,{\bar{\upsilon}_{k}(\mu)})}(t)\,:\,t\in[-a_{P},a_{P}]\,\big{\\}}.$

######  Lemma 9.1.

The sequence of sets $\widehat{L}_{(s,a,{\bar{\upsilon}_{k}(\mu)})}$ converges
to the parabola

(9.9) $y=x^{2}+\alpha(a)\,x+\beta(a,s,\mu),$

when $k\to\infty$. The convergence is $C^{r}$ uniform on compact sets of
$\mathbb{R}^{2}$.

Next remark will be used in Section 9.3. It relates the parabolas in Lemma
9.1, hence the foliation $\widetilde{\mathcal{D}}_{\bar{\upsilon}_{k}(\mu)}$,
and the Hénon-like maps. It will play a key role in the arguments for
controlling the size of unstable sets of blenders.

######  Remark 9.3.

Recall the reference domain $\Delta=[-4,4]^{2}\times[-40,22]$ in (5.6) of the
blender of $G_{\mu}$ and that

$\pi_{1,2}(G_{\mu}(x,y,z))=(y,\mu+y^{2}+\kappa\,y\,z+\eta\,z^{2}).$

Hence for each fixed $z_{0}$ we have that $\pi_{1,2}(G_{\mu}(x,y,z_{0}))$ is a
curve of the family in (9.9). Therefore given any $K>0$ there is $y(K)$ such
that for every $z_{0}\in[-40,22]$ and every $y\geqslant y(K)$ the angle
between the curve $\pi_{1,2}(G_{\mu}(x,y,z_{0}))$ and the parallel to the
$x$-axis at the point $(0,y)$ are strictly bigger than $2\arctan(K)$, see
Figure 6.

\begin{overpic}[scale={0.032}]{parabolas.jpg} \put(60.0,170.0){$(x,x^{2})$}
\put(141.0,49.0){$\pi_{1,2}\big{(}G_{\mu}(x,y,z_{0})\big{)}$}
\put(165.0,110.0){$(y,\mu+y^{2})$} \put(273.0,20.0){$\mu$}
\put(300.0,154.0){$2\arctan(K)$} \put(267.0,135.0){\small{$y(K)$}}
\put(258.0,130.0){\Huge{$\cdot$}} \put(257.5,15.0){\Huge{$\cdot$}}
\put(303.0,87.0){{$4$}} \end{overpic} Figure 6. The limit family of parabolas
of the sets $\widehat{L}_{(s,0,\bar{\mu}_{k}(\mu))}$ (left). Angles between
parabolas and horizontal lines (right)

###### Proof of Lemma 9.1.

We first give an explicit calculation of the coordinates of the curves
$\ell_{(s,a,\bar{\upsilon}_{k}(\mu))}$ in (9.4). Consider the parameterisation
of $\ell_{(s,a,\bar{\upsilon}_{k}(\mu))}\subset U_{\widetilde{Y}}$ given by

$\begin{split}\ell_{(s,a,\bar{\upsilon}_{k}(\mu))}(t)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}f^{N_{2}}_{\bar{\upsilon}_{k}(\mu)}\big{(}s,1+t+a,1+t-a\big{)}\\\
&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\Big{(}\ell^{1}_{(s,a,{\bar{\upsilon}_{k}(\mu)})}(t),\ell^{2}_{(s,a,{\bar{\upsilon}_{k}(\mu)})}(t),\ell^{3}_{(s,a,{\bar{\upsilon}_{k}(\mu)})}(t)\Big{)}.\end{split}$

Recalling the expression $f^{N_{2}}_{\bar{\upsilon}_{k}(\mu)}$ in (2.5),
Remark 6.3, and (6.5), we get

$\begin{split}\ell^{1}_{(s,a,{\bar{\upsilon}_{k}(\mu)})}(t)=\,\,&1+a_{1}s+(a_{2}-a_{3})a+(a_{2}+a_{3})t\\\
&+H_{1}(s,t+a,t-a)-\lambda^{m_{k}}_{P}\,a_{1},\\\
\ell^{2}_{(s,a,{\bar{\upsilon}_{k}(\mu)})}(t)=\,\,&b_{1}s+(b_{2}+b_{3}-b_{4})\,a^{2}+(b_{2}+b_{3}+b_{4})\,t^{2}+2(b_{2}-b_{3})\,a\,t\\\
&+H_{2}(s,t+a,t-a)+\sigma_{Q}^{-n_{k}}+\sigma_{Q}^{-2n_{k}}\,\sigma_{P}^{-2m_{k}}\,\mu-\lambda^{m_{k}}_{P}\,b_{1},\\\
\ell^{3}_{(s,a,{\bar{\upsilon}_{k}(\mu)})}(t)=\,\,&1+c_{1}s+(c_{2}+c_{3})t+H_{3}(s,t+a,t-a)-\lambda^{m_{k}}_{P}\,c_{1}.\end{split}$

Write

$\overline{\ell}_{(s,a,\bar{\upsilon}_{k}(\mu))}(t)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\Big{(}\overline{\ell}^{1}_{(s,a,\bar{\upsilon}_{k}(\mu))}(t),\overline{\ell}^{2}_{(s,a,{\bar{\upsilon}_{k}(\mu)})}(t)\Big{)}.$

From the definitions of $\overline{\ell}_{(s,a,\bar{\upsilon}_{k}(\mu))}(t)$
in (9.7) and of $\Phi_{k}^{-1}$ in Remark 7.1, we get

$\begin{split}\overline{\ell}^{1}_{(s,a,{\bar{\upsilon}_{k}(\mu)})}(t)=\,\,&c_{1}\,\varsigma_{2}\,\varsigma^{-1}_{5}\,\sigma_{Q}^{n_{k}}\,\sigma_{P}^{m_{k}}s+\frac{\beta_{2}(b_{2}+b_{3}+b_{4})}{\sqrt{2}}\,\sigma_{Q}^{n_{k}}\,\sigma_{P}^{m_{k}}\,t\\\
&+\,\varsigma_{2}\,\varsigma^{-1}_{5}\,\sigma_{Q}^{n_{k}}\,\sigma_{P}^{m_{k}}H_{3}(s,t+a,t-a)-\lambda^{m_{k}}_{P}\,\sigma_{Q}^{n_{k}}\,\sigma_{P}^{m_{k}}\,\varsigma_{2}\,\varsigma^{-1}_{5}\,c_{1},\\\
\overline{\ell}^{2}_{(s,a,{\bar{\upsilon}_{k}(\mu)})}(t)=\,\,&b_{1}\,\varsigma_{2}\,\sigma_{Q}^{2n_{k}}\,\sigma_{P}^{2m_{k}}s+\varsigma_{2}\,(b_{2}+b_{3}-b_{4})\,\sigma_{Q}^{2n_{k}}\,\sigma_{P}^{2m_{k}}a^{2}\\\
&+\frac{\beta^{2}_{2}(b_{2}+b_{3}+b_{4})^{2}}{2}\sigma_{Q}^{2n_{k}}\,\sigma_{P}^{2m_{k}}\,t^{2}+2(b_{2}-b_{3})\,\varsigma_{2}\,\sigma_{Q}^{2n_{k}}\,\sigma_{P}^{2m_{k}}\,a\,t\\\
&+\varsigma_{2}\,\sigma_{Q}^{2n_{k}}\,\sigma_{P}^{2m_{k}}\,H_{2}(s,t+a,t-a)+\mu-\lambda^{m_{k}}_{P}\,\sigma_{Q}^{2n_{k}}\,\sigma_{P}^{2m_{k}}\,\varsigma_{2}\,b_{1}.\end{split}$

Recalling the re-scaling maps
$\widehat{s}_{k},\widehat{a}_{k},\widehat{t}_{k}$, in (9.8) and performing the
corresponding substitutions, a straightforward calculation implies that

$\widehat{\ell}_{(s,a,{\bar{\upsilon}_{k}(\mu)})}(t)\
\to(t,t^{2}+\alpha({a})\,t+\beta(\mu,{s},{a})),$

where $\alpha$ and $\beta$ are as in (9.6). This ends the proof of the lemma.
∎

###  9.2. Estimates of angles and expansion for admissible iterations

We prove the lemma below, which is version of [23, Claim 1 in Chapter 6.4] in
our context. To state this lemma consider the projection
$\pi_{2,3}\colon\mathbb{R}^{3}\to\mathbb{R}^{2}$ defined by
$\pi_{2,3}(x,y,z)=(y,z)$ and recall the definitions of the foliation
$\mathcal{D}_{\bar{\upsilon}_{k}(\mu)}$ of $U_{\widetilde{Y}}$ in (9.4) and of
the set $U_{\bar{\upsilon}_{k}(\mu)}$ of points with an admissible itinerary
as in (8.1).

######  Lemma 9.2.

There is a constant $C>0$ such that for every $\mu\in(-10,-9)$ and every
sufficiently large $k$ the following holds:

Consider $K>0$, a $\bar{\upsilon}_{k}(\mu)$-admissible point $w\in
U_{\bar{\upsilon}_{k}(\mu)}$, and a vector $v\in T_{w}M$ such that

(9.10)
$\mathrm{angle}\big{(}v,\mathcal{F}^{{\mathrm{s}}}_{Q,\bar{\upsilon}_{k}(\mu)}\big{)}\geqslant
K\,\sigma_{Q}^{-n_{k}}\sigma_{P}^{-m_{k}}.$

Then

  1. (1)

$\|\pi_{2,3}\big{(}D_{w}(f_{\bar{\upsilon}_{k}(\mu)}^{N_{2}+m_{k}+N_{1}+n_{k}})(v)\big{)}\|\geqslant
C\,K\,\|v\|$,

  2. (2)

$\mathrm{angle}\big{(}D_{w}(f_{\bar{\upsilon}_{k}(\mu)}^{N_{2}+m_{k}+N_{1}+n_{k}})(v),\mathcal{F}^{{\mathrm{u}}}_{P,\bar{\upsilon}_{k}(\mu)}\big{)}=O(\lambda_{P}^{m_{k}}\sigma_{Q}^{n_{k}})\to
0$ as $k\to\infty$, and

  3. (3)

$\mathrm{angle}\big{(}D_{w}(f_{\bar{\upsilon}_{k}(\mu)}^{N_{2}+m_{k}+N_{1}+n_{k}})(v),\mathcal{D}_{\bar{\upsilon}_{k}(\mu)}\big{)}\to
0,\quad\mbox{as $k\to\infty$}.$

We begin with some preliminary estimates. For $R=P,Q$, consider the coordinate
vector fields
$\Big{\\{}\frac{\partial}{\partial\,x_{R}},\quad\frac{\partial}{\partial\,y_{R}},\quad\frac{\partial}{\partial\,z_{R}}\Big{\\}},$
defined on the neighbourhood $U_{R}$ and tangent to the corresponding
foliations $\mathcal{F}^{\ast}_{R,\bar{\upsilon}_{k}(\mu)}$,
$\ast={\mathrm{u}},{\mathrm{s}}$. The derivatives of the transition maps
$\mathfrak{T}_{1}=f^{N_{1}}$ in (2.4) and $\mathfrak{T}_{2}=f^{N_{2}}$ in
(2.6) satisfy

(9.11)
$\begin{split}D\mathfrak{T}_{1}\Big{(}\frac{\partial}{\partial\,x_{Q}}\Big{)}&=\alpha_{1}\frac{\partial}{\partial\,x_{P}},\\\
D\mathfrak{T}_{1}\Big{(}\frac{\partial}{\partial\,y_{Q}}\Big{)}&=\alpha_{2}\frac{\partial}{\partial\,x_{P}}+\beta_{2}\frac{\partial}{\partial\,y_{P}},\\\
D\mathfrak{T}_{1}\Big{(}\frac{\partial}{\partial\,z_{Q}}\Big{)}&=\alpha_{3}\frac{\partial}{\partial\,x_{P}}+\gamma_{3}\frac{\partial}{\partial\,z_{P}},\\\
D\mathfrak{T}_{2}\Big{(}\frac{\partial}{\partial\,x_{P}}\Big{)}&=a_{1}\frac{\partial}{\partial\,x_{Q}}+b_{1}\frac{\partial}{\partial\,y_{Q}}+c_{1}\frac{\partial}{\partial\,z_{Q}},\\\
D\mathfrak{T}_{2}\Big{(}\frac{\partial}{\partial\,y_{P}}\Big{)}&=a_{2}\frac{\partial}{\partial\,x_{Q}}+c_{2}\frac{\partial}{\partial\,z_{Q}},\\\
D\mathfrak{T}_{2}\Big{(}\frac{\partial}{\partial\,z_{P}}\Big{)}&=a_{3}\frac{\partial}{\partial\,x_{Q}}+c_{3}\frac{\partial}{\partial\,z_{Q}}.\end{split}$

Given a point $w\in U_{\bar{\upsilon}_{k}(\mu)}$ and a vector

(9.12)
$v=v_{x}\,\frac{\partial}{\partial\,x_{Q}}+v_{y}\,\frac{\partial}{\partial\,y_{Q}}+v_{z}\,\frac{\partial}{\partial\,z_{Q}},$

we give the explicit expression of
$D_{w}(f_{\bar{\upsilon}_{k}(\mu)}^{N_{2}+m_{k}+N_{1}+n_{k}})(v)$. It follows
the sketch of the step by step calculations of this derivative, for details
see [16, Section 7].

