#  Hénon-like families and blender-horseshoes at non-transverse
heterodimensional cycles

Lorenzo J. Díaz and Sebastián A. Pérez  Departamento de Matemática PUC-Rio,
Marquês de São Vicente 225, Gávea, Rio de Janeiro 225453-900, Brazil
<EMAIL_ADDRESS>Centro de Matemática da Universidade do Porto, Rua do
Campo Alegre, 687, 4169-007 Porto, Portugal<EMAIL_ADDRESS>
###### Abstract.

In dimension three and under certain regularity assumptions, we construct a
renormalisation scheme at the heterodimensional tangency of a non-transverse
heterodimensional cycle associated with a pair of saddle-foci whose limit
dynamic is a center-unstable Hénon-like family displaying blender-horseshoes.
As a consequence, the initial cycle can be approximated in higher regularity
topologies by diffeomorphisms having blender-horseshoes.

###### Key words and phrases:

Blender-horseshoe, Dominated splitting, Hénon-like families, Heterodimensional
cycle, Renormalisation scheme

###### 2000 Mathematics Subject Classification:

Primary:

This paper is part of the PhD thesis of SP (PUC-Rio) supported by CNPq
(Brazil). The authors thank the hospitality and support of Centro de
Matemática of Univ. of Porto (Portugal). LJD is partially supported by CNE-
Faperj and CNPq (Brazil) and SP is supported by CMUP (UID/MAT/00144/2013) and
PTDC/MAT-CAL/3884/2014, which are funded by FCT (Portugal) with national (MEC)
and European structural funds through the programs COMPTE and FEDER, under the
partnership agreement PT2020.

##  1\. Introduction

Homoclinic tangencies and heterodimensional cycles are the two main sources of
non-hyperbolic chaotic dynamics. Palis’ density conjecture claims the these
two types of bifurcations are the only obstructions to hyperbolicity: any non-
hyperbolic system can be approximated by diffeomorphisms displaying some of
these configurations, see [37]. In the terminology of [10] (see Preface),
homoclinic tangencies are in the core of the so-called _critical dynamics_
while heterodimensional cycles occur in _non-critical_ settings. Here
_heterodimensional_ refers to the fact that the cycle involves saddles whose
unstable manifolds have different dimensions. A crucial point is that critical
and non-critical dynamics may occur simultaneously in some bifurcation
scenarios and their effects may superimpose. This is precisely the setting of
this paper. We consider three-dimensional diffeomorphisms having
simultaneously a heterodimensional cycle and a heterodimensional tangency (see
part (b) of Figure 1) and study a renormalisation scheme leading to blender-
horseshoes. Our results continues the line of research started in [14], where
renormalisation schemes were introduced in this heterodimensional critical
setting. The main difference with [14] is that we consider here cycles
associated to a pair of saddle-foci instead a pair of saddles with real
multipliers. We note that the dynamical setting of this paper is in some sense
a “heterodimensional version” of the _equidimensional_ (meaning that all the
saddles in the bifurcation have unstable manifolds of the same dimension)
configuration studied in [27], see part (a) of Figure 1. We now proceed to
discuss more precisely the setting and the concepts involved in this paper.

\begin{overpic}[scale={.065}]{homovshete.jpg} \put(85.0,0.0){\small{(a)}}
\put(280.0,0.0){\small{(b)}} \end{overpic} Figure 1. (a) The non-transverse
equidimensional cycle in [27]. (b) A non-transverse heterodimensional cycle.

### Homoclinic bifurcations and renormalisation

Homoclinic bifurcations may occur only in dimension two or higher. Some
important dynamical phenomena associated to homoclinic tangencies are the
existence of horseshoes whose invariant sets exhibit persistent non-transverse
homoclinic intersections, [34, 36], the generic coexistence of infinitely many
sinks/sources, [35], and the occurrence of Hénon-like attractors/repellers,
[2, 32]. Initially, these phenomena were studied independently, but the study
in [40] allows a unifying approach via renormalisation methods, see [40,
Chapter 3.4]. Roughly, the quadratic family can be viewed as a limit dynamics
after a re-scaling process at a homoclinic tangency point. This approach
enables to translate persistent dynamical behaviours present in the quadratic
family to the setting of homoclinic bifurcations. This is an important outcome
of the renormalisation approach that can be used in another bifurcating
settings that is explored in this paper. We emphasise that the renormalisation
methods have two parts: a (limit) family exhibiting some persistent dynamical
“interesting” features and a sequence of “dynamically defined” maps
approaching the initial configuration whose “returns” (obtained by composing
the diffeomorphisms) tend to this family. In this way, properties of the limit
family are translated to the considered dynamical systems. Let us postpone for
a while the discussion about renormalisation and analyse heterodimensional
cycles.

### Heterodimensional cycles

First note that heterodimensional cycles can occur only on manifolds of
dimension three or higher. To keep our presentation as simple as possible,
while describing all the essential complexity of these cycles, in what follows
we will assume that the dimension of the ambient manifold is three. We recall
that the _index_ of a hyperbolic periodic point $P$ of a diffeomorphism $f$ is
the number of expanding eigenvalues of $Df^{\pi}(P)$, where $\pi$ is the
period or $P$. We say that a diffeomorphism $f$ has a _heterodimensional
cycle_ associated to two saddles of different indices (here, by dimension
constrains, the indices are necessarily two and one, or vice-versa) $P$ and
$Q$ if the their invariant manifolds meet cyclically. In what follows, again
for simplicity of the presentation, we will assume that $P$ and $Q$ are both
fixed points and that $P$ has index two and $Q$ has index one. Key aspects in
the analysis of these cycles is the shape of the intersections between these
invariant sets as well as the type of expanding eigenvalues of $Df(P)$ and
contracting eigenvalues of $Df(Q)$. First, due to dimension deficiency the
one-dimensional invariant sets $W^{\mathrm{s}}(P)$ and $W^{\mathrm{u}}(Q)$
cannot have transverse intersections. Second, due to dimension sufficiency the
two-dimensional invariant manifolds $W^{\mathrm{u}}(P)$ and
$W^{\mathrm{s}}(P)$ may have transverse and non-transverse intersections (and
this two types of intersections may occur simultaneously). The combination of
these aspects (types of eigenvalues of the saddles and types and shapes of the
intersection of the invariant manifolds) leads to a plethora of dynamical
configurations and features (going from hyperbolic to a completely non-
dominated dynamics) that we do not aim to discuss in their full
complexity111An interesting illustration of the complexity of these
configurations can be done following the lines in the discussion in [39,
Section 2]. Here the complexity is much higher: the shapes of the
intersections of the two-dimensional manifolds need to be considered..
Instead, we prefer to outline two “antipodal” cases leading to two very
different dynamical settings: partial hyperbolicity and non-dominated
dynamics. Before presenting these scenarios let us observe that we follow the
approach proposed in [38, Section 3], we fix a _neighbourhood of the cycle_
(roughly, an small open set containing the saddles of the cycle and
intersections of their invariant manifolds) and study the relative dynamics in
such a set. In what follows, we assume that the intersection of the one-
dimensional invariant sets $W^{\mathrm{s}}(P)$ and $W^{\mathrm{u}}(Q)$ is
_quasi-transverse_ (i.e., their tangent vectors at the intersection are not
colinear).

#### Partial hyperbolicity versus non-domination

A first (and certainly the simplest) configuration occurs when all the
eigenvalues of $P$ and $Q$ are real and different in modulus and the
intersection between the two-dimensional invariant sets $W^{\mathrm{u}}(P)$
and $W^{\mathrm{s}}(Q)$ is transverse and defines a heteroclinic curve
$\gamma$ with endpoints $P$ and $Q$. The property of the eigenvalues allows to
define the one-dimensional strong unstable foliation of $W^{\mathrm{u}}(P)$
and the one-dimensional strong unstable foliation of $W^{\mathrm{u}}(Q)$. The
“simplest” case occurs when $\gamma$ is transverse to these two foliations.
This dynamical configuration is depicted in part (a) in Figure 2.

\begin{overpic}[scale={.063}]{Cycle.jpg} \put(85.0,0.0){\small{(a)}}
\put(280.0,0.0){\small{(b)}} \put(85.0,95.0){{$\gamma$}} \end{overpic} Figure
2. Partial hyperbolic and non-dominated heterodimensional cycles.

This type of configuration leads to partially hyperbolic dynamics with a one-
dimensional center: in the neighbourhood of the cycle there is a
$Df$-splitting with three one-dimensional invariant directions
$E^{\mathrm{s}}\oplus E^{\mathrm{c}}\oplus E^{\mathrm{u}}$, $E^{\mathrm{s}}$
uniformly contracting, $E^{\mathrm{u}}$ uniformly expanding, and
$E^{\mathrm{c}}$ has contracting and expanding regions. For instance, this
bifurcation can generate a pair of disjoint hyperbolic sets (horseshoes) each
one containing one of the saddles in the cycle [17] or a transitive set
containing both saddles in the cycle and/or the phenomenon of intermingled
homoclinic classes222A _homoclinic class_ of a saddle is the closure of the
set of its transverse homoclinic points. In this particular case, the
homoclinic classes of $P$ and $Q$ coincide, since these saddles have different
indices one has a non-hyperbolic transitive set., see [13, 12, 18], but this
list does not exhaust all the possibilities of this configuration, see [19].

A second configuration occurs when either (i) the two-dimensional manifolds
$W^{\mathrm{u}}(P)$ and $W^{\mathrm{s}}(Q)$ have non-transverse intersections
or (ii) the saddles in the cycle have both a pair of non-real eigenvalues
($Df^{\pi}(P)$ a pair of non-real expanding eigenvalues and $Df^{\pi}(Q)$ a
pair of non-real contracting eigenvalues), see part (b) in Figure 2. These
configurations lead to non-dominated dynamics: there is no (non-trivial)
bundle defined on the neighbourhood of the cycle that is $Df$-invariant. The
second type of dynamics was studied in [4] (to get infinitely many
sinks/sources) and in [5] (to get universal dynamics), and the first type in
[15] (where _heterodimensional tangencies_ were introduced). In this paper, we
consider heterodimensional tangencies and heterodimensional cycles associated
to saddles with non-real eigenvalues, continuing the study started in [14],
where a similar configuration (with saddles having real eigenvalues) was
considered.