Recalling the linearising coordinates of $f$ at $Q$ in (2.1) and the
definitions of $\mathfrak{c}_{k},\mathfrak{s}_{k}$ in (6.6), we have

$D_{w}(f_{\bar{\upsilon}_{k}(\mu)}^{n_{k}})(v)=\lambda_{Q}^{n_{k}}(\mathfrak{c}_{k}-\mathfrak{s}_{k})\,v_{x}\,\frac{\partial}{\partial
x_{Q}}+\sigma_{Q}^{n_{k}}\,v_{y}\,\frac{\partial}{\partial
y_{Q}}+\lambda_{Q}^{n_{k}}(\mathfrak{c}_{k}+\mathfrak{s}_{k})\,v_{z}\,\frac{\partial}{\partial
z_{Q}}.$

Write

$D_{w}(f_{\bar{\upsilon}_{k}(\mu)}^{N_{1}+n_{k}})(v)=v_{x,{\bar{\upsilon}_{k}(\mu)}}\,\frac{\partial}{\partial
x_{P}}+v_{x,{\bar{\upsilon}_{k}(\mu)}}\,\frac{\partial}{\partial
y_{P}}+v_{z,{\bar{\upsilon}_{k}(\mu)}}\,\frac{\partial}{\partial z_{P}}.$

Using (9.11) we have that

(9.13)
$\begin{split}v_{x,{\bar{\upsilon}_{k}(\mu)}}&=\lambda_{Q}^{n_{k}}(\mathfrak{c}_{k}-\mathfrak{s}_{k})\,v_{x}\,\alpha_{1}+\sigma_{Q}^{n_{k}}\,v_{y}\,\alpha_{2}+\lambda_{Q}^{n_{k}}(\mathfrak{c}_{k}-\mathfrak{s}_{k})\,v_{z}\,\alpha_{3},\\\
v_{y,{\bar{\upsilon}_{k}(\mu)}}&=\sigma_{Q}^{n_{k}}\,v_{y}\,\beta_{2},\\\
v_{z,{\bar{\upsilon}_{k}(\mu)}}&=\lambda_{Q}^{n_{k}}(\mathfrak{c}_{k}+\mathfrak{s}_{k})\,v_{z}\,\gamma_{3}.\end{split}$

Write

$D_{w}(f_{\bar{\upsilon}_{k}(\mu)}^{m_{k}+N_{1}+n_{k}})(v)=\widetilde{v}_{x,{\bar{\upsilon}_{k}(\mu)}}\,\frac{\partial}{\partial
x_{P}}+\widetilde{v}_{y,{\bar{\upsilon}_{k}(\mu)}}\,\frac{\partial}{\partial
y_{P}}+\widetilde{v}_{z,{\bar{\upsilon}_{k}(\mu)}}\,\frac{\partial}{\partial
z_{P}}.$

Using (9.13), the linearising coordinates of $f$ at $P$ in (2.1), and the
definitions of $\tilde{\mathfrak{c}}_{k},\tilde{\mathfrak{s}}_{k}$ in (6.6),
we have that

(9.14)
$\begin{split}\widetilde{v}_{x,{\bar{\upsilon}_{k}(\mu)}}&=\lambda_{P}^{m_{k}}\big{(}\lambda_{Q}^{n_{k}}(\mathfrak{c}_{k}-\mathfrak{s}_{k})\,v_{x}\,\alpha_{1}+\sigma_{Q}^{n_{k}}\,v_{y}\,\alpha_{2}+\lambda_{Q}^{n_{k}}(\mathfrak{c}_{k}-\mathfrak{s}_{k})\,v_{z}\,\alpha_{3}\big{)},\\\
\widetilde{v}_{y,{\bar{\upsilon}_{k}(\mu)}}&=\sigma_{P}^{m_{k}}\,\sigma_{Q}^{n_{k}}\,\tilde{\mathfrak{c}}_{k}\,v_{y}\,\beta_{2}-\sigma_{P}^{m_{k}}\,\lambda_{Q}^{n_{k}}\,\tilde{\mathfrak{s}}_{k}\,(\mathfrak{c}_{k}+\mathfrak{s}_{k})\,v_{z}\,\gamma_{3},\\\
\widetilde{v}_{z,{\bar{\upsilon}_{k}(\mu)}}&=\sigma_{P}^{m_{k}}\,\sigma_{Q}^{n_{k}}\,\tilde{\mathfrak{s}}_{k}\,v_{y}\,\beta_{2}+\sigma_{P}^{m_{k}}\,\lambda_{Q}^{n_{k}}\,\tilde{\mathfrak{c}}_{k}\,(\mathfrak{c}_{k}+\mathfrak{s}_{k})\,v_{z}\,\gamma_{3}.\end{split}$

Finally, we write

$D_{w}(f_{\bar{\upsilon}_{k}(\mu)}^{N_{2}+m_{k}+N_{1}+n_{k}})(v)=\widehat{v}_{x,{\bar{\upsilon}_{k}(\mu)}}\,\frac{\partial}{\partial
x_{Q}}+\widehat{v}_{x,{\bar{\upsilon}_{k}(\mu)}}\,\frac{\partial}{\partial
y_{Q}}+\widehat{v}_{z,{\bar{\upsilon}_{k}(\mu)}}\,\frac{\partial}{\partial
z_{Q}}.$

Using (9.11) and (9.14) we get

(9.15)
$\begin{split}\widehat{v}_{x,{\bar{\upsilon}_{k}(\mu)}}&=a_{1}\,\widetilde{v}_{x,{\bar{\upsilon}_{k}(\mu)}}+a_{2}\,\widetilde{v}_{y,{\bar{\upsilon}_{k}(\mu)}}+a_{3}\,\widetilde{v}_{z,{\bar{\upsilon}_{k}(\mu)}},\\\
\widehat{v}_{y,{\bar{\upsilon}_{k}(\mu)}}&=b_{1}\,\widetilde{v}_{x,{\bar{\upsilon}_{k}(\mu)}},\\\
\widehat{v}_{z,{\bar{\upsilon}_{k}(\mu)}}&=c_{1}\,\widetilde{v}_{x,{\bar{\upsilon}_{k}(\mu)}}+c_{2}\,\widetilde{v}_{y,{\bar{\upsilon}_{k}(\mu)}}+c_{3}\,\widetilde{v}_{z,{\bar{\upsilon}_{k}(\mu)}}.\end{split}$

###### Proof of Lemma 9.2.

Take an unitary vector $v$, write it as in (9.12), and let
$\alpha_{k,\mu}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathrm{angle}\big{(}v,\mathcal{F}^{{\mathrm{s}}}_{Q,\bar{\upsilon}_{k}(\mu)}\big{)}$.
Note that by hypothesis (9.10)

(9.16)
$|v_{y}|=\sin(\alpha_{k,\mu})\geqslant\sin(K\,\sigma_{Q}^{-n_{k}}\,\sigma_{P}^{-m_{k}})\approx
K\,\sigma_{Q}^{-n_{k}}\sigma_{P}^{-m_{k}}.$

To prove item (1), recall the definitions of
$\widehat{v}_{z,{\bar{\upsilon}_{k}(\mu)}}$ in (9.15) and of
$\widetilde{v}_{x,{\bar{\upsilon}_{k}(\mu)}}$,
$\widetilde{v}_{y,{\bar{\upsilon}_{k}(\mu)}}$, and
$\widetilde{v}_{z,{\bar{\upsilon}_{k}(\mu)}}$ in (9.14) and note that

$\dfrac{\widehat{v}_{z,{\bar{\upsilon}_{k}(\mu)}}}{\sigma_{P}^{m_{k}}\,\sigma_{Q}^{n_{k}}}=\dfrac{c_{1}\,\widetilde{v}_{x,{\bar{\upsilon}_{k}(\mu)}}}{\sigma_{P}^{m_{k}}\,\sigma_{Q}^{n_{k}}}+\dfrac{c_{2}\,\widetilde{v}_{y,{\bar{\upsilon}_{k}(\mu)}}}{\sigma_{P}^{m_{k}}\,\sigma_{Q}^{n_{k}}}+\dfrac{c_{3}\,\tilde{v}_{z,{\bar{\upsilon}_{k}(\mu)}}}{\sigma_{P}^{m_{k}}\,\sigma_{Q}^{n_{k}}}.$

Recalling that $\sigma_{P},\sigma_{Q}>1$,
$\sigma_{P}^{m_{k}}\,\lambda_{Q}^{n_{k}}\to\tau^{-1}\xi$ (see (6.2)) and
$\tilde{\mathfrak{c}}_{k},\,\tilde{\mathfrak{s}}_{k}\to 1/\sqrt{2}$,
$\mathfrak{c}_{k}\to 0,\,\mathfrak{s}_{k}\to 1$ (see Remark 6.4), we get

$\dfrac{\,\widetilde{v}_{x,{\bar{\upsilon}_{k}(\mu)}}}{\sigma_{P}^{m_{k}}\,\sigma_{Q}^{n_{k}}}\to
0,\qquad\dfrac{\,\widetilde{v}_{y,{\bar{\upsilon}_{k}(\mu)}}}{\sigma_{P}^{m_{k}}\,\sigma_{Q}^{n_{k}}}\to\frac{\beta_{2}\,v_{y}}{\sqrt{2}},\qquad\dfrac{\,\widetilde{v}_{z,{\bar{\upsilon}_{k}(\mu)}}}{\sigma_{P}^{m_{k}}\,\sigma_{Q}^{n_{k}}}\to\frac{\beta_{2}\,v_{y}}{\sqrt{2}}.$

Thus, for every large enough $k$, we get

(9.17)
$|\widehat{v}_{z,{\bar{\upsilon}_{k}(\mu)}}|\approx\sqrt{2}\,\sigma_{P}^{m_{k}}\,\sigma_{Q}^{n_{k}}\,|c_{2}+c_{3}|\,|\beta_{2}|\,|v_{y}|.$

Note that $c_{2}+c_{3}\neq 0$, see (2.7) and (2.6), and $\beta_{2}\neq 0$, see
(2.4). Finally, from (9.16) and (9.17) it follows

(9.18)
$\begin{split}\|\pi_{2,3}(D_{w}(f_{\bar{\upsilon}_{k}(\mu)}^{N_{2}+m_{k}+N_{1}+n_{k}})(v))\|&\geqslant|\widehat{v}_{z,{\bar{\upsilon}_{k}(\mu)}}|\\\
&\approx\sqrt{2}\,\sigma_{P}^{m_{k}}\,\sigma_{Q}^{n_{k}}\,|c_{2}+c_{3}|\,|\beta_{2}|\,|v_{y}|\\\
&\geqslant C\,K,\end{split}$

where $C=\sqrt{2}\,|\beta_{2}|\,|c_{2}+c_{3}|$. This proves the first item in
the lemma.

To prove item (2), let $\beta_{k,\mu}$ be the angle between the vector
$D_{w}(f_{\bar{\upsilon}_{k}(\mu)}^{m_{k}+N_{1}+n_{k}})(v)$ and the unstable
foliation $\mathcal{F}^{u}_{P,\bar{\upsilon}_{k}(\mu)}$ (tangent to
$\frac{\partial}{\partial y_{P}}$ and $\frac{\partial}{\partial z_{P}}$). Then

$\sin(\beta_{k,\mu})=\frac{|\widetilde{v}_{x,{\bar{\upsilon}_{k}(\mu)}}|}{\|D_{w}(f_{\bar{\upsilon}_{k}(\mu)}^{m_{k}+N_{1}+n_{k}})(v)\|}\leqslant\frac{|\widetilde{v}_{x,{\bar{\upsilon}_{k}(\mu)}}|}{\|\pi_{2,3}(D_{w}(f_{\bar{\upsilon}_{k}(\mu)}^{m_{k}+N_{1}+n_{k}})(v))\|}\lesssim\frac{\lambda_{P}^{m_{k}}\sigma_{Q}^{n_{k}}}{C\,K},$

where in the last inequality we use (9.18) and that
$\widetilde{v}_{x,{\bar{\upsilon}_{k}(\mu)}}={O}(\lambda_{P}^{m_{k}}\sigma_{Q}^{n_{k}})$,
see (9.14). Note that by (6.3)

$\lambda_{P}^{m_{k}}\sigma_{Q}^{n_{k}}\leqslant\lambda_{P}^{m_{k}}\sigma_{Q}^{2n_{k}}\sigma_{P}^{2m_{k}}\to
0.$

Therefore,

$\beta_{k,\mu}\approx\sin(\beta_{k,\mu})=O(\lambda_{P}^{m_{k}}\sigma_{Q}^{n_{k}})\to
0,$

proving the second item in the lemma.

To prove the last item in the lemma, recall again the definitions of
$\widetilde{v}_{y,{\bar{\upsilon}_{k}(\mu)}}$ and
$\widetilde{v}_{z,{\bar{\upsilon}_{k}(\mu)}}$ in (9.14) and note that

(9.19)
$\begin{split}\frac{|\widetilde{v}_{y,\bar{\upsilon}_{k}(\mu)}|}{|\widetilde{v}_{z,\bar{\upsilon}_{k}(\mu)}|}&=\frac{|\sigma_{P}^{m_{k}}\,\sigma_{Q}^{n_{k}}\,\tilde{\mathfrak{c}}_{k}\,v_{y}\,\beta_{2}-\sigma_{P}^{m_{k}}\,\lambda_{Q}^{n_{k}}\,\tilde{\mathfrak{s}}_{k}\,(\mathfrak{c}_{k}+\mathfrak{s}_{k})\,v_{z}\,\gamma_{3}|}{|\sigma_{P}^{m_{k}}\,\sigma_{Q}^{n_{k}}\,\tilde{\mathfrak{s}}_{k}\,v_{y}\,\beta_{2}+\sigma_{P}^{m_{k}}\,\lambda_{Q}^{n_{k}}\,\tilde{\mathfrak{c}}_{k}\,(\mathfrak{c}_{k}+\mathfrak{s}_{k})\,v_{z}\,\gamma_{3}|}\\\
&=\frac{\Big{|}\tilde{\mathfrak{c}}_{k}\,v_{y}\,\beta_{2}-\frac{\sigma_{P}^{m_{k}}\,\lambda_{Q}^{n_{k}}\,\tilde{\mathfrak{s}}_{k}\,(\mathfrak{c}_{k}+\mathfrak{s}_{k})\,v_{z}\,\gamma_{3}}{\sigma_{P}^{m_{k}}\,\sigma_{Q}^{n_{k}}}\Big{|}}{\Big{|}\tilde{\mathfrak{s}}_{k}\,v_{y}\,\beta_{2}+\frac{\sigma_{P}^{m_{k}}\,\lambda_{Q}^{n_{k}}\,\tilde{\mathfrak{c}}_{k}\,(\mathfrak{c}_{k}+\mathfrak{s}_{k})\,v_{z}\,\gamma_{3}}{\sigma_{P}^{m_{k}}\,\sigma_{Q}^{n_{k}}}\Big{|}}\to
1,\end{split}$

where for the limit we use again that
$\tilde{\mathfrak{c}}_{k},\,\tilde{\mathfrak{s}}_{k}\to 1/\sqrt{2}$ and
$\mathfrak{c}_{k}\to 0,\,\mathfrak{s}_{k}\to 1$.