Let us emphasise that in our setting the dynamics in the cycle is non-
dominated, thus _finding relevant hyperbolic sets and seeing how these sets
are embedded in the global dynamics_ is a complicated task. Let us now make
this sentence a bit more precise. In the case of homoclinic bifurcations,
relevant sets are the so-called _thick horseshoes_ which are the main
responsible of the persistent non-hyperbolic features of these bifurcations
(for an ample discussion see [40, Chapter 5]). In the case of
heterodimensional cycles, it is proved in [6, 8] that these cycles yield
_blenders_ (or more precisely, _blender-horseshoes_) and that these blenders
are in the core of most of the non-hyperbolic features related to these cycles
(as for instance, intermingled homoclinic classes and occurrence of robust
cycles and tangencies). Our result state that, in our setting, the bifurcating
diffeomorphisms $f$ can be approximated by diffeomorphisms exhibiting blender-
horseshoes, see Theorem 1 and Corollary 1. This approximation is done using a
renormalisation scheme converging to a center-unstable Hénon-like family (see
equations (2.1) and (2.2)) and using the fact that this family exhibits
blender-horseshoes, see [16] and [14] for a preliminary version. In this
approach, the Hénon-like family plays the role of the quadratic family in the
homoclinic setting.

### Blenders and Blender-horseshoes

Although blenders do not appear explicitly in our paper, they are a
fundamental object behind our constructions: the center-unstable Hénon-like
family provides these blenders as a side effect. Let us say a few words about
them. A blenders is just a transitive hyperbolic set that occurs in dimension
three or higher and whose (local) stable set geometrically behaves as a set of
dimension larger than the one of its stable bundle. For an informal
introduction to blenders we refer to [3], for some discussions on the notion
and properties of blenders see [11, Chapter 6.2]. Blender-horseshoes are a
special kind of blenders, they are locally maximal hyperbolic sets conjugate
to the complete shift in two symbols, for the precise definition of a blender-
horseshoe see [8]. Comparing with the blenders in [7], an important advantage
of blender-horseshoes is that they can be used to get robust cycles and
tangencies, as in [6, 8]. An important property of blender-horseshoes, that we
will use here in Corollary 1, is their $C^{r}$-persistence, see [8, Lemma
3.9].

### Regularity

So far, we have deliberately omitted (for simplicity of the discussion) any
reference to the regularity of the perturbations of the systems. This will be
a key point in the following discussion. To put the statements as simple as
possible (and slightly oversimplifying), the occurrence of thick horseshoes in
homoclinic bifurcations demands $C^{2}$-regularity while the occurrence of
blenders at heterodimensional cycles is obtained for $C^{1}$-perturbations.
This means that (partially) the two theories are developed considering
different degrees of differentiability. Here it is important to recall the
recent paper [1] where robust homoclinic tangencies are obtained in some
specific partially hyperbolic settings333This is a higher-dimensional result
where the ambient space is four-dimensional: to get the tangencies one needs a
“central direction” of dimension at least two (besides some stable and
unstable directions). It is also important to recall that in dimension two
there are no $C^{1}$-robust homoclinic tangencies, see [33].. A natural goal
is to get the occurrence of robust heterodimensional cycles and homoclinic
tangencies in $C^{2}$-settings. The rough idea (that we will explore) is that
one can go from $C^{1}$-regularity to higher regularity if the _dynamics of
the intersection is rich enough_ and _there are plenty of heteroclinic
intersections_ between the saddles of the cycle. It seems that the sort of
bifurcations that we consider here provide such a rich dynamics (this is an
ongoing project, see [41]). Here two problems arise: the generation of
blenders and their insertion in the global dynamics. One aims to “relate”
these blenders to the two saddles in the initial cycle, but this is not always
possible, see [9]. A part of this problem (concerning the generation of
blenders) was considered and solved in [14], but the embedding of these
blenders in the dynamics in [14] is still not completely adequate. The results
in [14] have three parts, it is proved that: (i) There is a center-unstable
Hénon-like family displaying (for appropriate range of parameters) blender-
horseshoes (see also [16]); (ii) There is renormalisation scheme converging to
that Hénon-like family; (iii) Some heteroclinic-like relations between the
blender and the saddles in the cycle are obtained for appropriate
$C^{1+\alpha}$, $\alpha<1$, perturbations. Here we see how (i) and (ii) hold
in our setting and in a forthcoming paper we deal with the problem of the
dynamical embedding of the obtained blender and the heteroclinic relations,
see also [41].

### Final comments

We now discuss briefly some results in the literature related to our setting.
As observed, there are interesting scenarios where critical and non-critical
dynamics are intermingled. In the equidimensional context let us recall the
homoclinic and heteroclinic two-dimensional non-transverse cycles in [43, 25,
28] and, in dimension three, the series of papers [42, 20, 21, 22, 27], where
homoclinic and heteroclinic non-transverse cycles were studied. These papers
involve renormalisation-like schemes leading to families of Hénon-like maps or
some variations: the logistic map in [43], generalisations of the Hénon map in
[25], Mira maps in [20, 21, 42], and three-dimensional Hénon maps in [24]. See
also the discussion in [26, page 226]. We observe that there are families of
three-dimensional Hénon-like maps that exhibit new types of strange
attractors, as for example the so-called wild-hyperbolic attractors444Roughly,
a wild-hyperbolic attractors possess the following two distinctive properties:
(1) wild-hyperbolic attractors allow homoclinic tangencies; (2) every such
attractor and all nearby attractors (in the $C^{r}$-topology with $r\geq 2$)
have no stable periodic orbits., for details see [44, 23, 24].

On the other hand, renormalisation methods do not seem to be sufficiently
exploited in the heterodimensional context. As far as we know, the first
example in this direction was given in [14]. Here we provide a
renormalisations scheme for a non-transverse cycle associated to pair of
saddle-foci points in the spirit of [14]. Finally, since the configurations in
this paper can be viewed as a “heterodimensional version” of [27], a natural
question is if they lead to families of Hénon-like maps with wild-hyperbolic
attractors.

Finally, let us recall that in the heterodimensional setting, critical
bifurcations were explored in [31], where heterodimensional tangencies and
non-transverse cycles lead to $C^{2}$-robust heterodimensional tangencies
(again, this is related to non-dominated dynamics). In [30] it is shown how
these tangencies lead to strange attractors.

##  2\. Statements of results

###  2.1. Center-unstable Hénon-like families

We consider the center-unstable Hénon-like family of endomorphisms
$G_{\xi,\mu,\kappa_{1},\kappa_{2}}:\mathbb{R}^{3}\to\mathbb{R}^{3}$ defined by

(2.1)
$G_{\xi,\mu,\kappa_{1},\kappa_{2}}(x,y,z)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{(}y,\mu+y^{2}+\kappa_{1}\,z^{2}+\kappa_{2}\,yz,\xi z+y\big{)}.$

By [16, Theorem 1], the family $G_{\xi,\mu,\kappa,\eta}$ has a blender-
horseshoe for all $(\xi,\mu,\kappa,\eta)$ in the open set

$\mathcal{O}=\mathcal{O}_{\varepsilon}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(1.18,1.19)\times(-10,-9)\times(-\varepsilon,\varepsilon)^{2},$

for some $\varepsilon>0$. See [14] for a version of this result for blenders
(instead of blender-horseshoes) and [29] for a complete numerical analysis of
this family including the study of the creation and annihilation of blenders.

Our results also involve the endomorphisms family
$E_{\xi,\mu,\bar{\varsigma}}:\mathbb{R}^{3}\to\mathbb{R}^{3}$ defined by

(2.2)
$E_{\xi,\mu,\bar{\varsigma}}(x,y,z)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(\xi\,x+\varsigma_{1}\,y,\,\mu+\varsigma_{2}\,y^{2}+\varsigma_{3}\,x^{2}+\varsigma_{4}\,x\,y,\,\varsigma_{5}\,y),$

where $\bar{\varsigma}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(\varsigma_{1},\varsigma_{2},\varsigma_{3},\varsigma_{4},\varsigma_{5})$.

Observe that if $\varsigma_{1}\varsigma_{2}\varsigma_{5}\neq 0$ then the
families of endomorphisms

$\big{(}\mu,E_{\xi,\mu,\bar{\varsigma}}\big{)}\quad\mbox{and}\quad\big{(}\mu,G_{\xi,\mu,\kappa,\eta}\big{)},\quad\mbox{where}\quad\kappa=\varsigma_{1}^{2}\varsigma_{3}\varsigma_{2}^{-1},\quad\eta=\varsigma_{1}\varsigma_{4}\varsigma_{2}^{-1},$

are conjugate by the change of coordinates

$\Theta:\mathbb{R}^{4}\to\mathbb{R}^{4},\quad\Theta(\mu,x,y,z)=(\varsigma_{2}^{-1}\mu,\varsigma_{2}^{-1}\varsigma_{1}z,\varsigma_{2}^{-1}y,\varsigma_{2}^{-1}\varsigma_{5}x).$

###  2.2. Bifurcation setting and main result

In what follows, $M$ denotes a closed boundaryless three manifold and
$\operatorname{Diff}^{r}(M)$, $r\geq 1$, the space of $C^{r}$-diffeomorphisms
of $M$ endowed with the norm $\|\cdot\|_{C^{r}}$ of the uniform
$C^{r}$-convergence.

We consider bifurcations of diffeomorphisms $f\in\operatorname{Diff}^{r}(M)$
having simultaneously a _heterodimensional cycle_ and a _heterodimensional
tangency._ We assume that $f$ has a pair of (periodic) saddles $P$ and $Q$ of
_indices_ (dimension of the unstable bundle) two and one, respectively, such
that conditions (A)-(C) below it holds:

(A) _Saddle-focus periodic points_ : Let $\pi(P)$ and $\pi(Q)$ be the periods
of $P$ and $Q$. Then $Df^{\pi(P)}(P)$ has a pair of non-real expanding
eigenvalues and $Df^{\pi(Q)}(Q)$ has a pair of non-real contracting
eigenvalues. We assume that the restrictions of $f^{\pi(P)}$ and $f^{\pi(Q)}$
in small neighbourhoods of $P$ and $Q$ are $C^{r}$-linearisable. Some open and
non-degenerate relations involving the eigenvalues of $Df^{\pi(P)}(P)$ and
$Df^{\pi(Q)}(Q)$ are assumed, see (3.2).

(B) _Quasi-transverse intersection_ : The one-dimensional invariant manifolds
of $P$ and $Q$ intersect quasi-transversely along the orbit of a heteroclinic
point $X$, i.e., $X\in W^{\mathrm{s}}(P,f)\cap W^{\mathrm{u}}(Q,f)$ and

$T_{X}W^{\mathrm{s}}(P,f)+T_{X}W^{\mathrm{u}}(Q,f)=T_{X}W^{\mathrm{s}}(P,f)\oplus
T_{X}W^{\mathrm{u}}(Q,f).$

Associated to the heteroclinic point $X$ there is a transition map
corresponding to some iterate of $f$, going from a neighbourhood $U_{Q}$ of
$Q$ to a neighbourhood $U_{P}$ of $P$ following the orbit of $X$. Some
conditions on this transition are given in (3.12).

(C) _Heterodimensional tangency_ : The two-dimensional invariant manifolds of
$P$ and $Q$ intersect along the orbit of a heteroclinic point $Y$ that is a
heterodimensional tangency, i.e., the orbit of $Y$ is contained in the set

$\big{(}W^{\mathrm{u}}(P,f)\cap
W^{\mathrm{s}}(Q,f)\big{)}\setminus\big{(}W^{\mathrm{u}}(P,f)\pitchfork
W^{\mathrm{s}}(Q,f)\big{)}.$

Associated to the heteroclinic point $Y$ there is a transition map
corresponding to some iterate of $f$, going from a neighbourhood $U_{P}$ of
$P$ to a neighbourhood $U_{Q}$ of $Q$ following the orbit of $Y$. Some
conditions on this transition are given in (3.22).

We are ready to state our main result.

######  Theorem 1.

Let $f:M\to M$ be a $C^{r}$-diffeomorphism, $r\geq 2$, having a cycle
associate to a pair of saddle-foci periodic points $P$ and $Q$ satisfying
conditions $\mathbf{(A)}$-$\mathbf{(C)}$. Then there is a unfolding family
$\mathfrak{F}=\\{f_{\bar{\upsilon}}\\}_{\bar{\upsilon} ̄\in\mathbb{R}^{8}}$ in
$\operatorname{Diff}^{r}(M)$ with $f_{\bar{0}}=f$ satisfying the following
properties:

For every $\xi>0$ there exists a renormalisation scheme
$\mathcal{R}(\xi,\mathfrak{F},f)$ consisting of

  * $\bullet$

two sequences of natural numbers $(m_{k})$ and $(n_{k})$ with
$(m_{k}),(n_{k})\to+\infty$ as $k\to+\infty$;

  * $\bullet$

a sequence of $C^{r}$-parameterisations $\Psi_{k}:\mathbb{R}^{3}\rightarrow
M$;

  * $\bullet$

a sequence of $C^{\infty}$-functions
$\bar{\upsilon}_{k}:\mathbb{R}\times[-\pi,\pi]^{2}\rightarrow\mathbb{R}^{8}$
parameterising the bifurcation parameter of the family
$\\{f_{\bar{\upsilon}}\\}_{\bar{\upsilon} ̄\in\mathbb{R}^{8}}$;

  * $\bullet$

a sequence of rescaled diffeomorphisms
$\mathcal{R}_{k}\big{(}f_{\bar{\upsilon}_{k}}\big{)}:M\rightarrow M$, defined
by

$\mathcal{R}_{k}\big{(}f_{\bar{\upsilon}_{k}}\big{)}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{(}f_{\bar{\upsilon}_{k}}\big{)}^{N_{2}+m_{k}+N_{1}+n_{k}},$

where $N_{1}$ and $N_{2}$ are natural numbers independent of $k$ and $\xi$,

satisfying the following convergence properties:

  * $\bullet$

for every pair of compact sets $\Delta\subset\mathbb{R}^{3}$ and
$L\subset\mathbb{R}\times[-\pi,\pi]^{2}$ it holds

$\Psi_{k}(\Delta)\rightarrow\\{Y\\},\quad{\bar{\upsilon}}_{k}(L)\rightarrow\\{\textbf{\em{0}}\\}\quad\mbox{when}\quad
k\rightarrow+\infty,$

where $Y$ is the heterodimensional tangency point of $f$ in Condition
$\mathbf{(C)}$;

  * $\bullet$

if $\varphi_{R}$ is the argument of the non-real eigenvalue of
$Df^{\pi(R)}(R)$, $R=P,Q$, then the renormalised sequence

$\Psi^{-1}_{k}\circ\mathcal{R}_{k}\big{(}f_{\bar{\upsilon}_{k}}\big{)}\circ\Psi_{k},\quad\mbox{where}\quad\bar{\upsilon}_{k}={\bar{\upsilon}}_{k}(\mu,\varphi_{P},\varphi_{Q}),\quad\mu\in\mathbb{R},$

converges in the $C^{r}$-topology and on compact sets of $\mathbb{R}^{3}$ to
the endomorphism $E_{\xi,\mu,\bar{\varsigma}}$ in (2.2), where
$\bar{\varsigma}$ depends smoothly on $f$ and $\xi$.

Equation (8.1) provides the explicit formula for
$\bar{\varsigma}=\bar{\varsigma}(f,\xi)$.

The family of endomorphisms $E_{\xi,\mu,\bar{\varsigma}}$ is called the limit
dynamics of the renormalisation scheme $\mathcal{R}(\xi,\mathfrak{F},f)$.

######  Remark 2.1.

The unfolding family in Theorem 1 involves eight parameters. Three parameters
for each non-transverse heteroclinic orbit and other two parameters to control
the arguments of the saddle-foci.

######  Remark 2.2.

An adequate choice of $\bar{\varsigma}=\bar{\varsigma}(f,\xi)$ guarantees that
the diffeomorphisms $f_{\bar{\upsilon}_{k}}$ have blender-horseshoes for every
$k$ large enough. More precisely, recalling the observations in Section 2.1,
it follows that if
$\bar{\varsigma}=(\varsigma_{1},\varsigma_{2},\varsigma_{3},\varsigma_{4},\varsigma_{5})$
is such that

(2.3) $\varsigma_{1}\varsigma_{2}\varsigma_{5}\neq
0\,\,\,\mbox{and}\,\,\,\big{(}\xi,\mu,\varsigma_{1}^{2}\varsigma_{3}\varsigma_{2}^{-1},\varsigma_{1}\varsigma_{5}\varsigma_{2}^{-1}\big{)}\in\mathcal{O},$

then the initial cycle is $C^{r}$-approximated by diffeomorphisms having
blender-horseshoes arbitrarily close to the heterodimensional tangency of $f$.

Let us restate Remark 2.2 using the $C^{r}$-persistence of blender-horseshoes
in [8, Lemma 3.9].

######  Corollary 1.

Let $f:M\to M$ be a $C^{r}$ diffeomorphism as in Theorem 1. For each
$\xi\in(1.18,1.19)$ consider the renormalisation scheme
$\mathcal{R}(\xi,\mathfrak{F},f)$ of $f$ and its limit dynamics
$E_{\xi,\mu,\bar{\varsigma}}$. If $\bar{\varsigma}$ satisfies equation (2.3)
then $f_{\bar{\upsilon}_{k}}$ has a blender-horseshoe nearby the
heterodimensional tangency of $f$ for every $k$ large enough. In particular,
for every $k$ large enough, there is a $C^{r}$-open set $\mathcal{U}_{k}$ of
diffeomorphisms $C^{r}$-close to $f$ such that each $g\in\mathcal{U}_{k}$ has
a blender-horseshoe nearby the heterodimensional tangency of $f$.

We observe that there are diffeomorphisms $f$ satisfying conditions (A)-(C)
whose vector $\bar{\varsigma}=\bar{\varsigma}(f,\xi)$ satisfies the condition
(2.3) for every $\xi\in(1.18,1.19)$. This point will be proved in Lemma 9.1
after introducing the appropriate terminology.

### Organisation of the paper

Conditions (A)-(C) are precisely stated in Section 3. In Section 4, we
describe the perturbations in the renormalisation scheme of Theorem 1. The
unfolding family $\mathfrak{F}$ is defined in Section 5. In Section 6 we
analyse the dynamics of the returns in the renormalisation scheme (i.e.,
compositions of the form $f^{N_{2}}_{\bar{\upsilon}}\circ
f^{m}_{\bar{\upsilon}}\circ f^{N_{1}}_{\bar{\upsilon}}\circ
f^{n}_{\bar{\upsilon}}$ defined nearby the heterodimensional tangency). The
convergence of the renormalised sequence is obtained in Section 8 after
obtaining an explicit formula for these maps in Section 7. The existence of
diffeomorphisms satisfying Corollary 1 is proved in Section 9.

##  3\. Description of the cycle: Conditions (A)-(C)

In this section, we describe in precise form the conditions (A)-(C) in Section
2.

###  3.1. (A) Local dynamics at $P$ and $Q$

Without loss of generality, let us assume that $P$ and $Q$ are fixed points of
$f$. We assume the existence of local $C^{r}$-linearising charts
$U_{P},U_{Q}=(-10,10)^{3}$ at the saddles $P$ and $Q$ (here the corresponding
saddle is identified with the origin) such that the expression of $f$ in these
neighbourhoods is of the form

(3.1) $\begin{split}&f|_{U_{P}}=\left(\begin{array}[]{ccc}\lambda_{P}&0&0\\\
0&\sigma_{P}\cos(2\pi\varphi_{P})&-\sigma_{P}\sin(2\pi\varphi_{P})\\\
0&\sigma_{P}\sin(2\pi\varphi_{P})&\sigma_{P}\cos(2\pi\varphi_{P})\end{array}\right),\
\quad\mbox{and}\\\
&f|_{U_{Q}}=\left(\begin{array}[]{ccc}\lambda_{Q}\cos(2\pi\varphi_{Q})&0&-\lambda_{Q}\sin(2\pi\varphi_{Q})\\\
0&\sigma_{Q}&0\\\
\lambda_{Q}\sin(2\pi\varphi_{Q})&0&\lambda_{Q}\cos(2\pi\varphi_{Q})\end{array}\right),\end{split}$

where $\varphi_{P},\varphi_{Q}\in[0,1]$, $\varphi_{P}\neq\varphi_{Q}$, and
$\lambda_{P},\lambda_{Q},\sigma_{P},\sigma_{Q}\in\mathbb{R}$ are such that

$0<|\lambda_{P}|,\,|\lambda_{Q}|<1<|\sigma_{P}|,\,|\sigma_{Q}|.$

We also assume the following condition called spectral condition of the cycle,

(3.2)
$0<\left|\big{|}|\lambda_{P}|^{\frac{1}{2}}\,\sigma_{P}\big{|}^{\eta}\sigma_{Q}\right|<1,\quad\mbox{where}\quad\eta=\dfrac{\log|\lambda_{Q}^{-1}|}{\log|\sigma_{P}|}.$

This condition plays a key role in the convergence of the renormalisation
scheme. Lemma 3.1 claims that this spectral condition is open and non-empty.

####  3.1.1. Spectral conditions on the cycle

Taking the square of $f$, if necessary, we can assume that
$\lambda_{P},\sigma_{P},\lambda_{Q}$ and $\sigma_{Q}$ are all positive. With
this in mind, in the next proposition we show that the condition (3.3) is non
degenerate. As consequence the set of diffeomorphisms $f$ such that its
eigenvalues satisfy the spectral condition (3.3) is an open set in
$\operatorname{Diff}^{r}(M)$.