This implies that the angle between
$D_{w}(f_{\bar{\upsilon}_{k}(\mu)}^{m_{k}+N_{1}+n_{k}})(v)$ and the diagonal
foliation $\mathcal{D}$ in (9.1) tends to $0$ as $k$ goes to infinity. Hence,
by the definition of $\mathcal{D}_{\bar{\upsilon}_{k}(\mu)}$, the angle
between $D_{w}(f_{\bar{\upsilon}_{k}(\mu)}^{N_{2}+m_{k}+N_{1}+n_{k}})(v)$ and
$\mathcal{D}_{\bar{\upsilon}_{k}(\mu)}$ also tends to $0$ as $k$ goes to
infinity. This ends the proof of the lemma. ∎

###  9.3. Separatrices of the saddles of blenders nearby heterodimensional
tangencies

We now study the stable and strong unstable separatrices of the reference
saddle $P^{+}_{\bar{\upsilon}_{k}(\mu)}$ of the blenders
$\Lambda_{\bar{\upsilon}_{k}(\mu)}$ of $F_{\bar{\upsilon}_{k}(\mu)}$ in
$\mathbb{R}^{3}$, recall Notation 7.3. Our goal are the angular estimates in
Lemma 9.3.

Let us go to the details. Consider first blenders for the endomorphisms
$G_{\mu}$. Take $\mu\in(-10,-9)$ and the blender
$\Lambda_{\mu}=\Lambda_{\xi,\mu,\bar{\eta}}$ of $G_{\mu}$ and its reference
fixed point $P^{+}_{\mu}=(p_{\mu}^{+},p_{\mu}^{+},\widetilde{p}_{\mu}^{+})$ in
(5.7).

Denote by $\sigma_{\mu}^{{\mathrm{u}}{\mathrm{u}}}$ the separatrix999That is,
one of the connected components of
$W^{{\mathrm{u}}{\mathrm{u}}}(P^{+}_{\mu})\setminus\\{P^{+}_{\mu}\\}$ of
$W^{{\mathrm{u}}{\mathrm{u}}}(P^{+}_{\mu})$ contained in $\\{y\geqslant
p_{\mu}^{+}\\}$ and by $\sigma_{\mu}^{{\mathrm{s}}}$ the separatrix of
$W^{{\mathrm{s}}}(P^{+}_{\mu})$ contained in $\\{x\geqslant
p_{\mu}^{+},\,y=p_{\mu}^{+},\,z=\widetilde{p}_{\mu}^{+}\\}$. We consider the
curve
$\widetilde{\sigma}_{\mu}^{{\mathrm{s}}}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\\{P^{+}_{\mu}\\}\cup\sigma_{\mu}^{{\mathrm{s}}}$. We now introduce
the ingredients of our construction which are depicted in Figure 7.

\begin{overpic}[scale={0.08}]{separatistas.jpg}
\put(156.0,95.0){\Large{$P^{+}_{\mu}$}}
\put(210.0,117.0){\Large$\sigma_{\mu}^{{\mathrm{s}}{\mathrm{s}}}$}
\put(162.0,115.0){$\bullet$} \put(240.0,52.0){\Large$x$}
\put(220.0,83.0){\Large$y$} \put(185.0,47.5){\Large{$z$}}
\put(117.0,188.0){$\bullet$} \put(127.0,190.0){$y(K)$}
\put(173.0,191.0){$\bullet$} \put(185.0,206.0){$y^{\star}_{\mu}(K)$}
\put(152.0,225.0){\Large$\sigma_{\mu}^{{\mathrm{u}}{\mathrm{u}}}$}
\end{overpic} Figure 7. Separatrices of the reference saddles of the blender

$\bullet$ Strong unstable separatrices. Given $K>0$ define $y(K)$ as in Remark
9.3 and consider the point

$y^{\star}_{\mu}(K)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\sigma_{\mu}^{{\mathrm{u}}{\mathrm{u}}}\cap\\{y=y(K)\\}.$

Let $\sigma_{K,\mu}^{{\mathrm{u}}{\mathrm{u}}}$ be the segment of
$\sigma_{\mu}^{{\mathrm{u}}{\mathrm{u}}}$ bounded by $y^{\star}_{\mu}(K)$ and
$P^{+}_{\mu}$ and consider its fundamental domain

$\widetilde{\sigma}_{K,\mu}^{{\mathrm{u}}{\mathrm{u}}}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\sigma_{K,\mu}^{{\mathrm{u}}{\mathrm{u}}}\setminus
G^{-1}_{\mu}(\sigma_{K,\mu}^{{\mathrm{u}}{\mathrm{u}}}).$

After increasing $K$, if necessary, we can assume that the angles between the
curve $\pi_{1,2}(\widetilde{\sigma}_{K,\mu}^{{\mathrm{u}}{\mathrm{u}}})$ and
the lines parallel to the $x$-axis are strictly bigger than $\arctan(K)$.

For $\mu\in(-10,-9)$ and large $k$, we consider the continuations of the
objects defined above for $G_{\mu}$:

  * •

the separatrices $\sigma^{{\mathrm{u}}{\mathrm{u}}}_{\bar{\upsilon}_{k}(\mu)}$
of $W^{{\mathrm{u}}{\mathrm{u}}}(P^{+}_{\bar{\upsilon}_{k}(\mu)})$,

  * •

the points
$y^{\star}_{\bar{\upsilon}_{k}(\mu)}(K)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\sigma^{{\mathrm{u}}{\mathrm{u}}}_{\bar{\upsilon}_{k}(\mu)}\cap\\{y=y(K)\\},$

  * •

the curves $\sigma_{K,\bar{\upsilon}_{k}(\mu)}^{{\mathrm{u}}{\mathrm{u}}}$ of
$\sigma_{\bar{\upsilon}_{k}(\mu)}^{{\mathrm{u}}{\mathrm{u}}}$ bounded by
$y^{\star}_{{\bar{\upsilon}_{k}(\mu)}}(K)$ and
$P^{+}_{{\bar{\upsilon}_{k}(\mu)}}$, and

  * •

the fundamental domain

$\widetilde{\sigma}_{K,\bar{\upsilon}_{k}(\mu)}^{{\mathrm{u}}{\mathrm{u}}}=\sigma_{K,\bar{\upsilon}_{k}(\mu)}^{{\mathrm{u}}{\mathrm{u}}}\setminus
F_{\bar{\upsilon}_{k}(\mu)}^{-1}(\sigma_{K,\bar{\upsilon}_{k}(\mu)}^{{\mathrm{u}}{\mathrm{u}}}).$

By Remark 9.3 and continuity, for every $\mu\in(-10,-9)$ and large $k$, the
angles between the curve
$\pi_{1,2}(\widetilde{\sigma}_{K,\bar{\upsilon}_{k}(\mu)}^{{\mathrm{u}}{\mathrm{u}}})$
and the lines parallel to the $x$-axis are strictly bigger than $\arctan(K)$.

$\bullet$ Stable separatrices. For large $k$, define the continuations
$\widetilde{\sigma}_{\bar{\upsilon}_{k}(\mu)}^{{\mathrm{s}}}$ of
$\widetilde{\sigma}_{\mu}^{{\mathrm{s}}}$ for $F_{\bar{\upsilon}_{k}(\mu)}$
contained in a separatrix of
$W^{{\mathrm{s}}}(P^{+}_{\bar{\upsilon}_{k}(\mu)})$ and whose boundary
contains $P^{+}_{\bar{\upsilon}_{k}(\mu)}$.

$\bullet$ Angles between the separatrices. Note that, by equation (5.8), the
tangent space of $W^{{\mathrm{u}}{\mathrm{u}}}(P^{+}_{\mu})$ at $P^{+}_{\mu}$
is contained in a cone field transverse to the horizontal line (parallel to
the $x$-axis). In particular, equation (5.8) implies that for every
$\mu\in(-10,-9)$ the angle between $\sigma^{{\mathrm{u}}{\mathrm{u}}}_{\mu}$
and $\widetilde{\sigma}^{{\mathrm{s}}}_{\mu}$ at $P^{+}_{\mu}$ is bounded from
below by $\pi/7$. Thus, for every $k$ large enough, the angle between
$\sigma_{\bar{\upsilon}_{k}(\mu)}^{{\mathrm{u}}{\mathrm{u}}}$ and
$\widetilde{\sigma}_{\bar{\upsilon}_{k}(\mu)}^{{\mathrm{s}}}$ at
$P_{\bar{\upsilon}_{k}(\mu)}^{+}$ is bigger than $\pi/8$.

For the next lemma recall the definition of the curves
$\widetilde{\ell}_{(s,a,\bar{\upsilon}_{k}(\mu))}$ in (9.5) and of the
parameter interval $J_{k}$ in (9.3).

######  Lemma 9.3.

For every $K>0$ there is $k_{0}=k_{0}(K)\geqslant 1$ such that for every
$\mu\in(-10,-9)$ and every $k\geqslant k_{0}$ the angles between

  * •

the lines parallel to the $x$-axis in $\mathbb{R}^{2}$ and the curves
$\pi_{1,2}(\widetilde{\sigma}^{{\mathrm{u}}{\mathrm{u}}}_{K,\bar{\upsilon}_{k}(\mu)})$
are at least $\arctan(K)$,

  * •

the curves
$\big{(}\widetilde{\ell}_{(s,a,\bar{\upsilon}_{k}(\mu))}\big{)}_{a\in[-a_{P},a_{P}],\,s\in
J_{k}}$ and
$\pi_{1,2}(\widetilde{\sigma}^{{\mathrm{s}}}_{\bar{\upsilon}_{k}(\mu)})$ are
at least $\frac{\pi}{8}$,

  * •

the lines parallel to the $x$-axis in $\mathbb{R}^{2}$ and
$\pi_{1,2}(\widetilde{\sigma}^{{\mathrm{s}}}_{\bar{\upsilon}_{k}(\mu)})$ are
at most $K^{-1}$,

  * •

the curves
$\big{(}\widetilde{\ell}_{(s,a,\bar{\upsilon}_{k}(\mu))}\big{)}_{a\in[-a_{P},a_{P}],\,s\in
J_{k}}$ and the curves
$\pi_{1,2}(\widetilde{\sigma}^{{\mathrm{u}}{\mathrm{u}}}_{K,\bar{\upsilon}_{k}(\mu)})$
are at most $K^{-1}$.

###### Proof.

The first three items of the lemma follow from the discussion before the
lemma. The last item follows from Lemma 9.1 and Remark 9.3. ∎

###  9.4. Estimates of angles and expansion for iterations of
$F_{{\bar{\upsilon}_{k}(\mu)}}$

By the formula of $\Phi_{k}$ in Remark 7.1, we have that $D_{w}\Phi_{k}$ is a
diagonal matrix that does not depend on the point $w$. Hence we will omit this
dependence. Thus, after identifying the tangent spaces in $U_{\widetilde{Y}}$
with $\mathbb{R}^{3}$, we have that

$\begin{split}||D_{w}F_{\bar{\upsilon}_{k}(\mu)}(v)||&=||D\Phi_{k}^{-1}\big{(}D_{\Phi_{k}(w)}f^{N_{2}+m_{k}+N_{1}+n_{k}}_{\bar{\upsilon}_{k}(\mu)}(D{\Phi_{k})}(v)\big{)}||\\\
&=||D_{\Phi_{k}(w)}f^{N_{2}+m_{k}+N_{1}+n_{k}}_{\bar{\upsilon}_{k}(\mu)}(v)||.\end{split}$

Observe also that the angles in Lemma 9.3 are taken with respect the
coordinates in the $xy$-plane in $\mathbb{R}^{3}$. To get these angles in
$U_{\widetilde{Y}}$ we need to replace each angle $\alpha$ by
$\arctan(\sigma_{P}^{-m_{k}}\sigma_{Q}^{-n_{k}}\tan(\alpha))$.

We have the following consequence of Lemma 9.2. For that recall also the
definition of the foliation
$\widetilde{\mathcal{D}}_{\bar{\upsilon}_{k}(\mu)}$ in 9.5:

######  Lemma 9.4.

There is a constant $C>0$ such that for every $\mu\in(-10,-9)$ and every
sufficiently large $k$ the following holds:

Take $K>0$, a point $w\in\Phi_{k}^{-1}(U_{{\bar{\upsilon}_{k}(\mu)}})$, and a
unitary vector $v=(v_{1},v_{2},v_{3})\in\mathbb{R}^{3}$ such that

(9.20) $|v_{2}|\geqslant\arctan(K).$

Then

  1. (1)

$\|\pi_{1,2}\big{(}D_{w}F_{\bar{\upsilon}_{k}(\mu)}(v)\big{)}\|\geqslant
C\,K\,\|v\|$,

  2. (2)

$\mathrm{angle}\big{(}D_{w}F_{\bar{\upsilon}_{k}(\mu)}(v),\Phi_{k}^{-1}(\mathcal{F}^{{\mathrm{u}}}_{P,\bar{\upsilon}_{k}(\mu)})\big{)}=\arctan\big{(}O(\lambda_{P}^{m_{k}}\sigma_{Q}^{2n_{k}}\sigma_{P}^{m_{k}})\big{)}\to
0$ as $k\to\infty$, and

  3. (3)

$\mathrm{angle}\big{(}D_{w}F_{\bar{\upsilon}_{k}(\mu)}(v),\widetilde{\mathcal{D}}_{\bar{\upsilon}_{k}(\mu)}\big{)}\to
0,\quad\mbox{as $k\to\infty$}.$

###### Proof.

The first two items of the lemma are direct translations of the corresponding
items of Lemma 9.2. For the convergence to zero in the second item we use
(6.3). The third item follows from (9.19) and the definition of $\Phi_{k}$. ∎

###  9.5. End of the proof of Proposition 9.1

Using the local coordinates in $U_{Q}$, fixed small $\delta>0$ consider the
neighbourhood

$U_{X}(\delta)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}[-\delta,\delta]\times[1-\delta,1+\delta]\times[-\delta,\delta]$

of the heteroclinic point $X=(0,1,0)$ and the two-disc (see Figure 8)

$S(\delta)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}[-\delta,\delta]\times\\{{1-\delta}\\}\times[-\delta,\delta]\subset\partial
U_{X}(\delta).$

For large $k>0$ and $\mu\in(-10,-9)$ define the set

$\widehat{S}_{\bar{\upsilon}_{k}(\mu)}(\delta)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}C\Big{(}f^{-n_{k}}_{\bar{\upsilon}_{k}(\mu)}(0,1-\delta,0),f^{-n_{k}}_{\bar{\upsilon}_{k}(\mu)}(S(\delta))\cap
U_{Q}\Big{)},$

recall that $C(x,A)$ is the connected component of the set $A$ containing the
point $x$. We finally let

$\widetilde{S}_{\bar{\upsilon}_{k}(\mu)}(\delta)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\widehat{S}_{\bar{\upsilon}_{k}(\mu)}(\delta)\cap
U_{\widetilde{Y}}\quad\mbox{and}\quad
S_{\bar{\upsilon}_{k}(\mu)}(\delta)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\Phi_{k}^{-1}(\widetilde{S}_{\bar{\upsilon}_{k}(\mu)}(\delta))\subset\mathbb{R}^{3}.$

######  Lemma 9.5.

For every $\delta>0$ there is $k_{0}$ such that for every $k\geqslant k_{0}$
and every $\mu\in(-10,-9)$ it holds

$W^{{\mathrm{u}}{\mathrm{u}}}(P_{\bar{\upsilon}_{k}(\mu)}^{+},F_{\bar{\upsilon}_{k}(\mu)})\pitchfork
S_{\bar{\upsilon}_{k}(\mu)}(\delta)\neq\emptyset.$

###### Proof.

Let $C>0$ be the constant in Lemma 9.4 and take large $K>0$ with
$C\,K=\tau>1$. Applying Lemma 9.3 to $K$, we get $k_{0}$ such that the angles
between the lines parallel to the $x$-axis in $\mathbb{R}^{2}$ and
$\pi_{1,2}(\widetilde{\sigma}^{{\mathrm{u}}{\mathrm{u}}}_{K,\bar{\upsilon}_{k}(\mu)})$
are at least $\arctan(K)$. Thus, after increasing $k_{0}$, we get the angular
condition (9.20) in Lemma 9.4. Hence

$\mathrm{lenght}\big{(}F_{\bar{\upsilon}_{k}(\mu)}(\widetilde{\sigma}^{{\mathrm{u}}{\mathrm{u}}}_{K,\bar{\upsilon}_{k}(\mu)})\big{)}\geqslant\tau\,\mathrm{lenght}\big{(}\widetilde{\sigma}^{{\mathrm{u}}{\mathrm{u}}}_{K,\bar{\upsilon}_{k}(\mu)}\big{)}.$

The curve
$F_{\bar{\upsilon}_{k}(\mu)}(\widetilde{\sigma}^{{\mathrm{u}}{\mathrm{u}}}_{K,\bar{\upsilon}_{k}(\mu)})$
is contained in the strong unstable separatrix
$\sigma_{\bar{\upsilon}_{k}(\mu)}^{{\mathrm{u}}{\mathrm{u}}}$ of
$W^{{\mathrm{u}}{\mathrm{u}}}(P^{+}_{{\bar{\upsilon}_{k}(\mu)}})$. By
construction, the curve
$F_{\bar{\upsilon}_{k}(\mu)}(\widetilde{\sigma}^{{\mathrm{u}}{\mathrm{u}}}_{K,\bar{\upsilon}_{k}(\mu)})$
also satisfies the angular condition (9.20). The proof now follows
inductively: the curves
$F^{n}_{k}(\widetilde{\sigma}^{{\mathrm{u}}{\mathrm{u}}}_{K,\bar{\upsilon}_{k}(\mu)})$
are contained in $\sigma_{\bar{\upsilon}_{k}(\mu)}^{{\mathrm{u}}{\mathrm{u}}}$
and their lengths grow exponentially. This implies that this separatrix
transversally intersects the two-disc $S_{\bar{\upsilon}_{k}(\mu)}(\delta)$. ∎

Note that, for every sufficiently large $k$, there are defined the
continuations
$Z^{\pm}_{\bar{\upsilon}_{k}(\mu)}=Z^{\pm}_{\varepsilon,\bar{\upsilon}_{k}(\mu)}$
of the transverse homoclinic point $Z^{\pm}_{\varepsilon}$ of $Q$ in
Proposition 4.1. These points are transverse homoclinic points of $Q$ for
$f_{\bar{\upsilon}_{k}(\mu)}$ and (in the coordinates in $U_{Q}$) are of the
form

$Z^{\pm}_{\bar{\upsilon}_{k}(\mu)}=(0,1\pm\zeta^{\pm}_{\bar{\upsilon}_{k}(\mu)},0),\qquad\zeta^{\pm}_{\bar{\upsilon}_{k}(\mu)}\in(0,\delta).$

For each $k$ and $\mu$, we can also consider a disc
$W^{\pm}_{\bar{\upsilon}_{k}(\mu)}$ in
$W^{\mathrm{s}}(Q,f_{\bar{\upsilon}_{k}(\mu)})$ centred at
$Z^{\pm}_{\bar{\upsilon}_{k}(\mu)}$ which has uniform size and is uniformly
transverse to $W^{\mathrm{u}}_{\mathrm{loc}}(Q,f_{\bar{\upsilon}_{k}(\mu)})$,
meaning that the angles between $W^{\pm}_{\bar{\upsilon}_{k}(\mu)}$ and
$W^{\mathrm{u}}_{\mathrm{loc}}(Q,f_{\bar{\upsilon}_{k}(\mu)})$ at
$Z^{\pm}_{\bar{\upsilon}_{k}(\mu)}$ are uniformly bounded from below.

\begin{overpic}[scale={0.1}]{2-D.jpg}
\put(210.0,163.0){\Large{$\widetilde{\gamma}^{{\mathrm{u}}{\mathrm{u}}}_{\bar{\upsilon}_{k}(\mu)}$}}
\put(42.0,153.0){\Large{$W^{{\mathrm{s}}}(Q)$}} \put(121.5,95.0){$\bullet$}
\put(125.0,85.0){\Large$Z^{-}_{\bar{\upsilon}_{k}(\mu)}$}
\put(121.0,136.0){$\bullet$} \put(105.0,132.0){\Large$X$}
\put(32.0,103.0){\Large$S(\delta)$} \end{overpic} Figure 8. Two-dimensional
connection between the blender and the saddle $Q$.

We are now ready to conclude the proof of Proposition 9.1. Let
$\gamma^{{\mathrm{u}}{\mathrm{u}}}_{\bar{\upsilon}_{k}(\mu)}$ be the segment
of $\sigma^{{\mathrm{u}}{\mathrm{u}}}_{\bar{\upsilon}_{k}(\mu)}$ joining
$P_{\bar{\upsilon}_{k}(\mu)}^{+}$ and $S_{\bar{\upsilon}_{k}(\mu)}(\delta)$
and consider (see Figure 8)

$\widetilde{\gamma}^{{\mathrm{u}}{\mathrm{u}}}_{\bar{\upsilon}_{k}(\mu)}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}f^{n_{k}}_{\bar{\upsilon}_{k}(\mu)}(\Phi_{k}(\gamma^{{\mathrm{u}}{\mathrm{u}}}_{\bar{\upsilon}_{k}(\mu)})).$

Consider the domain $\Delta_{k}(\mu)$ of the blender
$\Upsilon_{\bar{\upsilon}_{k}(\mu)}$ in (7.3). The calculations in [16, Step
A, eq. (35)] imply that for every large $k$ the coordinates $(x,1+y,z)\in
U_{Q}$ of the points in $f^{n_{k}}_{\bar{\upsilon}_{k}(\mu)}(\Delta_{k}(\mu))$
are close to $X=(0,1,0)$ and they have Landau symbols

$x=O({\lambda_{Q}}^{n_{k}}),\quad
y=O(\sigma_{P}^{-2m_{k}}\,{\sigma_{Q}}^{-n_{k}}),\quad{z}=O({\lambda_{Q}}^{n_{k}}).$

Hence $f^{n_{k}}_{\bar{\upsilon}_{k}(\mu)}(P^{+}_{\bar{\upsilon}_{k}(\mu)})$
converges to $X$ as $k\to\infty$. It follows now from Lemma 9.5 that the
curves
$\widetilde{\gamma}^{{\mathrm{u}}{\mathrm{u}}}_{\bar{\upsilon}_{k}(\mu)}$
accumulate to the disc

$\\{0\\}\times[1-\delta,1]\times\\{0\\}\subset
W^{{\mathrm{u}}}_{{\mathrm{loc}}}(Q,f^{n_{k}}_{\bar{\upsilon}_{k}(\mu)}).$

This implies that for sufficiently large $k$ the curve
$\widetilde{\gamma}^{{\mathrm{u}}{\mathrm{u}}}_{\bar{\upsilon}_{k}(\mu)}$
transversely intersects the disc
$W^{-}_{\varepsilon,\bar{\upsilon}_{k}(\mu)}\subset
W^{{\mathrm{s}}}_{{\mathrm{loc}}}(Q,f_{\bar{\upsilon}_{k}(\mu)})$. As the
curve
$\widetilde{\gamma}^{{\mathrm{u}}{\mathrm{u}}}_{\bar{\upsilon}_{k}(\mu)}$ is
contained in the unstable manifold of the blender this ends the proof of
Proposition 9.1.

##  10\. Proof of Theorem 1.1: Intersections between one-dimensional manifolds

We now prove that the stable manifold the blender
$\Upsilon_{\varepsilon,k,\mu}$ of $g_{\varepsilon,k,\mu}$ robustly intersects
the unstable manifold of the saddle $Q$.

######  Proposition 10.1.

For every small $\varepsilon>0$, large $k$, and $\mu\in(-10,-9)$, there is a
$C^{r}$ neighbourhood $\mathcal{V}_{\varepsilon,k,\mu}$ of
$g_{\varepsilon,k,\mu}$ consisting of diffeomorphisms $g$ such that

$W^{\mathrm{s}}\big{(}{\Upsilon}_{g},g\big{)}\cap
W^{\mathrm{u}}\big{(}Q_{g},g\big{)}\neq\emptyset,$

where $\Upsilon_{g}$ and $Q_{g}$ are the continuations of
$\Upsilon_{\varepsilon,k,\mu}$ and $Q$. Moreover, this intersection can be
chosen quasi-transverse.

By Lemmas 5.1 and 5.2 this proposition follows from the next result:

######  Lemma 10.1.

For every small $\varepsilon>0$, large $k$, and $\mu\in(-10,-9)$, the unstable
manifold $W^{\mathrm{u}}(Q,g_{\varepsilon,k,\mu})$ contains a
${\mathrm{u}}{\mathrm{u}}$-disc in the superposition region of
$\Upsilon_{\varepsilon,k,\mu}$.

###  10.1. Proof of Lemma 10.1

We import some ingredients from Section 5.3. Consider the disc $L$ in (5.9).
By Remark 7.4, for sufficiently large $k$ the set

$\Phi_{k}^{-1}\circ\mathcal{R}_{\varepsilon,k,\mu}(g_{\varepsilon,k,\mu})\circ\Phi_{k}\big{(}L\big{)}\subset\mathbb{R}^{3}$

contains a disc in the superposition region of the blender
$\Lambda_{\varepsilon,k,\mu}$ in (7.5).

Take small $\delta=\delta(\varepsilon)>0$ and the segment
$L^{\mathrm{u}}_{1,\varepsilon}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}L^{\mathrm{u}}_{1,\varepsilon}(\delta)\subset
W^{\mathrm{u}}(Q,f_{\varepsilon})$ in (4.12). By the definition of
$g_{\varepsilon,k,\mu}$, this disc is contained in
$W^{\mathrm{u}}(Q,g_{\varepsilon,k,\mu})$. Thus

(10.1)
$J^{\mathrm{u}}_{\varepsilon,k,\mu}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}g_{\varepsilon,k,\mu}^{N_{1}}\big{(}L^{\mathrm{u}}_{1,\varepsilon}\big{)}\subset
W^{\mathrm{u}}(Q,g_{\varepsilon,k,\mu}).$

Note that the transition $g_{\varepsilon,k,\mu}^{N_{1}}$ does not depend on
$\mu$, thus in what follows we will omit the dependence on $\mu$ of the sets
$J^{\mathrm{u}}_{\varepsilon,k,\mu}$ writing just
$J^{\mathrm{u}}_{\varepsilon,k}$. We will prove that there is a compact
subdisc $\widehat{J}^{\mathrm{u}}_{\varepsilon,k}$ of
$J^{\mathrm{u}}_{\varepsilon,k}$ such that the $C^{r}$ distance between the
discs

$\Phi_{k}^{-1}\circ
g_{\varepsilon,k,\mu}^{N_{2}+m_{k}}(\widehat{J}^{\mathrm{u}}_{\varepsilon,k})\quad\mbox{and}\quad\Phi_{k}^{-1}\circ\mathcal{R}_{\varepsilon,k,\mu}(g_{\varepsilon,k,\mu})\circ\Phi_{k}(L)$

goes to zero as $k\to\infty$. As the latter set contains a disc in the
superposition region of the blender, this implies the lemma (see Figure 9). We
now go to the details of the proof.

\begin{overpic}[scale={0.1}]{1-D.jpg} \put(171.0,73.0){\Large{$\Delta$}}
\put(100.0,105.0){\Large{$L$}} \put(70.0,40.0){$\bullet$}
\put(160.0,103.0){$\bullet$}
\put(110.0,183.0){\large$\Phi_{k}^{-1}\circ\mathcal{R}_{\varepsilon,k,\mu}(g_{\varepsilon,k,\mu})\circ\Phi_{k}(L)$}
\put(-10.0,180.0){\large$\Phi_{k}^{-1}\circ
g_{\varepsilon,k,\mu}^{N_{2}+m_{k}}(\widehat{J}^{\mathrm{u}}_{\varepsilon,k})$}
\end{overpic} Figure 9. One-dimensional connection between the blender and the
saddle $Q$.

Recall the definition of $\Phi_{k}=\Psi_{k}\circ\Theta_{\bar{\varsigma}}$ in
(7.2) and consider the parameterisation of $\Phi_{k}(L)$ given by

$\gamma_{k}:[-4,4]\to
M,\quad\gamma_{k}(t)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\Psi_{k}\circ\Theta_{\bar{\varsigma}}(0,t,0)=\Psi_{k}(0,\varsigma_{2}^{-1}t,0).$

We will provide a parameterisation $\gamma_{\varepsilon,k}:[-4,4]\to M$ of
$\widehat{J}_{\varepsilon,k}^{\mathrm{u}}$ such that

(10.2) $\lim_{k\to\infty}\,\big{\|}\big{(}\Phi_{k}^{-1}\circ
g_{\varepsilon,k,\mu}^{N_{2}+m_{k}+N_{1}+n_{k}}\circ\gamma_{k}\,-\,\Phi_{k}^{-1}\circ
g_{\varepsilon,k,\mu}^{N_{2}+m_{k}}\circ\gamma_{\varepsilon,k}\big{)}|_{[-4,4]}\big{\|}_{r}=0.$

As the set $\Psi_{k}^{-1}\circ
g_{\varepsilon,k,\mu}^{N_{2}+m_{k}+N_{1}+n_{k}}\circ\gamma_{k}([-4,4])$
contains a disc in the superposition region of the blenders (see Remark 7.4)
this proves Lemma 10.1.