######  Lemma 3.1.

Let $\mathcal{P}$ be the set of points
$(\tilde{\lambda},\tilde{\sigma},\lambda,\sigma)\in\mathbb{R}^{4}$ such that
$0<\lambda,\tilde{\lambda}<1$ and $\tilde{\sigma},\sigma>1$ and

(3.3)
$0<(\tilde{\lambda}^{\frac{1}{2}}\,\tilde{\sigma})^{\eta}\sigma<1,\quad\mbox{where}\quad\eta=\frac{\log\lambda^{-1}}{\log\tilde{\sigma}}.$

The set $\mathcal{P}$ is non-empty and open.

###### Proof.

The set $\mathcal{P}$ is open by definition. Thus it remains to show the
existence of numbers satisfying these inequalities. For this, consider the set

(3.4) $\widetilde{\mathcal{Z}}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{\\{}(\tilde{\lambda},\tilde{\sigma})\in(0,1)\times(1,+\infty):0<\tilde{\lambda}^{\frac{1}{2}}\,\tilde{\sigma}<1\big{\\}}\subset\mathbb{R}^{2}.$

The lemma follows the next claim.

######  Claim 3.2.

For
$(\tilde{\lambda},\tilde{\sigma},\lambda)\in\widetilde{\mathcal{Z}}\times(0,1)$
there is a interval $I_{(\lambda,\tilde{\lambda},\tilde{\sigma})}$, such that
every
$(\tilde{\lambda},\tilde{\sigma},\lambda,\sigma)\in\widetilde{\mathcal{Z}}\times(0,1)\times
I_{(\lambda,\tilde{\lambda},\tilde{\sigma})}$ satisfies (3.3).

###### Proof.

The equation (3.3) is equivalent to

$\dfrac{\log\lambda^{-1}}{\log\tilde{\sigma}}\,\log(\tilde{\lambda}^{\frac{1}{2}}\,\tilde{\sigma})+\log\sigma<0.$

Note that every vector $(\tilde{\lambda},\tilde{\sigma},\lambda)$ in
$\widetilde{\mathcal{Z}}\times(0,1)$ satisfies the inequality

$\dfrac{\log\lambda^{-1}}{\log\tilde{\sigma}}\,\log(\tilde{\lambda}^{\frac{1}{2}}\,\tilde{\sigma})<0.$

Thus for every $\sigma>1$ such that

(3.5)
$\frac{\log\lambda^{-1}}{\log\tilde{\sigma}}\log(\tilde{\lambda}^{\frac{1}{2}}\,\tilde{\sigma})<-\log\sigma,$

we get that $(\tilde{\lambda},\tilde{\sigma},\lambda,\sigma)$ satisfies (3.3).
Now it is enough to take
$I_{(\lambda,\tilde{\lambda},\tilde{\sigma})}=(1,\sigma^{*}_{(\lambda,\tilde{\lambda},\tilde{\sigma})})$,
where $\sigma^{*}_{(\lambda,\tilde{\lambda},\tilde{\sigma})}$ is the supremum
of the $\sigma>1$ satisfying (3.5). ∎

The proof of the lemma is now complete. ∎

###  3.2. (B) Quasi-transverse intersection and transition from $Q$ to $P$.

We now describe the transition from $Q$ to $P$ along the orbit of the quasi-
transverse intersection point $X\in W^{\mathrm{s}}(P,f)\cap
W^{\mathrm{u}}(Q,f)$.

Using Condition (A), we can consider the local invariant manifolds of $P$ and
$Q$ defined by

$W^{\mathrm{s}}_{\mathrm{loc}}(P,f)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(-10,10)\times\\{(0,0)\\},\qquad
W^{\mathrm{u}}_{\mathrm{loc}}(P,f)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\\{0\\}\times(-10,10)^{2}$

and

$W^{\mathrm{s}}_{\mathrm{loc}}(Q,f)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(-10,10)\times\\{0\\}\times(-10,10),\qquad
W^{\mathrm{u}}_{\mathrm{loc}}(Q,f)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\\{0\\}\times(-10,10)\times\\{0\\}.$

Replacing $X$ by some backward iterate, if necessary, we can assume that $X\in
W^{\mathrm{u}}_{\mathrm{loc}}(Q,f)$. Rescalling the segment
$W^{\mathrm{u}}_{\mathrm{loc}}(Q,f)$, if necessary, we can assume that
$X=(0,1,0)\in U_{Q}$. By hypothesis, there exists $N_{1}\in\mathbb{N}$ such
that $\widetilde{X}=f^{N_{1}}(X)\in W^{\mathrm{s}}_{\mathrm{loc}}(P,f)\subset
U_{P}$. We choose $N_{1}$ so that $f^{N_{1}-1}({X})\notin U_{P}$ (i.e.
$\widetilde{X}$ is the first iterated of the orbit of $X$ meeting $U_{P}$).
Arguing as before, we can take $\widetilde{X}=(1,0,0)\in U_{P}$. We assume
that there are small neighbourhoods $U_{X}$ of $X$ in $U_{Q}$ and
$U_{\widetilde{X}}$ of $\widetilde{X}$ in $U_{P}$ such that the map
$f^{N_{1}}|_{U_{X}}\colon U_{X}\to U_{\widetilde{X}}$ is given by

(3.12) $\displaystyle f^{N_{1}}\left(\begin{array}[]{cc}x\\\ y+1\\\
z\end{array}\right)=\left(\begin{array}[]{ccc}1+\alpha_{1}x+\alpha_{2}y+\alpha_{3}z+\widetilde{H}_{1}(x,y,z)\\\
\beta_{1}x+\beta_{2}y+\beta_{3}z+\widetilde{H}_{2}(x,y,z)\\\
\gamma_{1}x+\gamma_{2}y+\gamma_{3}z+\widetilde{H}_{3}(x,y,z)\end{array}\right),$

where $\alpha_{i},\beta_{i},\gamma_{i}$, $i=1,2,3$ are constants such that

(3.13) $\beta_{1}=\beta_{3}=\gamma_{1}=\gamma_{2}=0$

and the maps $\widetilde{H}_{i}$, $i=1,2,3$, are higher order terms satisfying

(3.14) $\widetilde{H}_{i}(\textbf{0})=\frac{\partial}{\partial
x}\widetilde{H}_{i}(\textbf{0})=\frac{\partial}{\partial
y}\widetilde{H}_{i}(\textbf{0})=\frac{\partial}{\partial
z}\widetilde{H}_{i}(\textbf{0})=0.$

Note that as $f^{N_{1}}$ is a (local) diffeomorphism it holds that

(3.15) $\alpha_{1}\,\beta_{2}\,\gamma_{3}\neq 0.$

We call $f^{N_{1}}$ and $N_{1}$ transition map and _transition time from $Q$
to $P$_, respectively.

######  Remark 3.3.

The vector $\bar{u}=(0,1,0)$ spans $T_{X}W^{\mathrm{u}}(Q,f)$ and
$Df^{N_{1}}(X)(\bar{u})=(\alpha_{2},\beta_{2},0)$, Since, by (3.15),
$\beta_{2}\neq 0$, this vector does not belong to to
$T_{\tilde{X}}W^{\mathrm{s}}(P,f)$ (which is spanned by $(1,0,0)$). Thus $X$
is a quasi-transverse heteroclinic point.

###  3.3. (C) Heterodimensional tangency and transition from $P$ to $Q$

We now describe the transition from $P$ to $Q$ along the orbit of the
heterodimensional tangency point $Y\in W^{\mathrm{u}}(P,f)\cap
W^{\mathrm{s}}(Q,f)$. By replacing $Y$ by some backward iterate we can assume
that $Y\in W^{\mathrm{u}}_{\mathrm{loc}}(P,f)$. By hypothesis, there is
$N_{2}\in\mathbb{N}$ such that $\widetilde{Y}=f^{N_{2}}(Y)\in
W^{\mathrm{s}}_{\mathrm{loc}}(Q,f)$. We choose $N_{2}$ so that
$f^{N_{2}-1}({Y})\notin U_{Q}$. By some linear coordinate change in $U_{P}$
and in $U_{Q}$, one can assume $Y=(0,1,1)\in U_{P}$ and
$\widetilde{Y}=(1,0,1)\in U_{Q}$. Note that this coordinate change can be done
without changing the previous choice of $X$ and $\widetilde{X}$. We assume
that there are small neighbourhoods $U_{Y}$ of $Y$ in $U_{P}$ and
$U_{\widetilde{Y}}$ of $\widetilde{Y}$ in $U_{Q}$ such that the map
$f^{N_{2}}|_{U_{Y}}\colon U_{Y}\to U_{\widetilde{Y}}$ is given by

(3.22) $\displaystyle\quad f^{N_{2}}\left(\begin{array}[]{cc}x\\\ 1+y\\\
1+z\end{array}\right)=\
\left(\begin{array}[]{cc}1+a_{1}x+a_{2}y+a_{3}z+H_{1}(x,y,z)\\\
b_{1}x+b_{2}y^{2}+b_{3}z^{2}+b_{4}yz+H_{2}(x,y,z)\\\
1+c_{1}x+c_{2}y+c_{3}z+H_{3}(x,y,z)\end{array}\right),$

where $a_{i},b_{i},c_{i}$, $i=1,2,3$, are constants with

(3.23) $c_{2}=c_{3}$

and the maps $H_{i}$, $i=1,2,3$, are higher order terms satisfying the
following conditions:

(3.24) $\begin{split}&\quad H_{i}(\textbf{0})=\frac{\partial}{\partial
x}H_{i}(\textbf{0})=\frac{\partial}{\partial
y}H_{i}(\textbf{0})=\frac{\partial}{\partial z}H_{i}(\textbf{0})=0,\\\
&\quad\frac{\partial^{2}}{\partial
y^{2}}H_{2}(\textbf{0})=\frac{\partial^{2}}{\partial
z^{2}}H_{2}(\textbf{0})=\frac{\partial^{2}}{\partial y\,\partial
z}H_{2}(\textbf{0})=0.\\\ \end{split}$

Note that since $f^{N_{2}}$ is a (local) diffeomorphism it follows that

(3.25) $b_{1}\,c_{2}\,(a_{3}-a_{2})\neq 0.$

We call $f^{N_{2}}$ and $N_{2}$ transition map and _transition time from $P$
to $Q$_, respectively.

Finally, we assume the following condition on the parameters $a_{2},a_{3}$,
and $\gamma_{3}$ above

(3.26) $\gamma_{3}\,(a_{3}-a_{2})>0.$

Note that the choices of $a_{2},a_{3},$ and $\gamma$ are compatible with
(3.25) and (3.15).