To get equation (10.2) we observe that

$\begin{split}\big{\|}\,&\Phi_{k}^{-1}\circ
g_{\varepsilon,k,\mu}^{N_{2}+m_{k}+N_{1}+n_{k}}\circ\gamma_{k}\,-\,\Phi_{k}^{-1}\circ
g_{\varepsilon,k,\mu}^{N_{2}+m_{k}}\circ{\gamma}_{\varepsilon,k}\,\big{\|}_{r}=\\\
&\,\,\big{\|}\,{\Theta}_{\bar{\varsigma}}^{-1}\circ\Psi_{k}^{-1}\circ
g_{\varepsilon,k,\mu}^{N_{2}+m_{k}+N_{1}+n_{k}}\circ\gamma_{k}\,-\,{\Theta}_{\bar{\varsigma}}^{-1}\circ\Psi_{k}^{-1}\circ
g_{\varepsilon,k,\mu}^{N_{2}+m_{k}}\circ\gamma_{\varepsilon,k}\,\big{\|}_{r}\leqslant\\\
&\,\,\,C\,\big{\|}\,\Psi_{k}^{-1}\circ
g_{\varepsilon,k,\mu}^{N_{2}+m_{k}+N_{1}+n_{k}}\circ\gamma_{k}\,-\,\Psi_{k}^{-1}\circ
g_{\varepsilon,k,\mu}^{N_{2}+m_{k}}\circ\gamma_{\varepsilon,k}\,\big{\|}_{r},\end{split}$

where $C$ is an upper bound of the $C^{r}$ norm of
${\Theta}_{\bar{\varsigma}}^{-1}$ in the cube $\Delta$. Thus to conclude the
proof of the proposition it is enough to prove the following:

######  Lemma 10.2.

There is a parameterisation ${\gamma}_{\varepsilon,k}:[-4,4]\to M$ of
$\widehat{J}_{\varepsilon,k}^{\mathrm{u}}$ such that

$\lim_{k\to\infty}\,\big{\|}\big{(}\Psi_{k}^{-1}\circ
g_{\varepsilon,k,\mu}^{N_{2}+m_{k}+N_{1}+n_{k}}\circ\gamma_{k}\,-\,\Psi_{k}^{-1}\circ
g_{\varepsilon,k,\mu}^{N_{2}+m_{k}}\circ\gamma_{\varepsilon,k}\big{)}|_{[-4,4]}\big{\|}_{r}=0.$

We begin by giving the explicit parameterisation of $\gamma_{\varepsilon,k}$
of $\widehat{J}_{\varepsilon,k}^{\mathrm{u}}$.

####  10.1.1. The parameterisations $\gamma_{\varepsilon,k}$

We first write the segment $J^{\mathrm{u}}_{\varepsilon,k}$ in (10.1) in local
coordinates. Recalling the definitions of $L^{\mathrm{u}}_{1,\varepsilon}$ in
(4.10),
$\mathfrak{T}_{1,1,\varepsilon}=f_{\varepsilon}^{N_{1}}|_{U_{1,\varepsilon}}$
in (4.8), and $\theta_{\varepsilon,k}$ in (7.8) (where $\bar{\tau}_{k}$ is
defined), we have

$\begin{split}&J^{\mathrm{u}}_{\varepsilon,k}=\big{\\{}\widetilde{X}_{1,\varepsilon}+A\big{(}t\,\textbf{e}_{2}+\bar{\rho}_{1,\varepsilon}(t)\big{)}+\widetilde{H}^{1}_{\varepsilon}\big{(}t\,\textbf{e}_{2}+\bar{\rho}_{1,\varepsilon}(t)\big{)}+\\\
&\qquad+\Pi_{\delta}\big{(}\,A\big{(}t\,\textbf{e}_{2}+\bar{\rho}_{1,\varepsilon}(t)\big{)}+\widetilde{H}^{1}_{\varepsilon}\big{(}t\,\textbf{e}_{2}+\bar{\rho}_{1,\varepsilon}(t)\big{)}\,\big{)}\bar{\tau}_{k}:|t|<\delta=\delta(\varepsilon)\big{\\}}.\end{split}$

Using (4.11), (4.9), $\bar{\rho}_{1,\varepsilon}(0)=0$, and Remark 4.5, we get
$\widehat{\delta}\in(0,\delta)$ such that for every $|t|<\widehat{\delta}$ it
holds

$A\big{(}t\,\textbf{e}_{2}+\bar{\rho}_{1,\varepsilon}(t)\big{)}+\widetilde{H}^{1}_{\varepsilon}\big{(}t\,\textbf{e}_{2}+\bar{\rho}_{1,\varepsilon}(t))\in
B(\mathbf{0},\delta).$

As the map $\Pi_{\delta}$ is igual to $1$ in $B(\mathbf{0},\delta)$ (recall
(3.2)), we can consider the subdisc

$\widetilde{J}^{\mathrm{u}}_{\varepsilon,k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{\\{}\widetilde{X}_{1,\varepsilon}+A\big{(}t\,\textbf{e}_{2}+\bar{\rho}_{1,\varepsilon}(t)\big{)}+\widetilde{H}^{1}_{\varepsilon}\big{(}t\,\textbf{e}_{2}+\bar{\rho}_{1,\varepsilon}(t)\big{)}+\bar{\tau}_{k}:|t|\leqslant\widehat{\delta}\,\big{\\}}\subset
J^{\mathrm{u}}_{\varepsilon,k}.$

For convenience, we write $\widetilde{J}^{\mathrm{u}}_{\varepsilon,k}$ in the
following compact form:

$\widetilde{J}^{\mathrm{u}}_{\varepsilon,k}=\big{\\{}\widetilde{X}_{1,\varepsilon}+t\,A(\mathbf{v}_{1,\varepsilon})+\widetilde{\rho}_{1,\varepsilon}(t)+\bar{\tau}_{k}:|t|\leqslant\widehat{\delta}\,\big{\\}},$

where

$\begin{split}\mathbf{v}_{1,\varepsilon}&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\textbf{e}_{2}+A^{-1}D\widetilde{H}^{1}_{\varepsilon}(\textbf{0})\,\textbf{e}_{2},\\\
\widetilde{\rho}_{1,\varepsilon}(t)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}A\big{(}\bar{\rho}_{1,\varepsilon}(t)\big{)}+\widetilde{H}^{1}_{\varepsilon}\big{(}t\,\textbf{e}_{2}+\bar{\rho}_{1,\varepsilon}(t)\big{)}-t\,D\widetilde{H}^{1}_{\varepsilon}(\textbf{0})\,\textbf{e}_{2}.\end{split}$

Note that, by (4.11), we have that

$\frac{d}{dt}(\widetilde{\rho}_{1,\varepsilon})(0)=\widetilde{\rho}_{1,\varepsilon}(0)=\textbf{0}.$

We now consider the subdisc $\widehat{J}^{\mathrm{u}}_{\varepsilon,k}$ of
$\widetilde{J}^{\mathrm{u}}_{\varepsilon,k}$ obtained by rescaling the
parameter $t$ by the factor
$\sigma_{P}^{-2m_{k}}{\sigma_{Q}}^{-n_{k}}\,|\,\varsigma_{2}^{-1}|\ll\frac{\widehat{\delta}}{4}<1$
as follows:

$\widehat{J}^{\mathrm{u}}_{\varepsilon,k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{\\{}\widetilde{X}_{1,\varepsilon}+\sigma_{P}^{-2m_{k}}{\sigma_{Q}}^{-n_{k}}\,\varsigma_{2}^{-1}\,t\,A(\mathbf{v}_{1,\varepsilon})+\widehat{\rho}_{1,\varepsilon,k}(t)+\bar{\tau}_{k}:|t|\leqslant
4\big{\\}},$

where

$\widehat{\rho}_{1,\varepsilon,k}(t)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\widetilde{\rho}_{1,\varepsilon}\big{(}\sigma_{P}^{-2m_{k}}{\sigma_{Q}}^{-n_{k}}\,\varsigma_{2}^{-1}\,t\big{)}.$

To rewrite the set $\widehat{J}^{\mathrm{u}}_{\varepsilon,k}$ in a compact
form, let

(10.3)
$\begin{split}(\widetilde{w}^{1,\varepsilon}_{1},\widetilde{w}^{1,\varepsilon}_{2},\widetilde{w}^{1,\varepsilon}_{3})&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}D\widetilde{H}^{1}_{\varepsilon}(\textbf{0})\,\textbf{e}_{2},\\\
\widehat{\rho}^{\ell}_{1,\varepsilon,k}(t)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\widetilde{\rho}^{\ell}_{1,\varepsilon}\big{(}\sigma_{P}^{-2m_{k}}{\sigma_{Q}}^{-n_{k}}\,\varsigma_{2}^{-1}\,t\big{)},\qquad\ell=1,2,3,\end{split}$

where $\widetilde{\rho}^{\ell}_{1,\varepsilon}$ is the $\ell$-th coordinate of
$\widetilde{\rho}_{1,\varepsilon}$.

######  Remark 10.2.

Remark 4.5 implies that
$(\widetilde{w}^{1,\varepsilon}_{1},\widetilde{w}^{1,\varepsilon}_{2},\widetilde{w}^{1,\varepsilon}_{3})\to(0,0,0)$
as $\varepsilon\to 0$.

Recalling that
$\widetilde{X}_{1,\varepsilon}=(1+\widetilde{x}_{1,\varepsilon},0,0)$, see
(4.6) and (4.7), and the definitions of $A$ in (2.4) and of $\bar{\tau}_{k}$
in (7.7), we can write

$\widehat{J}^{\mathrm{u}}_{\varepsilon,k}=\big{\\{}\big{(}1+\widetilde{x}_{1,\varepsilon}+{x}_{\varepsilon,k}(t),{y}_{\varepsilon,k}(t),{z}_{\varepsilon,k}(t)\big{)}:|t|\leqslant
4\big{\\}},$

where

(10.4)
$\begin{split}{x}_{\varepsilon,k}(t)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\sigma_{P}^{-2m_{k}}{\sigma_{Q}}^{-n_{k}}\,\big{(}\alpha_{2}+\widetilde{w}^{1,\varepsilon}_{1})\,\varsigma_{2}^{-1}\,t+\widehat{\rho}^{1}_{1,\varepsilon,k}(t),\\\
{y}_{\varepsilon,k}(t)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\sigma_{P}^{-2m_{k}}{\sigma_{Q}}^{-n_{k}}\,\big{(}\beta_{2}+\widetilde{w}^{1,\varepsilon}_{2}\big{)}\,\varsigma_{2}^{-1}\,t+\widehat{\rho}^{2}_{1,\varepsilon,k}(t)\\\
&\,\,\,\,\,\,\,\,+\sigma^{-m_{k}}_{P}(\tilde{\mathfrak{c}}_{k}+\tilde{\mathfrak{s}}_{k}),\\\
{z}_{\varepsilon,k}(t)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\sigma_{P}^{-2m_{k}}{\sigma_{Q}}^{-n_{k}}\,\widetilde{w}^{1,\varepsilon}_{3}\,\varsigma_{2}^{-1}\,t+\widehat{\rho}^{3}_{1,\varepsilon,k}(t)+\sigma^{-m_{k}}_{P}(\tilde{\mathfrak{c}}_{k}-\tilde{\mathfrak{s}}_{k}).\end{split}$

The announced parameterisation of $\widehat{J}_{\varepsilon,k}^{\mathrm{u}}$
is given by

(10.5) $\gamma_{\varepsilon,k}:[-4,4]\to
M,\quad\gamma_{\varepsilon,k}(t)=(1+\widetilde{x}_{1,\varepsilon}+x_{\varepsilon,k}(t),y_{\varepsilon,k}(t),z_{\varepsilon,k}(t)).$

####  10.1.2. End of the proof of Lemma 10.2 (thus of Proposition 10.1)

We calculate separately the two terms in the lemma. This involves some
explicit calculations in the renormalisation scheme borrowed from [16] which
are stated in Section 13.

$\bullet$ The term $\Psi_{k}^{-1}\circ
g_{\varepsilon,k,\mu}^{N_{2}+m_{k}+N_{1}+n_{k}}\big{(}\gamma_{k}(t)\big{)}$.
Write

(10.6)
$\begin{split}(\overline{x}_{\varepsilon,k,\mu}(t),\overline{y}_{\varepsilon,k,\mu}(t),\overline{z}_{\varepsilon,k,\mu}(t))&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\Psi_{k}^{-1}\circ
g_{\varepsilon,k,\mu}^{N_{2}+m_{k}+N_{1}+n_{k}}\big{(}\gamma_{k}(t)\big{)}\\\
&=\Psi^{-1}_{k}\circ
g_{\varepsilon,k,\mu}^{N_{2}+m_{k}+N_{1}+n_{k}}\circ\Psi_{k}(\Theta_{\bar{\varsigma}}(0,t,0)).\end{split}$

Using the formula in (13.1), replacing $f_{\bar{\upsilon}_{k}(\mu),\rho}$ by
$g_{\varepsilon,k,\mu}=\theta_{\varepsilon,k}\circ
f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}$ (this leads to a
dependence on $\varepsilon$ of the next expressions), and considering the
curve $\Theta_{\bar{\varsigma}}(0,t,0)=(0,\varsigma_{2}^{-1}t,0)$ (see (5.5)),
from equation (13.3) we get

$\begin{split}\overline{x}_{\varepsilon,k,\mu}(t)&=\Big{(}a_{1}\,\lambda_{P}^{m_{k}}\,\sigma_{P}^{-m_{k}}\,\alpha_{2}+\big{(}\tilde{\mathfrak{c}}_{k}\,a_{2}+\tilde{\mathfrak{s}}_{k}\,a_{3}\big{)}\,\beta_{2}\Big{)}\varsigma_{2}^{-1}\,t+{\sigma_{P}}^{m_{k}}\,\sigma_{Q}^{n_{k}}\,\mathrm{hot}^{x}_{\varepsilon,k,\mu}(t),\\\
\overline{y}_{\varepsilon,k,\mu}(t)&=\varsigma_{2}^{-1}\,\mu+b_{1}\,\lambda_{P}^{m_{k}}\,{\sigma_{Q}}^{n_{k}}\,\alpha_{2}\,\varsigma_{2}^{-1}\,t+\Big{(}\tilde{\mathfrak{c}}_{k}^{2}\,b_{2}+\tilde{\mathfrak{s}}_{k}^{2}\,b_{3}+\tilde{\mathfrak{c}}_{k}\,\tilde{\mathfrak{s}}_{k}\,b_{4}\Big{)}\,\beta_{2}^{2}\,\varsigma_{2}^{-2}\,t^{2}+\\\
&\qquad+{\sigma_{P}}^{2m_{k}}\,\sigma_{Q}^{2n_{k}}\,\mathrm{hot}^{y}_{\varepsilon,k,\mu}(t),\\\
\overline{z}_{\varepsilon,k,\mu}(t)&=\Big{(}c_{1}\,\lambda_{P}^{m_{k}}\,\sigma_{P}^{-m_{k}}\,\alpha_{2}+\big{(}\tilde{\mathfrak{c}}_{k}\,c_{2}+\tilde{\mathfrak{s}}_{k}\,c_{3}\big{)}\,\beta_{2}\Big{)}\,\varsigma_{2}^{-1}\,t+{\sigma_{P}}^{m_{k}}\,\sigma_{Q}^{n_{k}}\,\mathrm{hot}^{z}_{\varepsilon,k,\mu}(t),\end{split}$

where $\mathrm{hot}^{\ast}_{\varepsilon,k,\mu}(t)$, $\ast=x,y,z$, are high
order terms101010In [16] these high order terms are denoted by
$\mathrm{h.o.t.}^{*}$, $\mathrm{h.o.t.}^{**}$, $\mathrm{h.o.t.}^{***}$..