######  Remark 3.4.

The vectors $\bar{v}=(0,1,0)$ and $\bar{w}=(0,0,1)$ span
$T_{Y}W^{\mathrm{u}}(P,f)$ and $Df^{N_{2}}(Y)(\bar{v})=(a_{2},0,c_{2})$ and
$Df^{N_{2}}(Y)(\bar{w})=(a_{3},0,c_{3})$. Since these vectors belong to
$T_{\widetilde{Y}}W^{\mathrm{s}}(Q,f)$ (which is spanned by $(1,0,0)$ and
$(0,0,1)$), the point $Y$ corresponds to a heterodimensional tangency.

##  4\. Translation and rotation-like perturbations

In this section, we describe the two types of perturbations (translation and
rotation like) involved in the renormalisation scheme of Theorem 1.

We consider auxiliary $C^{r}$-bump functions
$b_{\rho}:\mathbb{R}\to\mathbb{R}$, $\rho>0$, satisfying

$\left\\{\begin{array}[]{c l}b_{\rho}(x)=0,&\mbox{if}\quad|x|\geq\rho,\\\
0<b_{\rho}(x)<1,&\mbox{if}\quad\dfrac{\rho}{2}<|x|<\rho,\\\
b_{\rho}(x)=1,&\mbox{if}\quad|x|\leq\dfrac{\rho}{2},\end{array}\right.$

and their associated $C^{r}$-bump functions
$B_{\rho}:\mathbb{R}^{3}\to\mathbb{R}$ defined by

$B_{\rho}(x,y,z)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}b_{\rho}(x)\,b_{\rho}(y)\,b_{\rho}(z).$

###  4.1. Translation-like perturbations

For $Z_{0}\in\mathbb{R}^{3}$ denote by
$\mathbb{B}_{\rho}(Z_{0})\subset\mathbb{R}^{3}$ the open ball of radius $\rho$
and center $Z_{0}$. Given $Z_{0}\in\mathbb{R}^{3}$, consider the family of
$C^{r}$-maps

(4.1)
$T_{Z_{0},\bar{w}}:\mathbb{R}^{3}\to\mathbb{R}^{3},\quad\bar{w}\in\mathbb{R}^{3},$

defined by

  * •

if $Z+Z_{0}\in\mathbb{B}_{\rho}(Z_{0})$, then
$T_{Z_{0},\bar{w}}(Z+Z_{0})=Z+Z_{0}+B_{\rho}(Z)\bar{w}$,

  * •

if $Z\not\in\mathbb{B}_{\rho}(Z_{0})$, then $T_{Z_{0},\bar{w}}(Z)=Z$.

Note that

$\big{\|}T_{Z_{0},\bar{w}}-\mathrm{id}\big{\|}_{C^{r}}\leq\big{\|}B_{\rho}\big{\|}_{C^{r}}\cdot||\bar{w}||.$

Thus, for every $||\bar{w}||$ small enough, the map $T_{Z_{0},\bar{w}}$ is a
$C^{r}$-perturbation of the identity supported in $\mathbb{B}_{\rho}(Z_{0})$.
Note also that, by construction,

$T_{Z_{0},\bar{w}}(\mathbb{B}_{\rho}(Z_{0}))=\mathbb{B}_{\rho}(Z_{0}).$

We call $T_{Z_{0},\bar{w}}$ a translation-like perturbation of the identity.

###  4.2. Rotation-like perturbations

Consider the families of linear maps

$I^{x}_{\omega},\,I^{y}_{\omega}\colon\mathbb{R}^{3}\to\mathbb{R}^{3},\quad\omega\in[-\pi,\pi],$

given by

$\displaystyle I^{x}_{\omega}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\left(\begin{array}[]{ccc}1&0&0\\\
0&\cos(2\pi\omega)&-\sin(2\pi\omega)\\\
0&\sin(2\pi\omega)&\cos(2\pi\omega)\end{array}\right),\quad
I^{y}_{\omega}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\left(\begin{array}[]{ccc}\cos(2\pi\omega)&0&-\sin(2\pi\omega)\\\
0&1&0\\\ \sin(2\pi\omega)&0&\cos(2\pi\omega)\end{array}\right).$

Observe that if $\omega=0$ then $I^{x}_{0}=I^{y}_{0}=\mathrm{id}$.

Consider the families of $C^{r}$-diffeomorphisms

(4.3)
$\begin{split}&R^{x}_{{\omega},\rho}\colon\mathbb{R}^{3}\to\mathbb{R}^{3},\quad
R^{x}_{{\omega,\rho}}(W)=I^{x}_{\omega\,b_{\rho}(||W||)}\cdot W^{T},\\\
&R^{y}_{{\omega,\rho}}\colon\mathbb{R}^{3}\to\mathbb{R}^{3},\quad
R^{y}_{{\omega,\rho}}(W)=I^{y}_{\omega\,b_{\rho}(||W||)}\cdot
W^{T}\end{split}$

where $W^{T}$ denotes the transpose of the vector $W\in\mathbb{R}^{3}$.

Note that the restriction of $R^{x}_{{\omega,\rho}}$ to the set
$[-\frac{\rho}{2},\frac{\rho}{2}]^{3}$ coincides with $I^{x}_{\omega}$ and
$R^{x}_{{\omega,\rho}}$ is the identity map in the complement of
$[-\rho,\rho]^{3}$. Analogously, $R^{y}_{{\omega,\rho}}=I^{y}_{\omega}$ in
$[-\frac{\rho}{2},\frac{\rho}{2}]^{3}$ and $R^{y}_{{\omega,\rho}}=\mathrm{id}$
in $\mathbb{R}^{3}\setminus[-\rho,\rho]^{3}$. Moreover,

$R^{x}_{{\omega,\rho}}\big{(}[-\rho,\rho]^{3}\big{)}=R^{y}_{{\omega,\rho}}\big{(}[-\rho,\rho]^{3}\big{)}=[-\rho,\rho]^{3}.$

It is not hard to see that there are constants $C_{\rho}$ and
$C^{\prime}_{\rho}$ with

$\big{\|}R^{x}_{{\omega,\rho}}-\mathrm{id}\big{\|}_{C^{r}}<C_{\rho}|{\omega}|\quad\mbox{and}\quad\big{\|}R^{y}_{{\omega,\rho}}-\mathrm{id}\big{\|}_{C^{r}}<C^{\prime}_{\rho}|{\omega}|.$

Thus, for every $\omega$ small enough, the maps $R^{x}_{{\omega,\rho}}$ and
$R^{y}_{{\omega,\rho}}$ are $C^{r}$-perturbations of identity supported in
$[-\rho,\rho]^{3}$.

We call $R^{x}_{{\omega,\rho}}$ and $R^{y}_{{\omega,\rho}}$ rotation-like
perturbations of the identity.

##  5\. The unfolding family $\mathfrak{F}$

We now describe the $8$-parameter family
$\mathfrak{F}=\\{f_{\bar{\upsilon}}\\}_{\bar{\upsilon}}$ of
$C^{r}$-diffeomorphisms, $r\geq 2$, unfolding the cycle of $f$ at
$\bar{\upsilon}=\bar{0}\in\mathbb{R}^{8}$ (i.e., $f_{\bar{0}}=f$) in Theorem
1. The cycle of $f$ has two parts with (say) “independent” unfoldings, one
associated to the heterodimensional tangency and another one associated to the
quasi-transverse heteroclinic point. Besides the unfolding of these
heteroclinic non-transverse orbits we need to consider slight adjustments on
the arguments of the eigenvalues of the saddles $P$ and $Q$. These adjustments
are given by the rotation-like perturbations in Section 4. In summary, the
family $\mathfrak{F}$ is obtained considering translation-like perturbations
nearby the heteroclinic points $\widetilde{X}$ and $\widetilde{Y}$ and
rotation-like perturbations corresponding to small changes of the arguments
$\varphi_{P}$ and $\varphi_{Q}$. More precisely, the unfolding family is of
the form

(5.1) $f_{\bar{\upsilon}}=\Gamma_{\bar{\upsilon}}\circ
f,\quad\mbox{with}\quad\bar{\upsilon}=(\bar{\mu},\bar{\nu},{\alpha},\beta)\in[-\epsilon,\epsilon]^{8},$

where $\Gamma_{\bar{\upsilon}}$ is the perturbation of the identity obtained
as follows. For $R=P,Q,\widetilde{X},\widetilde{Y}$ we take pairwise disjoint
neighbourhoods $V_{R}\subset U_{P}\cup U_{Q}$ and let

(5.2)
$\Gamma_{\bar{\upsilon}=(\bar{\mu},\bar{\nu},{\alpha},\beta)}(Z)=\left\\{\begin{split}&\Gamma^{P}_{\alpha}(Z)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}R^{x}_{\alpha,\rho}(Z),\quad\mbox{if $Z\in V_{P}$},\\\
&\Gamma^{\mathrm{qt}}_{\bar{\nu}}(Z)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}T_{\widetilde{X},\bar{\nu}}(Z),\quad\mbox{if $Z\in
V_{\widetilde{X}}$},\\\
&\Gamma^{Q}_{\beta}(Z)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}R^{y}_{\beta,\rho}(Z),\quad\mbox{if $Z\in V_{Q}$},\\\
&\Gamma^{\mathrm{ht}}_{\bar{\mu}}(Z)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}T_{\widetilde{Y},\bar{\mu}}(Z),\quad\mbox{if $Z\in
V_{\widetilde{Y}}$},\\\ &\mathrm{id}(Z),\quad\mbox{\,\,\,if $Z\not\in
V_{P}\cup V_{Q}\cup V_{X}\cup V_{Y}$},\end{split}\right.$

where $\rho>0$ is small $T_{Z,\bar{w}},R^{x}_{\omega,\rho}$ and
$R^{y}_{\omega,\rho}$ are as in Section 4. The precise choice of these
neighbourhoods $V_{R}$ is done below and the choice of $\rho$ in Sections 5.1
and 5.2. Properties of the map $\Gamma_{\bar{\upsilon}}$ are discussed in
Section 5.3. Finally, we call $\Gamma^{\mathrm{ht}}_{\bar{\mu}}$ and
$\Gamma^{\mathrm{qt}}_{\bar{\nu}}$ _unfolding perturbations_ and
$\Gamma^{P}_{\alpha}$ and $\Gamma^{Q}_{\beta}$ _rotating perturbations_.

We recall in Conditions (A)-(C), the definition of the neighbourhoods

$U_{R}=(-10,10)^{3}\ni R\quad\mbox{and}\quad
U_{\widetilde{Z}}\ni\widetilde{Z},\quad\mbox{where}\quad R=P,Q,\quad Z=X,Y.$

We recall that by the choice of $N_{1}$ and $N_{2}$ in Conditions (B)-(C), the
points $\widetilde{X}$ and $\widetilde{Y}$ satisfies
$f^{-1}(\widetilde{X})\notin U_{P}$ and $f^{-1}(\widetilde{Y})\notin U_{Q}$.

Consider the neighbourhoods $V_{R}=[-8,8]^{3}\subset U_{R}$ of $R=P,Q$.

###  5.1. The unfolding perturbations $\Gamma^{\mathrm{qt}}_{\bar{\mu}}$ and
$\Gamma^{\mathrm{ht}}_{\bar{\nu}}$

Consider a small enough $\rho>0$ such that

$\begin{split}&\mathbb{B}_{\rho}(\widetilde{X})\subset U_{\widetilde{X}},\quad
f^{-1}\Big{(}\mathrm{closure}\big{(}\mathbb{B}_{\rho}(\widetilde{X})\big{)}\Big{)}\cap
U_{P}=\emptyset,\quad\mbox{and}\\\ &\mathbb{B}_{\rho}(\widetilde{Y})\subset
U_{\widetilde{Y}},\quad
f^{-1}\Big{(}\mathrm{closure}\big{(}\mathbb{B}_{\rho}(\widetilde{Y})\big{)}\Big{)}\cap
U_{Q}=\emptyset.\end{split}$

In particular,

$f(\mathbb{B}_{\rho}(\widetilde{Z}))\cap\mathbb{B}_{\rho}(\widetilde{Z})=\emptyset,\quad
Z=X,Y;$

and

$P\notin\mathrm{closure}(\mathbb{B}_{\rho}(\widetilde{X}))\quad\mbox{and}\quad
Q\notin\mathrm{closure}({\mathbb{B}}_{\rho}(\widetilde{Y})).$

For $\bar{\mu}$ and $\bar{\nu}$ in $\mathbb{R}^{3}$ we define the unfolding
perturbations as

$\Gamma^{\mathrm{ht}}_{\bar{\mu}}\colon M\to
M,\quad\Gamma^{\mathrm{ht}}_{\bar{\mu}}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}T_{\widetilde{Y},\bar{\mu}}\quad\mbox{and}\quad\Gamma^{\mathrm{qt}}_{\bar{\nu}}\colon
M\to
M,\quad\Gamma^{\mathrm{qt}}_{\bar{\nu}}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}T_{\widetilde{X},\bar{\nu}},$

where $T_{Z_{0},\bar{w}}$ are the translation-like perturbations in (4.1). See
Figure 3.

Figure 3. The unfolding family $\mathfrak{F}$.

As was observed, if $||\bar{\mu}||$ and $||\bar{\nu}||$ are small enough the
maps $\Gamma^{\mathrm{ht}}_{\bar{\mu}}$ and $\Gamma^{\mathrm{qt}}_{\bar{\nu}}$
are $C^{r}$-perturbations of the identity supported in
$\mathbb{B}_{\rho}(\widetilde{Y})$ and $\mathbb{B}_{\rho}(\widetilde{X})$,
respectively, with

$\Gamma^{\mathrm{qt}}_{\bar{\nu}}(\mathbb{B}_{\rho}(\widetilde{X}))=\mathbb{B}_{\rho}(\widetilde{X})\quad\mbox{and}\quad\Gamma^{\mathrm{ht}}_{\bar{\mu}}(\mathbb{B}_{\rho}(\widetilde{Y}))=\mathbb{B}_{\rho}(\widetilde{Y}).$

###  5.2. The rotating perturbations $\Gamma^{P}_{{\alpha}}$ and
$\Gamma^{Q}_{\beta}$.

For $\alpha$ and $\beta$ in $\mathbb{R}$, we define the rotating perturbations

$\Gamma^{P}_{\alpha}\colon M\to
M,\quad\Gamma^{P}_{\alpha}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}R^{x}_{\alpha,8}\quad\mbox{and}\quad\Gamma^{Q}_{\beta}\colon M\to
M,\quad\Gamma^{Q}_{\beta}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}R^{y}_{\beta,8},$

where $R^{x}_{{\omega,\rho}}$ and $R^{y}_{{\omega,\rho}}$ are the rotation
like-perturbations in (4.3). As was observed, if $\alpha$ and $\beta$ are
small enough the maps $\Gamma^{P}_{\alpha}$ and $\Gamma^{Q}_{\beta}$ are
$C^{r}$-perturbations of the identity supported in $V_{P}$ and $V_{Q}$,
respectively, such that

$\Gamma^{P}_{\alpha}(V_{P})=V_{P}\quad\mbox{and}\quad\Gamma^{Q}_{\beta}(V_{Q})=V_{Q}.$

###  5.3. Properties of the unfolding family $\mathfrak{F}$

We now list some relevant properties satisfied by the family $\mathfrak{F}$.

######  Remark 5.1 (Properties of $\mathfrak{F}$).

By construction, $f_{\bar{0}}=f$ and for every small (in norm)
$\bar{\upsilon}=(\bar{\mu},\bar{\nu},\alpha,\beta)$ we have that
$f_{\bar{\upsilon}}$ is a diffeomorphism $C^{r}$-close to $f$ having a pair of
saddle points $P_{\bar{\upsilon}}=P$ and $Q_{\bar{\upsilon}}=Q$ such that (see
Figure 3):

  1. (1)

For every $W\in U_{P}\cap f^{-1}(V_{P})$, it holds
$f_{\bar{\upsilon}}(W)=\Gamma^{P}_{{\alpha}}\circ f(W).$

  2. (2)

For every $W\in U_{Q}\cap f^{-1}(V_{Q})$, it holds
$f_{\bar{\upsilon}}(W)=\Gamma^{Q}_{\beta}\circ f(W).$

  3. (3)

$f_{\bar{\upsilon}}\big{(}f^{-1}\big{(}\mathbb{B}_{\rho}(\widetilde{X})\big{)}\big{)}=\mathbb{B}_{\rho}(\widetilde{X})$
and
$f^{N_{1}}_{\bar{\upsilon}}(X)=f_{\bar{\nu}}\big{(}f^{-1}(\widetilde{X})\big{)}=\widetilde{X}+\bar{\nu}$.

  4. (4)

$f_{\bar{\upsilon}}\big{(}f^{-1}\big{(}\mathbb{B}_{\rho}(\widetilde{Y})\big{)}\big{)}=\mathbb{B}_{\rho}(\widetilde{Y})$
and
$f^{N_{2}}_{\bar{\upsilon}}(X)=f_{\bar{\mu}}\big{(}f^{-1}(\widetilde{Y})\big{)}=\widetilde{Y}+\bar{\mu}$.

  5. (5)

$f_{\bar{\upsilon}}=f$ in $M\setminus V_{P}\cup V_{Q}\cup
f^{-1}\big{(}\mathbb{B}_{\rho}(\widetilde{X})\big{)}\cup
f^{-1}\big{(}\mathbb{B}_{\rho}(\widetilde{Y})\big{)}$.

##  6\. Return maps at the heterodimensional tangency

The renormalisation scheme of $f$ in Theorem 1 involves return maps (defined
on a small neighbourhood of the heterodimensional tangency point
$\widetilde{Y}$) of the form

(6.1) $F^{m,n}_{\bar{\upsilon}}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}f^{N_{2}}_{\bar{\upsilon}}\circ f^{m}_{\bar{\upsilon}}\circ
f^{N_{1}}_{\bar{\upsilon}}\circ f^{n}_{\bar{\upsilon}},$

where $N_{1}$ and $N_{2}$ are the transition times between neighbourhoods of
the saddles in the cycle in Conditions B and C (see Sections 3.2 and 3.3). For
the next discussion recall the definitions of the quasi-transverse
heteroclinic point $\widetilde{X}$ and of the points $X\in U_{Q}$ with
$f^{N_{1}}(X)=\widetilde{X}\in U_{P}$ and $Y\in U_{P}$ with
$f^{N_{2}}(Y)=\widetilde{Y}\in U_{Q}$. For suitable choices of $n$ and $m$, we
are interested in iterations of points close to heterodimensional tangency
point $\widetilde{Y}$ by the map $f_{\bar{\upsilon}}$ that after $n$ iterates
in $U_{Q}$ land in a neighbourhood of $X$. These points are mapped by
$f^{N_{1}}_{\bar{\upsilon}}$ close to $\widetilde{X}$ in $U_{P}$. Thereafter,
they remain in $U_{P}$ during $m$ iterates of $f_{\bar{\upsilon}}$, landing in
a neighbourhood of $Y$. Finally, they return nearby $\widetilde{Y}$ by the
transition $f^{N_{2}}_{\bar{\upsilon}}$. The specially selected times $n$ and
$m$ are called _sojourn times_ in the neighbourhoods $U_{P}$ and $U_{Q}$.
These times “determine” the set nearby $\widetilde{Y}$ where this return is
defined.

The composition (6.1) also demands another cares. For instance, to guarantee
that after $m$ iterations certain points nearby $\widetilde{X}$ are mapped in
the domain of the transition from $P$ to $Q$ (i.e., in a small neighbourhood
of $Y$) it is necessary a small change $\varphi_{P}+{\alpha}_{m}(\varphi_{P})$
of the argument $\varphi_{P}$ of $Df(P)$, where ${\alpha}_{m}(\varphi_{P})$
depends on the sojourn time $m$. For the sojourn times $n$, we need to
consider similar adjustments ${\beta}_{n}(\varphi_{Q})$ of the argument
$\varphi_{Q}$ of $Df(Q)$. The arguments
$\varphi_{P}+{\alpha}_{m}(\varphi_{P})$ and
$\varphi_{Q}+{\beta}_{n}(\varphi_{Q})$ are called adapted arguments.

We divide the study of the renormalisation scheme in two parts: (i) choice of
adequate sojourn times $m$ and $n$ and adapted arguments, see Sections 6.1 and
6.2; (ii) choice of suitable sequences of unfolding parameters
$\bar{\upsilon}_{m,n}$ and charts $\Psi_{m,n}\colon\mathbb{R}^{3}\to M$, see
Section 6.3. The convergence of the renormalised sequence
$\Psi_{m,n}^{-1}\circ F^{m,n}_{\bar{\upsilon}_{m,n}}\circ\Psi_{m,n}$ will be
studied in Sections 7 and 8.

###  6.1. Sojourn times

Consider the set

$\mathcal{Z}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\\{(\sigma,\lambda)\in\mathbb{R}^{2}:0<\lambda<1<\sigma\\}.$

An application of the next lemma will provide the desired sojourn times.