######  Remark 10.3 (Lemma 8.3 in [16]).

The terms

${\sigma_{P}}^{m_{k}}\,\sigma_{Q}^{n_{k}}\,\mathrm{hot}^{x}_{\varepsilon,k,\mu}(t),\quad{\sigma_{P}}^{2m_{k}}\,\sigma_{Q}^{2n_{k}}\,\mathrm{hot}^{y}_{\varepsilon,k,\mu}(t),\quad{\sigma_{P}}^{m_{k}}\,\sigma_{Q}^{n_{k}}\,\mathrm{hot}^{z}_{\varepsilon,k,\mu}(t),$

go to zero in the $C^{r}$ topology as $k$ goes to infinity.

$\bullet$ The term $\Psi_{k}^{-1}\circ
g_{\varepsilon,k,\mu}^{N_{2}+m_{k}}\big{(}{\gamma}_{\varepsilon,k}(t)\big{)}$.

Recalling the parameterisation of ${\gamma}_{\varepsilon,k}(t)$ in (10.5),
write

(10.7)
$\begin{split}&(\widetilde{x}_{\varepsilon,k,\mu}(t),\widetilde{y}_{\varepsilon,k,\mu}(t),\widetilde{z}_{\varepsilon,k,\mu}(t))\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\Psi_{k}^{-1}\circ
g_{\varepsilon,k,\mu}^{N_{2}+m_{k}}\big{(}{\gamma}_{\varepsilon,k}(t)\big{)}\\\
&\quad\quad\quad=\Psi^{-1}_{k}\circ
g_{\varepsilon,k,\mu}^{N_{2}+m_{k}}\big{(}1+\widetilde{x}_{1,\varepsilon}+{x}_{\varepsilon,k}(t),{y}_{\varepsilon,k}(t),{z}_{\varepsilon,k}(t)\big{)}.\end{split}$

Using equation (10.4) and the linearity of $g_{\varepsilon,k,\mu}$ in $U_{P}$,
we get

$g_{\varepsilon,k,\mu}^{m_{k}}\big{(}1+\widetilde{x}_{1,\varepsilon}+{x}_{\varepsilon,k}(t),{y}_{\varepsilon,k}(t),{z}_{\varepsilon,k}(t)\big{)}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{(}\widehat{x}_{\varepsilon,k}(t),1+\widehat{y}_{\varepsilon,k}(t),1+\widehat{z}_{\varepsilon,k}(t)\big{)},$

where

$\begin{split}\widehat{x}_{\varepsilon,k}(t)&={\lambda_{P}}^{m_{k}}\,(1+\widetilde{x}_{1,\varepsilon})+{\lambda_{P}}^{m_{k}}\sigma_{P}^{-2m_{k}}{\sigma_{Q}}^{-n_{k}}\,\big{(}\alpha_{2}+\widetilde{w}^{1,\varepsilon}_{1}\big{)}\,\varsigma_{2}^{-1}\,t\\\
&\,\,\,\,\,\,\,\,+{\lambda_{P}}^{m_{k}}\widehat{\rho}^{1}_{1,\varepsilon,k}(t),\\\
\widehat{y}_{\varepsilon,k}(t)&=\big{(}\tilde{\mathfrak{c}}_{k}(\beta_{2}+\widetilde{w}^{1,\varepsilon}_{2})-\widetilde{w}^{1,\varepsilon}_{3}\,\tilde{\mathfrak{s}}_{k}\big{)}\,\sigma_{P}^{-m_{k}}\,{\sigma_{Q}}^{-n_{k}}\,\varsigma_{2}^{-1}\,t+{\sigma_{P}}^{m_{k}}\,u_{\varepsilon,k}(t),\\\
\widehat{z}_{\varepsilon,k}(t)&=\big{(}\tilde{\mathfrak{s}}_{k}(\beta_{2}+\widetilde{w}^{1,\varepsilon}_{2})+\widetilde{w}^{1,\varepsilon}_{3}\,\tilde{\mathfrak{c}}_{k}\big{)}\,\sigma_{P}^{-m_{k}}\,{\sigma_{Q}}^{-n_{k}}\,\varsigma_{2}^{-1}\,t+{\sigma_{P}}^{m_{k}}\,v_{\varepsilon,k}(t),\end{split}$

with

(10.8)
$\begin{split}u_{\varepsilon,k}(t)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\tilde{\mathfrak{c}}_{k}\,\widehat{\rho}^{2}_{1,\varepsilon,k}(t)-\tilde{\mathfrak{s}}_{k}\,\widehat{\rho}^{3}_{1,\varepsilon,k}(t),\\\
v_{\varepsilon,k}(t)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\tilde{\mathfrak{s}}_{k}\,\widehat{\rho}^{2}_{1,\varepsilon,k}(t)+\tilde{\mathfrak{c}}_{k}\,\widehat{\rho}^{3}_{1,\varepsilon,k}(t).\end{split}$

Recalling the expressions of $\widehat{\rho}^{2}_{1,\varepsilon,k}(t)$,
$\widehat{\rho}^{3}_{1,\varepsilon,k}(t)$ in (10.3), for $i=2,3$ we get

(10.9)
$O(u_{\varepsilon,k}(t))=O(v_{\varepsilon,k}(t))=O(\widehat{\rho}^{i}_{1,\varepsilon,k}(t))=O(\sigma_{P}^{-4m}{\sigma_{Q}}^{-2n_{k}}).$

Thus the Landau symbols of
$\widehat{x}_{\varepsilon,k}(t),\widehat{y}_{\varepsilon,k}(t)$, and
$\widehat{z}_{\varepsilon,k}(t)$ are of the form

$\widehat{x}_{\varepsilon,k}(t)=O({\lambda_{P}}^{m_{k}}),\qquad\widehat{y}_{\varepsilon,k}(t)=O(\sigma_{P}^{-m_{k}}{\sigma_{Q}}^{-n_{k}})=\widehat{z}_{\varepsilon,k}(t).$

Note that

$(\widetilde{x}_{\varepsilon,k,\mu}(t),\widetilde{y}_{\varepsilon,k,\mu}(t),\widetilde{z}_{\varepsilon,k,\mu}(t))=\Psi_{k}^{-1}\circ
g^{N_{2}}_{\varepsilon,k,\mu}\big{(}\widehat{x}_{\varepsilon,k}(t),1+\widehat{y}_{\varepsilon,k}(t),1+\widehat{z}_{\varepsilon,k}(t)\big{)}.$

Recalling the definitions of $\Psi_{k}$ in (6.9) and of
$f_{\bar{\upsilon}_{k}(\mu)}^{N_{2}}$ (see Remark 6.3) and that
$g_{\varepsilon,k,\mu}^{N_{2}}=f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}^{N_{2}}=f_{\bar{\upsilon}_{k}(\mu)}^{N_{2}}$
in the neighbourhood
$f_{\bar{\upsilon}_{k}(\mu)}^{-N_{2}}(B(\widetilde{Y},\rho(\varepsilon)))$ of
$Y$ that we are considering, we get

$\begin{split}\widetilde{x}_{\varepsilon,k,\mu}(t)&=a_{1}\,{\lambda_{P}}^{m_{k}}\sigma_{P}^{m_{k}}{{\sigma_{Q}}}^{n_{k}}\,\widetilde{x}_{1,\varepsilon}\\\
&\,\,\,\,\,\,+\Big{(}a_{1}\,{\lambda_{P}}^{m_{k}}{\sigma_{P}}^{-m_{k}}\,(\alpha_{2}+\widetilde{w}^{1,\varepsilon}_{1})+a_{2}\big{(}\tilde{\mathfrak{c}}_{k}\,(\beta_{2}+\widetilde{w}^{1,\varepsilon}_{2})-\widetilde{w}^{1,\varepsilon}_{3}\,\tilde{\mathfrak{s}}_{k}\big{)}\\\
&\,\,\,\,\,\,+a_{3}\big{(}\tilde{\mathfrak{s}}_{k}\,(\beta_{2}+\widetilde{w}^{1,\varepsilon}_{2})+\widetilde{w}^{1,\varepsilon}_{3}\,\tilde{\mathfrak{c}}_{k}\big{)}\Big{)}\,\,\varsigma_{2}^{-1}\,t+{\mathrm{Hot}}^{x}_{\varepsilon,k,\mu}(t);\\\
\widetilde{y}_{\varepsilon,k,\mu}(t)&=\varsigma_{2}^{-1}\,\mu+b_{1}\,{\lambda_{P}}^{m_{k}}\sigma_{P}^{2m_{k}}{{\sigma_{Q}}}^{2n_{k}}\,\widetilde{x}_{1,\varepsilon}+b_{1}\,{\lambda_{P}}^{m_{k}}\sigma_{Q}^{n_{k}}\,(\alpha_{2}+\widetilde{w}^{1,\varepsilon}_{1})\,\varsigma_{2}^{-1}\,t\\\
&\,\,\,\,\,\,+\Big{(}b_{2}\,\big{(}\tilde{\mathfrak{c}}_{k}(\beta_{2}+\widetilde{w}^{1,\varepsilon}_{2})-\widetilde{w}^{1,\varepsilon}_{3}\,\tilde{\mathfrak{s}}_{k}\big{)}^{2}+b_{3}\,\big{(}\tilde{\mathfrak{s}}_{k}(\beta_{2}+\widetilde{w}^{1,\varepsilon}_{2})+\widetilde{w}^{1,\varepsilon}_{3}\,\tilde{\mathfrak{c}}_{k}\big{)}^{2}\\\
&\,\,\,\,\,\,+b_{4}\,\big{(}\tilde{\mathfrak{c}}_{k}(\beta_{2}+\widetilde{w}^{1,\varepsilon}_{2})-\widetilde{w}^{1,\varepsilon}_{3}\,\tilde{\mathfrak{s}}_{k}\big{)}\big{(}\tilde{\mathfrak{s}}_{k}(\beta_{2}+\widetilde{w}^{1,\varepsilon}_{2})+\widetilde{w}^{1,\varepsilon}_{3}\,\tilde{\mathfrak{c}}_{k}\big{)}\Big{)}\,\varsigma^{-2}_{2}\,t^{2}\\\
&\,\,\,\,\,\,+{\mathrm{Hot}}^{y}_{\varepsilon,k,\mu}(t);\\\
\widetilde{z}_{\varepsilon,k,\mu}(t)&=c_{1}\,{\lambda_{P}}^{m_{k}}\sigma_{P}^{m_{k}}{{\sigma_{Q}}}^{n_{k}}\,\tilde{x}_{1,\varepsilon}\\\
&\,\,\,\,\,\,+\Big{(}c_{1}\,{\lambda_{P}}^{m_{k}}{\sigma_{P}}^{-m_{k}}\,(\alpha_{2}+\widetilde{w}^{1,\varepsilon}_{1})+c_{2}\big{(}\tilde{\mathfrak{c}}_{k}\,(\beta_{2}+\widetilde{w}^{1,\varepsilon}_{2})-\widetilde{w}^{1,\varepsilon}_{3}\,\tilde{\mathfrak{s}}_{k}\big{)}\\\
&\,\,\,\,\,\,+c_{3}\big{(}\tilde{\mathfrak{s}}_{k}\,(\beta_{2}+\widetilde{w}^{1,\varepsilon}_{2})+\widetilde{w}^{1,\varepsilon}_{3}\,\tilde{\mathfrak{c}}_{k}\big{)}\Big{)}\,\,\varsigma_{2}^{-1}\,t+\mathrm{Hot}^{z}_{\varepsilon,k,\mu}(t),\end{split}$

where $\mathrm{Hot}^{\ast}_{\varepsilon,k,\mu}(t)$, $\ast=x,y,z$, are high
order terms. Their explicit expressions can be found in Section 13.2.