######  Lemma 6.1 (Lemma 5.1 in [14]).

There is a residual subset $\mathcal{R}$ of $\mathcal{Z}$ consisting of points
$(\sigma,\lambda)$ satisfying the following property. For every $\epsilon>0$,
$\xi>0$, $\tau>0$ and $N_{0}>0$, with $\epsilon<\xi$, there exist integers
$m,n>N_{0}$ such that

$|\tau\sigma^{m}\lambda^{n}-\xi|<\epsilon\quad\mbox{and}\quad|m-n\eta-\tilde{\eta}|<1,$

where

$\eta=\frac{\log\lambda^{-1}}{\log\sigma}\quad\mbox{and}\quad\tilde{\eta}=\frac{\log(\tau\xi^{-1})}{\log\sigma}.$

In particular, there exit sequences $(m_{k}),(n_{k})\to+\infty$ as
$k\to+\infty$ such that

$\lim_{k\to+\infty}\,\sigma^{m_{k}}\,\lambda^{n_{k}}=\tau^{-1}\,\xi.$

######  Remark 6.2 (Choice of sojourn times).

A _sequence of sojourn times_ (adapted to $\sigma_{P},\lambda_{Q},\tau$ and
$\xi$) is any sequence $(\mathbf{s}_{k})_{k}$, with
$\mathbf{s}_{k}=(m_{k},n_{k})\in\mathbb{N}^{2}$, obtained by applying Lemma
6.1 to:

  * •

$(\sigma_{P},\lambda_{Q})\in\mathcal{Z}$, where $\lambda_{Q},\sigma_{P}$
satisfying the spectral condition of the cycle in (3.3)555As the spectral
condition (3.3) is open, we can suppose that
$(\sigma_{P},\lambda_{Q})\in\mathcal{R}$.;

  * •

${\tau}={\tau}(a_{2},a_{3},\gamma_{3})\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\dfrac{\gamma_{3}(a_{3}-a_{2})}{\sqrt{2}}$, where $a_{2}$, $a_{3}$
and $\gamma_{3}$ are the constants in the definition of the transition maps
between $P$ and $Q$, see (3.12) and (3.22). Recall that by (3.26) it holds
${\tau}(a_{2},a_{3},\gamma_{3})>0$;

  * •

$\xi>0$ is arbitrary but fixed.

As a consequence, for a sequence of sojourn times
$\mathbf{s}_{k}=(m_{k},n_{k})$ it holds

(6.2)
$\sigma_{P}^{m_{k}}\,\lambda_{Q}^{n_{k}}\rightarrow\tau^{-1}\xi=\left(\frac{\gamma_{3}(a_{3}-a_{2})}{\sqrt{2}}\right)^{-1}\xi,\quad
k\to+\infty.$

We say that the sequence ${\mathbf{s}}_{k}$ is _adapted to $\tau^{-1}\xi$._

###  6.2. Adapted arguments

We now discuss the choice of the “adjusting arguments”. Given a sequence
$\mathbf{s}_{k}=(m_{k},n_{k})$ of sojourn times consider a sequence
$\Theta_{k}=(\zeta_{m_{k}},\vartheta_{n_{k}})$ in $\mathbb{R}^{2}$ such that
$\Theta_{k}\to(0,0)$ as $k\to+\infty$. We call the pair
${\mathfrak{S}}_{k}=({\mathbf{s}}_{k},\Theta_{k})$ a _sequence of sojourn
times with associated arguments_. We define the _sequences
${\alpha}^{P}_{k}=\alpha^{P}_{{\mathfrak{S}}_{k}}$ and
$\beta^{Q}_{k}=\beta^{Q}_{{\mathfrak{S}}_{k}}$ of argument adjustment maps_ as
follows,

(6.3)
$\begin{split}&{\alpha}^{P}_{k}:[-\pi,\pi]\to[-\pi,\pi],\quad{\alpha}^{P}_{k}(\theta)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\frac{1}{2\pi m_{k}}\left(\frac{\pi}{4}-2\pi
m_{k}\theta+2\pi[m_{k}\theta]+\zeta_{m_{k}}\right);\\\
&{\beta}^{Q}_{k}:[-\pi,\pi]\to[-\pi,\pi],\quad{\beta}^{Q}_{k}(\omega)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\frac{1}{2\pi n_{k}}\left(\frac{\pi}{2}-2\pi
n_{k}\omega+2\pi[n_{k}\omega]+\vartheta_{n_{k}}\right),\end{split}$

where $[x]$ denotes the integer part of $x\in\mathbb{R}$. Since
$x-[x]\in[0,1)$ it follows that

${\alpha}^{P}_{k}(\theta)\to 0,\quad{\beta}^{Q}_{k}(\omega)\to 0,\quad
k\to+\infty,$

for every fixed $\theta,\omega\in[-\pi,\pi]$.

The pair of sequences

$\theta^{P}_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\theta+{\alpha}^{P}_{k}(\theta)\to\theta\quad\mbox{and}\quad\omega^{Q}_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\omega+{\beta}^{Q}_{k}(\omega)\to\omega,\quad k\to+\infty,$

is called sequence of arguments adapted to $\theta$ and $\omega$ associated to
${\mathfrak{S}}_{k}=({\mathbf{s}}_{k},\Theta_{k})$.

###  6.3. Elements of the renormalisation scheme

Consider $0<\lambda_{Q}<1<\sigma_{P}$ satisfying (3.3) and $\xi>0$. If
$(\sigma_{P},\lambda_{Q})\in\mathcal{Z}$, Lemma 6.1 provides a sequence of
sojourn times $\mathbf{s}_{k}=(m_{k},n_{k})\in\mathbb{N}^{2}$ adapted to
$\tau^{-1}\xi$, where $\tau$ is as Remark 6.2. Consider a sequence of sojourn
times with associated arguments
${\mathfrak{S}}_{k}=({\mathbf{s}}_{k},\Theta_{k})$, where
$\Theta_{k}=(\zeta_{m_{k}},\vartheta_{n_{k}})$.

The renormalisation scheme $\mathcal{R}(\xi,\mathfrak{F})$ of $f$ consists of
the following elements:

  1. (i)

$\Psi_{k}=\Psi_{\mathbf{s}_{k}}\colon\mathbb{R}^{3}\to M$ a sequence of
parameterisations on the manifold $M$;

  2. (ii)

$\bar{\upsilon}_{k}=\bar{\upsilon}_{{\mathfrak{S}}_{k}}:\mathbb{R}\times[-\pi,\pi]^{2}\to\mathbb{R}^{8}$
a sequence of bifurcation parameter maps of the family $\bar{\upsilon}\to
f_{\bar{\upsilon}}$ in (5.1);

  3. (iii)

$\mathcal{R}_{k}(f)=\mathcal{R}_{\bar{\upsilon}_{k}}(f)$ is the sequence in
$\operatorname{Diff}^{r}(M)$ defined by

$f_{\bar{\upsilon}_{k}}^{N_{2}}\circ f_{\bar{\upsilon}_{k}}^{m_{k}}\circ
f_{\bar{\upsilon}_{k}}^{N_{1}}\circ f_{\bar{\upsilon}_{k}}^{n_{k}}.$

This composition is the renormalised sequence of $f$.

We now give the precise definitions of the objects in the renormalisation
scheme. In what follows, we fix a sequence of sojourn times with arguments
$({\mathfrak{S}}_{k})=({\mathbf{s}}_{k},\Theta_{k})$, where
${\mathbf{s}}_{k}=(m_{k},n_{k})$ and
$\Theta_{k}=(\theta_{m_{k}},\vartheta_{n_{k}})$.

####  6.3.1. The parameterisations $\Psi_{k}=\Psi_{{\mathbf{s}}_{k}}$

Given ${\mathbf{s}}=(m,n)\in\mathbb{N}^{2}$, consider the map
$\Psi_{{\mathbf{s}}}\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$ defined by

$\Psi_{{\mathbf{s}}}(x,y,z)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(1+\sigma_{P}^{-m}\sigma_{Q}^{-n}\,x,\sigma_{Q}^{-n}+\sigma_{P}^{-2m}{\sigma_{Q}}^{-2n}\,y,1+\sigma_{P}^{-m}{\sigma_{Q}}^{-n}\,z).$

######  Remark 6.3.

Let $K\subset\mathbb{R}^{3}$ be any compact set. The sequence of compact sets
$(\Psi_{{\mathbf{s}}_{k}}(K))_{k}$ in $M$ satisfies

$\Psi_{{\mathbf{s}}_{k}}(K)\to\widetilde{Y}=(1,0,1)\in U_{Q},\quad\mbox{as
$k\to+\infty$},$

where the convergence is in the Hausdorff distance. In particular, there is
$k_{0}=k_{0}(K)$ such that $\Psi_{{\mathbf{s}}_{k}}(K)\subset U_{Q}$ for every
$k\geq k_{0}$. In what follows, for notational simplicity, we write
$\Psi_{k}=\Psi_{{\mathbf{s}}_{k}}$.

####  6.3.2. The bifurcation parameters
$\bar{\upsilon}_{k}=\bar{\upsilon}_{{\mathfrak{S}}_{k}}$

The sequence of maps
$\bar{\upsilon}_{{\mathfrak{S}}_{k}}:\mathbb{R}^{3}\to\mathbb{R}^{8}$ is of
the form

$\bar{\upsilon}_{{\mathfrak{S}}_{k}}(\mu,{\theta},\omega)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{(}\bar{\mu}_{{\mathbf{s}}_{k}}(\mu),\bar{\nu}_{{\mathfrak{S}}_{k}},\alpha^{P}_{{\mathfrak{S}}_{k}}({\theta}),\beta^{Q}_{{\mathfrak{S}}_{k}}(\omega)\big{)}$

where:

$\quad\bullet$
$\bar{\mu}_{{\mathbf{s}}_{k}}:\mathbb{R}\rightarrow\mathbb{R}^{3}$ is defined
by

(6.4)
$\bar{\mu}_{{\mathbf{s}}_{k}}(\mu)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(-\lambda_{P}^{m_{k}}a_{1},\sigma_{Q}^{-n_{k}}+\sigma_{Q}^{-2n_{k}}\sigma_{P}^{-2m_{k}}\mu-\lambda_{P}^{m_{k}}b_{1},-\lambda_{P}^{m_{k}}c_{1}),$

where $a_{1},b_{1},c_{1}$ are given in (3.22). Note that
$\bar{\mu}_{{\mathbf{s}}_{k}}(\mu)\to(0,0,0)$ as $k\to+\infty$ for every fixed
$\mu\in\mathbb{R}$. In what follows, for notational simplicity, we write
$\bar{\mu}_{k}=\bar{\mu}_{{\mathbf{s}}_{k}}$.

$\quad\bullet$ The maps
$\alpha^{P}_{k}(\theta)=\alpha^{P}_{{\mathfrak{S}}_{k}}(\theta)$ and
$\beta^{Q}_{k}(\omega)=\beta^{Q}_{{\mathfrak{S}}_{k}}(\omega)$ are defined in
(LABEL:e.alphakbetak).