$\bullet$ Comparing the terms in Lemma 10.2 We are now ready to estimate the
difference between the coordinates of the points in (10.6) and (10.7). For
that writing

$\delta^{w}_{\varepsilon,k,\mu}(t)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\widetilde{w}_{\varepsilon,k,\mu}(t)-\overline{w}_{\varepsilon,k,\mu}(t),\quad
w=x,y,z,$

we obtain

$\begin{split}\delta^{x}_{\varepsilon,k,\mu}(t)&=a_{1}\,{\lambda_{P}}^{m_{k}}\sigma_{P}^{m_{k}}{{\sigma_{Q}}}^{n_{k}}\,\widetilde{x}_{1,\varepsilon}+\Big{(}a_{1}\,{\lambda_{P}}^{m_{k}}{\sigma_{P}}^{-m_{k}}\,\widetilde{w}^{1,\varepsilon}_{1}+a_{2}\big{(}\tilde{\mathfrak{c}}_{k}\,\widetilde{w}^{1,\varepsilon}_{2}-\tilde{\mathfrak{s}}_{k}\,\widetilde{w}^{1,\varepsilon}_{3}\big{)}\\\
&+a_{3}\big{(}\tilde{\mathfrak{s}}_{k}\,\widetilde{w}^{1,\varepsilon}_{2}+\tilde{\mathfrak{c}}_{k}\,\widetilde{w}^{1,\varepsilon}_{3}\big{)}\Big{)}\,\,\varsigma_{2}^{-1}\,t+{\mathrm{Hot}}^{x}_{\varepsilon,k,\mu}(t)-\sigma^{m_{k}}_{P}\,\sigma^{n_{k}}_{Q}{\mathrm{hot}}^{x}_{k}(t);\end{split}$
$\begin{split}\delta^{y}_{\varepsilon,k,\mu}(t)&=b_{1}\,{\lambda_{P}}^{m_{k}}\sigma_{P}^{2m_{k}}{{\sigma_{Q}}}^{2n_{k}}\,\widetilde{x}_{1,\varepsilon}+b_{1}\,{\lambda_{P}}^{m_{k}}\sigma_{Q}^{n_{k}}\,\widetilde{w}^{1,\varepsilon}_{1}\,\varsigma_{2}^{-1}\,t\\\
&+\Big{(}b_{2}\,\big{[}2\,\beta_{2}\,\widetilde{w}^{1,\varepsilon}_{2}\,\tilde{\mathfrak{c}}_{k}^{2}+(\widetilde{w}^{1,\varepsilon}_{2})^{2}\,\tilde{\mathfrak{c}}_{k}^{2}-2\,\tilde{\mathfrak{c}}_{k}\,\tilde{\mathfrak{s}}_{k}\,\widetilde{w}^{1,\varepsilon}_{3}(\beta_{2}+\widetilde{w}^{1,\varepsilon}_{2})+(\widetilde{w}^{1,\varepsilon}_{3})^{2}\,\tilde{\mathfrak{s}}_{k}^{2}\big{]}\\\
&+b_{3}\,\big{[}2\,\beta_{2}\,\widetilde{w}^{1,\varepsilon}_{2}\,\tilde{\mathfrak{s}}_{k}^{2}+(\widetilde{w}^{1,\varepsilon}_{2})^{2}\,\tilde{\mathfrak{s}}_{k}^{2}+2\,\tilde{\mathfrak{c}}_{k}\,\tilde{\mathfrak{s}}_{k}\,\widetilde{w}^{1,\varepsilon}_{3}(\beta_{2}+\widetilde{w}^{1,\varepsilon}_{2})+(\widetilde{w}^{1,\varepsilon}_{3})^{2}\,\tilde{\mathfrak{c}}_{k}^{2}\big{]}\\\
&+b_{4}\,\big{[}2\,\beta_{2}\,\widetilde{w}^{1,\varepsilon}_{2}\,\tilde{\mathfrak{s}}_{k}\,\tilde{\mathfrak{c}}_{k}+(\widetilde{w}^{1,\varepsilon}_{2})^{2}\,\tilde{\mathfrak{s}}_{k}\,\tilde{\mathfrak{c}}_{k}\\\
&+(\tilde{\mathfrak{c}}_{k}^{2}-\tilde{\mathfrak{s}}_{k}^{2})\,\widetilde{w}^{1,\varepsilon}_{3}(\beta_{2}+\widetilde{w}^{1,\varepsilon}_{2})-(\widetilde{w}^{1,\varepsilon}_{3})^{2}\tilde{\mathfrak{s}}_{k}\,\tilde{\mathfrak{c}}_{k}\big{]}\Big{)}\,\varsigma^{-2}_{2}\,t^{2}\\\
&+{\mathrm{Hot}}^{y}_{\varepsilon,k,\mu}(t)-\sigma_{P}^{2m_{k}}{{\sigma_{Q}}}^{2n_{k}}\,{\mathrm{hot}}^{y}_{k}(t);\end{split}$
$\begin{split}\delta^{z}_{\varepsilon,k,\mu}(t)&=c_{1}\,{\lambda_{P}}^{m_{k}}\sigma_{P}^{m_{k}}{{\sigma_{Q}}}^{n_{k}}\,\widetilde{x}_{1,\varepsilon}+\Big{(}c_{1}\,{\lambda_{P}}^{m_{k}}{\sigma_{P}}^{-m_{k}}\,\widetilde{w}^{1,\varepsilon}_{1}+c_{2}\big{(}\tilde{\mathfrak{c}}_{k}\,\widetilde{w}^{1,\varepsilon}_{2}-\tilde{\mathfrak{s}}_{k}\,\widetilde{w}^{1,\varepsilon}_{3}\big{)}\\\
&+c_{3}\big{(}\tilde{\mathfrak{s}}_{k}\,\widetilde{w}^{1,\varepsilon}_{2}+\tilde{\mathfrak{c}}_{k}\,\widetilde{w}^{1,\varepsilon}_{3}\big{)}\Big{)}\,\,\varsigma_{2}^{-1}\,t+{\mathrm{Hot}}^{z}_{\varepsilon,k,\mu}(t)-\sigma^{m_{k}}_{P}\,\sigma^{n_{k}}_{Q}{\mathrm{hot}}^{z}_{k}(t).\end{split}$

We now prove that
$\big{\|}\delta^{x}_{\varepsilon,k,\mu}\,|_{[-4,4]}\big{\|}_{r}\to 0$. The
proofs of
$\big{\|}\delta^{\ast}_{\varepsilon,k,\mu}\,|_{[-4,4]}\big{\|}_{r}\to 0$,
$\ast=y,z$, are similar and hence omitted. For that write

$\delta^{x}_{\varepsilon,k,\mu}(t)=\mathcal{A}_{\varepsilon,k,\mu}(t)+{\mathrm{Hot}}^{x}_{\varepsilon,k,\mu}(t)-\sigma^{m_{k}}_{P}\,\sigma^{n_{k}}_{Q}{\mathrm{hot}}^{x}_{k}(t),$

where $\mathcal{A}_{\varepsilon,k,\mu}(t)$ denotes the affine part of
$\delta^{x}_{\varepsilon,k,\mu}(t)$.

######  Claim 10.4.

  1. (1)

$\lim_{k\to\infty}\,\big{\|}\
\mathcal{A}_{\varepsilon,k,\mu}\,|_{[-4,4]}\big{\|}_{r}=0$,

  2. (2)

$\lim_{k\to\infty}\,\big{\|}\sigma^{m_{k}}_{P}\,\sigma^{n_{k}}_{Q}\,{\mathrm{hot}}^{x}_{\varepsilon,k,\mu}\,|_{[-4,4]}\big{\|}_{r}=0,$

  3. (3)

$\lim_{k\to\infty}\,\big{\|}{\mathrm{Hot}}^{x}_{\varepsilon,k,\mu}\,|_{[-4,4]}\big{\|}_{r}=0$.

Clearly, this claim implies Lemma 10.2.

###### Proof of Claim 10.4.

For the first item recall that by (6.3) we get
$\lambda_{P}^{m_{k}}\sigma_{P}^{2m_{k}}{{\sigma_{Q}}}^{2n_{k}}\to 0$ and that
by Remark 10.2 the norm of
$(\widetilde{w}^{1,\varepsilon}_{1},\widetilde{w}^{1,\varepsilon}_{2},\widetilde{w}^{1,\varepsilon}_{3})$
is small. The second item was stated in Remark 10.3. The proof of last item is
postponed to Section 13.2.1. ∎

The proof of Proposition 10.1 is now complete.

##  11\. Proof of Theorem 1.1: homoclinic relations

We now prove that blender $\Upsilon_{\varepsilon,k,\mu}$ and the saddle $P$
are homoclinically related.

######  Proposition 11.1.

For every small $\varepsilon>0$, large $k$, and $\mu\in(-10,-9)$ there is $g$
arbitrarily $C^{r}$ close to $g_{\varepsilon,k,\mu}$ such that

$W^{\mathrm{u}}\big{(}{\Upsilon}_{g},g\big{)}\pitchfork
W^{\mathrm{s}}\big{(}P,g\big{)}\neq\emptyset\quad\mbox{and}\quad
W^{\mathrm{s}}\big{(}{\Upsilon}_{g},g\big{)}\pitchfork
W^{\mathrm{u}}\big{(}P,g\big{)}\neq\emptyset.$

\begin{overpic}[scale={0.07}]{tudo.jpg} \put(65.0,53.0){\Large$P$}
\put(50.0,49.0){$\bullet$} \put(162.0,152.0){\Large$\Upsilon_{g}$}
\put(290.0,165.0){\Large$Q$} \put(283.0,146.0){$\bullet$} \end{overpic} Figure
10. Robust homoclinic tangencies

###### Proof.

Note that Propositions 9.1 and 10.1 hold for perturbations $g$ of
$g_{\varepsilon,k,\mu}$. Recall the definition of the quasi-transverse
heteroclinic point $X_{2,\varepsilon}$ in Proposition 4.1 and observe that
this intersection point is preserved by the local perturbations we have
considered. Therefore $X_{2,\varepsilon}$ is also a heteroclinic point of
$g_{\varepsilon,k,\mu}$. Note also that after a perturbation $g$ of
$g_{\varepsilon,k,\mu}$ (that does not destroy the heteroclinic points) we can
assume that the argument of $Dg(Q)$ is irrational. Thus the hypotheses of
Lemma 4.1 hold.

By Proposition 9.1, $W^{\mathrm{u}}\big{(}{\Upsilon}_{g},g\big{)}\pitchfork
W^{\mathrm{s}}\big{(}Q,g\big{)}\neq\emptyset.$ Thus, after a perturbation,
that we continue denoting by $g$ and does not change the argument of the
complex eigenvalue of $Dg(Q)$, we can assume that there is a disc $S\subset
W^{\mathrm{u}}({\Upsilon}_{g},g)$ that transversely intersects
$W^{\mathrm{s}}_{\mathrm{loc}}(Q,g)$ along a curve $\gamma$ with nontrivial
radial projection. Lemma 4.1 now implies that $W^{\mathrm{s}}(P,g)$
transversally intersects $S\subset W^{\mathrm{u}}(\Upsilon_{g},g)$, proving
the first item of the proposition. Note also that $g$ has a robust cycle
associated to $Q$ and $\Upsilon_{g}$.

To prove the second intersection, note that $W^{\mathrm{u}}(P,g)$ and
$W^{\mathrm{s}}_{\mathrm{loc}}(Q,g)$ interesects transversely along a curve
with a nontrivial radial projection. Note also that by Proposition 10.1,
$W^{\mathrm{s}}({\Upsilon}_{g},g)$ and $W^{\mathrm{u}}\big{(}Q,g\big{)}$
intersects quasi-transversely at some point $Z$. Consider a curve $D\subset
W^{\mathrm{s}}({\Upsilon}_{g},g)$ and containing the point $Z$ in its
interior. Arguing as above and applying Lemma 4.1, we have that the negative
iterates of $D$ (contained in $W^{\mathrm{s}}({\Upsilon}_{g},g)$ transversally
intersects $W^{\mathrm{s}}(P,g)$. This completes the proof of the proposition.
∎

##  12\. Proof of Theorem 1.1: homoclinic tangencies

In this section, we consider perturbations $g_{\varepsilon,k,\mu}$ of the
diffeomorphisms $f$ in $\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{e}^{+}}(M)$.
Next proposition implies the part of Theorem 1.1 about homoclinic tangencies.

######  Proposition 12.1.

For every small $\varepsilon>0$, large $k$, and $\mu\in(-10,-9)$ there is $g$
arbitrarily $C^{r}$ close to $g_{\varepsilon,k,\mu}$ with a $C^{r}$ robust
homoclinic tangency associated to blender-horseshoe ${\Upsilon}_{g}$.

###### Proof.

Note that close to the original heterodimensional tangency, the manifold
$W^{\mathrm{u}}\big{(}P,g_{\varepsilon,k,\mu}\big{)}$ intersect
$W^{\mathrm{s}}\big{(}Q,g_{\varepsilon,k,\mu}\big{)}$ in closed curve denoted
by $C_{\varepsilon,k,\mu}$. Let $S_{\varepsilon,k,\mu}$ be the two-dimensional
compact disc contained in
$W^{\mathrm{u}}\big{(}P,g_{\varepsilon,k,\mu}\big{)}$ bounded by
$C_{\varepsilon,k,\mu}$. By the $\lambda$-lemma, the forward iterates
$g^{i}_{\varepsilon,k,\mu}(S_{\varepsilon,k,\mu})$ of $S_{\varepsilon,k,\mu}$
accumulated to the unstable manifold of $Q$. By Lemma 10.1, the unstable
manifold of $Q$ contains a disc in the superposition region of the blender
${\Upsilon}_{\varepsilon,k,\mu}$. Thus there are infinitely many iterates of
$S_{\varepsilon,k,\mu}$ containing ${\mathrm{uu}}$-tubes in the superposition
region of the blender, see Figure 10 and recall Definition 5.1. Corollary 5.1
implies that the manifold
$W^{\mathrm{u}}\big{(}P,g_{\varepsilon,k,\mu}\big{)}$ and
$W^{\mathrm{s}}_{\mathrm{loc}}\big{(}\Upsilon_{\varepsilon,k,\mu},g_{\varepsilon,k,\mu}\big{)}$
have a $C^{r}$ robust tangency. The proposition follows noting that by
Proposition 11.1 the point $P$ and the blender $\Upsilon_{\varepsilon,k,\mu}$
are homoclinically related. ∎

##  13\. Calculations in the renormalisation scheme

We collect some calculations from [16] that we used in the previous sections.

###  13.1. The renormalisation formula

First, recall the perturbations $f_{\bar{\upsilon},\rho}$ of
$f\in\mathcal{H}_{{\mathrm{BH}}}^{r}(M^{3})$ in (6.1). For that, we borrow
from [16, Section 7.3] the explicit formula for compositions of the form

(13.1) $\Psi_{k}^{-1}\circ
f_{\bar{\upsilon}_{k}(\mu),\rho}^{N_{2}+m_{k}+N_{1}+n_{k}}\circ\Psi_{k}(x,y,z)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(\check{x}_{k,\mu,\rho},\check{y}_{k,\mu,\rho},\check{z}_{k,\mu,\rho}).$

Here, the iterates corresponding to $n_{k}$ occurs in $U_{Q}$, the $N_{1}$
iterates correspond to the transition $\mathfrak{T}_{1}$, the iterates
corresponding to $m_{k}$ occurs in $U_{P}$, and the $N_{2}$ iterates
correspond to the transition $\mathfrak{T}_{2}$.

Consider the heteroclinic points $X,Y$ in the cycle (see Section 2.1.2) and
write

(13.2)
$\begin{split}\textbf{x}_{k,\mu,\rho}(x,y,z)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}f^{n_{k}}_{\bar{\upsilon}_{k}(\mu),\rho}\circ\Psi_{k}(x,y,z)-X,\\\
\widehat{\textbf{x}}_{k,\mu,\rho}(x,y,z)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}f^{m_{k}+N_{1}+n_{k}}_{\bar{\upsilon}_{k}(\mu),\rho}\circ\Psi_{k}(x,y,z)-Y.\end{split}$

Using the notation in Section 6, the composition in (13.1) reads as follows:

(13.3)
$\begin{split}\breve{x}_{k,\mu,\rho}&=a_{1}\,\lambda_{P}^{m_{k}}\,{\lambda_{Q}}^{n_{k}}\,\big{(}\alpha_{1}\,(\mathfrak{c}_{k}\,x-\mathfrak{s}_{k}\,z)+\alpha_{3}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)\big{)}\\\
&\quad+a_{1}\,\lambda_{P}^{m_{k}}\,\sigma_{P}^{-m_{k}}\,\alpha_{2}\,y+\big{(}\tilde{\mathfrak{c}}_{k}\,a_{2}+\tilde{\mathfrak{s}}_{k}\,a_{3}\big{)}\,\beta_{2}\,y\\\
&\quad+\sigma_{P}^{m_{k}}\,\lambda_{Q}^{n_{k}}\,\gamma_{3}\,\big{(}\tilde{\mathfrak{c}}_{k}\,a_{3}-\tilde{\mathfrak{s}}_{k}\,a_{2}\big{)}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)\\\
&\quad+{\sigma_{P}}^{m_{k}}\,\sigma_{Q}^{n_{k}}\,\mathrm{hot}^{{x}}_{k,\mu,\rho},\\\
\breve{y}_{k,\mu,\rho}&=\mu+b_{1}\,\lambda_{P}^{m_{k}}\,{\sigma_{Q}}^{n_{k}}\,\alpha_{2}\,y\\\
&\quad+b_{1}\,{\lambda_{P}}^{m_{k}}\,{\sigma_{P}}^{m_{k}}\,\lambda_{Q}^{n_{k}}\,\sigma_{Q}^{n_{k}}\,\big{(}\alpha_{1}\,(\mathfrak{c}_{k}\,x-\mathfrak{s}_{k}\,z)\alpha_{3}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)\big{)}\\\
&\quad+\Big{(}\tilde{\mathfrak{c}}_{k}^{2}\,b_{2}+\tilde{\mathfrak{s}}_{k}^{2}\,b_{3}+\tilde{\mathfrak{c}}_{k}\,\tilde{\mathfrak{s}}_{k}\,b_{4}\Big{)}\,\beta_{2}^{2}\,y^{2}\\\
&\quad+\sigma_{P}^{2m_{k}}\,\lambda_{Q}^{2n_{k}}\Big{(}\tilde{\mathfrak{s}}_{k}^{2}\,b_{2}+\tilde{\mathfrak{c}}_{k}^{2}\,b_{3}-\tilde{\mathfrak{c}}_{k}\,\tilde{\mathfrak{s}}_{k}\,b_{4}\Big{)}\,\gamma_{3}^{2}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)^{2}\\\
&\quad+\sigma_{P}^{m_{k}}\,\lambda_{Q}^{n_{k}}\,\Big{(}2\tilde{\mathfrak{c}}_{k}\,\tilde{\mathfrak{s}}_{k}\,(b_{3}-b_{2})(\tilde{\mathfrak{c}}_{k}^{2}\,-\tilde{\mathfrak{s}}_{k}^{2})\,b_{4}\Big{)}\beta_{2}\,\gamma_{3}(\mathfrak{s}_{k}\,x\,y+\mathfrak{c}_{k}\,y\,z)\\\
&\quad+{\sigma_{P}}^{2m_{k}}\,\sigma_{Q}^{2n_{k}}\,\mathrm{hot}^{y}_{k,\mu,\rho},\\\
\breve{z}_{k,\mu,\rho}&=c_{1}\,\lambda_{P}^{m_{k}}\,{\lambda_{Q}}^{n_{k}}\,\big{(}\alpha_{1}\,(\mathfrak{c}_{k}\,x-\mathfrak{s}_{k}\,z)+\alpha_{3}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)\big{)}\\\
&\quad+c_{1}\,\lambda_{P}^{m_{k}}\,\sigma_{P}^{-m_{k}}\,\alpha_{2}\,y+\big{(}\tilde{\mathfrak{c}}_{k}\,c_{2}+\tilde{\mathfrak{s}}_{k}\,c_{3}\big{)}\,\beta_{2}\,y\\\
&\quad+\sigma_{P}^{m_{k}}\,\lambda_{Q}^{n_{k}}\,\gamma_{3}\,\big{(}\tilde{\mathfrak{c}}_{k}\,c_{3}-\tilde{\mathfrak{s}}_{k}\,c_{2}\big{)}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)\\\
&\quad+{\sigma_{P}}^{m_{k}}\,\sigma_{Q}^{n_{k}}\,\mathrm{hot}^{z}_{k,\mu,\rho},\end{split}$

where
$\mathrm{hot}^{\ast}_{k,\mu,\rho}=\mathrm{hot}^{\ast}_{k,\mu,\rho}(x,y,z)$,
$\ast=x,y,z$, are higher order terms whose Laundau’s symbols satisfy the
following conditions (see [16, Lemma 8.3]). Write

$\widehat{H}_{2}(\textbf{x}_{k,\mu,\rho})\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\widetilde{H}_{2}(\textbf{x}_{k,\mu,\rho})-\lambda_{Q}^{2n_{k}}\,\tilde{\rho}_{2,k},$

where $\tilde{\rho}_{2,k}$ is defined in (6.7). Then

  * (i)

${O}\big{(}\mathrm{hot}^{x}_{k,\mu,\rho}(x,y,z)\big{)}={O}\big{(}{\lambda_{P}}^{m_{k}}\,\widetilde{H}_{1}(\textbf{x}_{k,\mu,\rho})\big{)}+{O}\big{(}\sigma_{P}^{m_{k}}\,\widehat{H}_{2}(\textbf{x}_{k,\mu,\rho})\big{)}\\\
+{O}\big{(}H_{1}(\widehat{\textbf{x}}_{k,\mu,\rho})\big{)},$

  * (ii)

${O}\big{(}\mathrm{hot}^{y}_{k,\mu,\rho}(x,y,z)\big{)}=O({\lambda_{P}}^{m_{k}})+O\big{(}\widetilde{H}_{1}(\textbf{x}_{k,\mu,\rho})\big{)}+O\big{(}\big{(}\sigma_{P}^{m_{k}}\,\widehat{H}_{2}(\textbf{x}_{k,\mu,\rho})\big{)}^{2}\big{)}\\\
+O\big{(}\sigma_{Q}^{-n_{k}}\,\widehat{H}_{2}(\textbf{x}_{k,\mu,\rho})\big{)}+O\big{(}H_{2}(\widehat{\textbf{x}}_{k,\mu,\rho})\big{)}$,

  * (iii)

${O}\big{(}\mathrm{hot}^{z}_{k,\mu,\rho}(x,y,z)\big{)}=O\big{(}{\lambda_{P}}^{m_{k}}\widetilde{H}_{1}(\textbf{x}_{k,\mu,\rho})\big{)}+O\big{(}\sigma_{P}^{m_{k}}\,\widehat{H}_{2}(\textbf{x}_{k,\mu,\rho})\big{)}\newline
+O\big{(}H_{3}(\widehat{\textbf{x}}_{k,\mu,\rho})\big{)}.$

###  13.2. The high order terms of $\Psi_{k}^{-1}\circ
g_{\varepsilon,k,\mu}^{N_{2}+m_{k}}\big{(}{\gamma}_{\varepsilon,k}(t)\big{)}$

Recall the definition of $\gamma_{\varepsilon,k}$ in (10.5). We now provide an
explicit expression of the high order terms
$\mathrm{Hot}^{\ast}_{\varepsilon,k,\mu}(t)$ in $\Psi_{k}^{-1}\circ
g_{\varepsilon,k,\mu}^{N_{2}+m_{k}}\big{(}{\gamma}_{\varepsilon,k}(t)\big{)}$.
Define $\textbf{x}_{\varepsilon,k,\mu}$ and
$\widehat{\textbf{x}}_{\varepsilon,k,\mu}$ as in (13.2), where
$f_{\bar{\upsilon}_{k}(\mu)}$ is replaced by
$f_{\varepsilon,\bar{\upsilon}_{k}(\mu)}$ (this is why the subscript
$\varepsilon$ appears). Write

$\textbf{w}_{\varepsilon,k,\mu}(t)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\textbf{x}_{\varepsilon,k,\mu}\circ\Theta_{\bar{\varsigma}}(0,t,0),\qquad\widehat{\textbf{w}}_{\varepsilon,k,\mu}(t)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\widehat{\textbf{x}}_{\varepsilon,k,\mu}\circ\Theta_{\bar{\varsigma}}(0,t,0),$

where $\Theta_{\bar{\varsigma}}$ is an in (5.5). We have

$\begin{split}{\mathrm{Hot}}^{x}_{\varepsilon,k,\mu}(t)&=a_{1}\,{\lambda_{P}}^{m_{k}}\sigma_{P}^{m_{k}}{{\sigma_{Q}}}^{n_{k}}\widehat{\rho}^{1}_{\varepsilon,k}(t)+a_{2}\,\sigma_{P}^{2m_{k}}{{\sigma_{Q}}}^{n_{k}}u_{\varepsilon,k}(t)\\\
&\quad+a_{3}\,{\sigma_{P}}^{2m_{k}}{{\sigma_{Q}}}^{n_{k}}v_{\varepsilon,k}(t)+\sigma_{P}^{m_{k}}{{\sigma_{Q}}}^{n_{k}}H_{1}\big{(}\widehat{\textbf{w}}_{\varepsilon,k,\mu}(t)\big{)};\\\
{\mathrm{Hot}}^{y}_{\varepsilon,k,\mu}(t)&=b_{1}\,{\lambda_{P}}^{m_{k}}\sigma_{P}^{2m_{k}}{{\sigma_{Q}}}^{2n_{k}}\widehat{\rho}^{1}_{\varepsilon,k}(t)\\\
&\quad+\sigma_{P}^{2m_{k}}{\sigma_{Q}}^{n_{k}}\,\Big{(}2b_{2}\big{(}\tilde{\mathfrak{c}}_{k}(\alpha_{2}+\widetilde{w}^{1,\varepsilon}_{2})-\widetilde{w}^{1,\varepsilon}_{3}\tilde{\mathfrak{s}}_{k}\big{)}\\\
&\qquad\qquad+b_{4}\big{(}\tilde{\mathfrak{s}}_{k}(\alpha_{2}+\widetilde{w}^{1,\varepsilon}_{2})+\widetilde{w}^{1,\varepsilon}_{3}\,\tilde{\mathfrak{c}}_{k}\big{)}\Big{)}\varsigma_{2}^{-1}\,t\,u_{\varepsilon,k}(t)\\\
&\quad+\sigma_{P}^{2m_{k}}{\sigma_{Q}}^{n_{k}}\,\Big{(}2b_{3}\big{(}\tilde{\mathfrak{s}}_{k}(\alpha_{2}+\widetilde{w}^{1,\varepsilon}_{2})+\widetilde{w}^{1,\varepsilon}_{3}\tilde{\mathfrak{c}}_{k}\big{)}\\\
&\qquad\qquad+b_{4}\big{(}\tilde{\mathfrak{c}}_{k}(\alpha_{2}+\widetilde{w}^{1,\varepsilon}_{2})-\widetilde{w}^{1,\varepsilon}_{3}\,\tilde{\mathfrak{s}}_{k}\big{)}\Big{)}\varsigma_{2}^{-1}\,t\,v_{\varepsilon,k}(t)\\\
&\quad+\sigma_{P}^{4m_{k}}{\sigma_{Q}}^{2n_{k}}\,\Big{(}b_{2}\,\big{(}u_{\varepsilon,k}(t)\big{)}^{2}+b_{3}\big{(}v_{\varepsilon,k}(t)\big{)}^{2}\\\
&\qquad\qquad+b_{4}\,u_{\varepsilon,k}(t)\,v_{\varepsilon,k}(t)\Big{)}+\sigma_{P}^{2m_{k}}{{\sigma_{Q}}}^{2n_{k}}H_{2}\big{(}\widehat{\textbf{w}}_{\varepsilon,k,\mu}(t)\big{)};\\\
{\mathrm{Hot}}^{z}_{\varepsilon,k,\mu}(t)&=c_{1}\,{\lambda_{P}}^{m_{k}}\sigma_{P}^{m_{k}}{{\sigma_{Q}}}^{n_{k}}\widehat{\rho}^{1}_{\varepsilon,k}(t)+c_{2}\,\sigma_{P}^{2m_{k}}{{\sigma_{Q}}}^{n_{k}}u_{\varepsilon,k}(t)\\\
&\quad+c_{3}\,{\sigma_{P}}^{2m_{k}}{{\sigma_{Q}}}^{n_{k}}v_{\varepsilon,k}(t)+\sigma_{P}^{m_{k}}{{\sigma_{Q}}}^{n_{k}}H_{3}\big{(}\widehat{\textbf{w}}_{\varepsilon,k,\mu}(t)\big{)}.\end{split}$

####  13.2.1. Proof of item (3) in Claim 10.4

We claim that

$\big{\|}\big{(}\mathrm{Hot}^{x}_{\varepsilon,k,\mu}\,-\,\sigma_{P}^{m_{k}}{{\sigma_{Q}}}^{n_{k}}H_{1}\circ\widehat{\textbf{w}}_{\varepsilon,k,\mu}\big{)}\,|_{[-4,4]}\big{\|}_{r}\to
0.$

For this just note that

  * •

$u_{\varepsilon,k}(t)$, $v_{\varepsilon,k}(t)$ and
$\widehat{\rho}^{\ell}_{\varepsilon,k}(t)$, $\ell=2,3$, have the same symbol
of Landau $O(\sigma_{P}^{-4m_{k}}\,\sigma_{Q}^{-2n_{k}})$, see (10.8) and
(10.9),

  * •

$\widehat{\rho}_{\varepsilon,k}^{1}$ is bounded and (6.3).

Finally, the convergence

$\lim_{k\to\infty}\big{\|}\sigma_{P}^{m_{k}}{{\sigma_{Q}}}^{n_{k}}H_{1}\circ\widehat{\textbf{w}}_{\varepsilon,k,\mu}\,|_{[-4,4]}\big{\|}_{r}=0$

follows exactly as in [16, Claim 8.4].

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