$\quad\bullet$ The sequence $\bar{\nu}_{{\mathfrak{S}}_{k}}\in\mathbb{R}^{3}$
is defined as follows. For the sequence of arguments adapted to $\varphi_{P}$
and $\varphi_{Q}$ (associated to ${\mathfrak{S}}_{k}$), we write

$\varphi_{P,k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(\varphi_{P})^{P}_{k}=\varphi_{P}+{\alpha}^{P}_{k}(\varphi_{P})\quad\mbox{and}\quad\varphi_{Q,k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(\varphi_{Q})^{Q}_{k}=\varphi_{Q}+{\beta}^{Q}_{k}(\varphi_{Q})$

and define the following sequences

(6.5)
$\begin{split}\tilde{\mathfrak{c}}_{k}&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\cos\big{(}2\pi
m_{k}(\varphi_{P,k})\big{)},\quad\tilde{\mathfrak{s}}_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\sin\big{(}2\pi m_{k}(\varphi_{P,k})\big{)},\\\
\mathfrak{c}_{k}&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\cos\big{(}2\pi
n_{k}(\varphi_{Q,k})\big{)},\quad\mathfrak{s}_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\sin\big{(}2\pi n_{k}(\varphi_{Q,k})\big{)},\end{split}$

and

(6.6)
$\begin{split}\tilde{\rho}_{2,k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}\widetilde{H}_{2}(\textbf{0})(\mathfrak{c}_{k}-\mathfrak{s}_{k})^{2}+\frac{1}{2}\frac{\partial^{2}}{\partial
z^{2}}\widetilde{H}_{2}(\textbf{0})(\mathfrak{s}_{k}+\mathfrak{c}_{k})^{2},\\\
\tilde{\rho}_{3,k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}\widetilde{H}_{3}(\textbf{0})(\mathfrak{s}_{k}-\mathfrak{c}_{k})^{2}+\frac{1}{2}\frac{\partial^{2}}{\partial
z^{2}}\widetilde{H}_{3}(\textbf{0})(\mathfrak{s}_{k}+\mathfrak{c}_{k})^{2}.\end{split}$

We now let

(6.7)
$\begin{split}\bar{\nu}_{{\mathfrak{S}}_{k}}&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\Big{(}-\lambda_{Q}^{n_{k}}\big{(}\alpha_{1}\,(\mathfrak{c}_{k}-\mathfrak{s}_{k})+\alpha_{3}\,(\mathfrak{s}_{k}+\mathfrak{c}_{k})\big{)},\sigma_{P}^{-m_{k}}(\tilde{\mathfrak{c}}_{k}+\tilde{\mathfrak{s}}_{k})-{\lambda_{Q}}^{2n_{k}}\tilde{\rho}_{2,k},\\\
&\qquad\qquad\sigma_{P}^{-m_{k}}(\tilde{\mathfrak{c}}_{k}-\tilde{\mathfrak{s}}_{k})-\lambda_{Q}^{n_{k}}\,\gamma_{3}\,(\mathfrak{c}_{k}+\mathfrak{s}_{k})-\lambda_{Q}^{2n_{k}}\tilde{\rho}_{3,k}\Big{)}.\end{split}$

In what follows, for notational simplicity, we write
$\bar{\nu}_{k}=\bar{\nu}_{{\mathfrak{S}}_{k}}$.

######  Claim 6.4.

$\bar{\nu}_{k}\to(0,0,0)\in\mathbb{R}^{3}$ as $k\rightarrow+\infty$.

###### Proof.

Note that

(6.8)
$\begin{split}&\tilde{\mathfrak{c}}_{k}=\cos\Big{(}\frac{\pi}{4}+\zeta_{m_{k}}\Big{)}\to\frac{1}{\sqrt{2}},\quad\tilde{\mathfrak{s}}_{k}=\sin\Big{(}\frac{\pi}{4}+\zeta_{m_{k}}\Big{)}\to\frac{1}{\sqrt{2}},\\\
&\mathfrak{c}_{k}=\cos\left(\frac{\pi}{2}+\vartheta_{n_{k}}\right)\to
0,\quad\mathfrak{s}_{k}=\sin\left(\frac{\pi}{2}+\vartheta_{n_{k}}\right)\to
1,\end{split}$

when $k\to+\infty$. Therefore,

$\tilde{\rho}_{2,k}\to\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}\widetilde{H}_{2}(\textbf{0})+\frac{1}{2}\frac{\partial^{2}}{\partial
z^{2}}\widetilde{H}_{2}(\textbf{0}),\quad\tilde{\rho}_{3,k}\to\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}\widetilde{H}_{3}(\textbf{0})+\frac{1}{2}\frac{\partial^{2}}{\partial
z^{2}}\widetilde{H}_{3}(\textbf{0}).$

The claim follows immediately from $|\lambda_{P}|,|\lambda_{Q}|<1$ and
$|\sigma_{P}|,|\sigma_{Q}|>1$. ∎

######  Remark 6.5 (Expression of
$f_{\bar{\upsilon}_{{\mathfrak{S}}_{k}}(\mu,\theta,\omega)}=f_{\bar{\upsilon}_{k}(\mu,\theta,\omega)}$).

Recall that $f_{\bar{\upsilon}}=\Gamma_{\bar{\upsilon}}\circ f$, see (5.1).
Recalling the definitions of $\Gamma_{\bar{\upsilon}}$ in (5.2), of
$\Gamma^{\mathrm{qt}}_{\bar{\mu}}$ and $\Gamma^{\mathrm{ht}}_{\bar{\nu}}$ in
Section 5.1, and of $\Gamma^{P}_{{\alpha}}$ and $\Gamma^{Q}_{\beta}$ in
Section 5.2, we have that

(6.9)
$f_{\bar{\upsilon}_{k}(\mu,\theta,\omega)}=\left\\{\begin{split}&\Gamma^{P}_{\alpha^{P}_{k}(\theta)}\circ
f\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}f_{\alpha^{P}_{k}(\theta)},\quad\mbox{in $V_{P}$,}\\\
&\Gamma^{\mathrm{ht}}_{\bar{\mu}_{k}(\mu)}\circ
f\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}f_{\bar{\mu}_{k}(\mu)},\quad\mbox{in
$V_{f^{-1}(\widetilde{Y})}$,}\\\ &\Gamma^{Q}_{\beta^{Q}_{k}(\omega)}\circ
f\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}f_{\beta^{Q}_{k}(\omega)},\quad\mbox{in $V_{Q}$,}\\\
&\Gamma^{\mathrm{qt}}_{\bar{\nu}_{k}}\circ
f\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}f_{\bar{\nu}_{k}},\quad\mbox{in $V_{f^{-1}(\widetilde{X})}$,}\\\
&f,\quad\mbox{in $M\setminus V_{P}\cup V_{Q}\cup V_{f^{-1}(\widetilde{X})}\cup
V_{f^{-1}(\widetilde{Y})}$},\end{split}\right.$

where $V_{f^{-1}(\widetilde{Z})}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}f^{-1}\big{(}\mathbb{B}_{\rho}(\widetilde{Z})\big{)}$, $Z=X,Y$. See
Remark 5.1.

######  Remark 6.6.

By definition of the arguments adapted to $\varphi_{P}$ and $\varphi_{Q}$ (see
Subsection 6.2) it follows that the argument of $f_{\alpha_{k}(\varphi_{P})}$
in the neighbourhoods $V_{P}$ of $U_{P}$ is given by
$2\pi(\varphi_{P}+\alpha^{P}_{k}(\varphi_{P}))=2\pi(\varphi_{P,k})\to
2\pi(\varphi_{P})$. Thus, recalling the local form of $f$ in the
neighbourhoods of $P$ and $Q$ in equation (3.1),

$(f_{\alpha^{P}_{k}(\varphi_{P})}|_{V_{P}})^{m_{k}}=\left(\begin{array}[]{ccc}\lambda_{P}^{m_{k}}&0&0\\\
0&\sigma_{P}^{m_{k}}\tilde{\mathfrak{c}}_{k}&-\sigma_{P}^{m_{k}}\tilde{\mathfrak{s}}_{k}\\\
0&\sigma_{P}^{m_{k}}\tilde{\mathfrak{s}}_{k}&\sigma_{P}^{m_{k}}\tilde{\mathfrak{c}}_{k}\end{array}\right),$

where $\tilde{\mathfrak{c}}_{k},\tilde{\mathfrak{s}}_{k}$, are as in (6.5).
Analogously, it holds

$(f_{\beta^{Q}_{k}(\varphi_{Q})}|_{V_{Q}})^{n_{k}}=\left(\begin{array}[]{ccc}\lambda_{Q}^{n_{k}}{\mathfrak{c}}_{k}&0&-\lambda_{Q}^{n_{k}}{\mathfrak{s}}_{k}\\\
0&\sigma_{Q}^{n_{k}}&0\\\
\lambda_{Q}^{n_{k}}{\mathfrak{s}}_{k}&0&\lambda_{Q}^{n_{k}}{\mathfrak{c}}_{k}\end{array}\right).$

######  Remark 6.7 (Convergence to $f$).

By construction, given any $(\mu,\theta,\omega)\in\mathbb{R}^{3}$, the
sequence $f_{\bar{\upsilon}_{k},\Theta_{k}}$ with
$\bar{\upsilon}_{k}=\bar{\upsilon}_{k}(\mu,\theta,\omega)$ converges to $f$ in
the $C^{r}$-topology.

##  7\. The renormalised sequence of maps

Fixed the renormalisation scheme $\mathcal{R}(\xi,\mathfrak{F})$ of $f$
associated to the sequence
$({\mathfrak{S}}_{k})=({\mathbf{s}}_{k},\Theta_{k})$, where
${\mathbf{s}}_{k}=(m_{k},n_{k})$ and
$\Theta_{k}=(\theta_{m_{k}},\vartheta_{n_{k}})$, and a compact set
$K\subset\mathbb{R}^{3}$, we now study the sequence of maps

$\Psi_{k}^{-1}\circ\mathcal{R}_{\bar{\upsilon}_{k}}(f)\circ\Psi_{k}:K\subset\mathbb{R}^{3}\rightarrow\mathbb{R}^{3},\quad\bar{\upsilon}_{k}=\bar{\upsilon}_{k}(\mu,\theta,\omega).$

Using the notation of the previous section, we begin by considering parameters
of the form

$\bar{\upsilon}_{k}(\mu,\varphi_{P},\varphi_{Q})=\big{(}\bar{\mu}_{k}(\mu),\bar{\nu}_{k},\alpha^{P}_{k}(\varphi_{P}),\beta^{Q}_{k}(\varphi_{Q})\big{)},$

For this choice of parameters, the renormalisation scheme involves the adapted
arguments $\varphi_{P}$ and $\varphi_{Q}$, the parameter $\bar{\nu}_{k}$
(which depends on the choice of these arguments, see (6.7)), and a “free”
parameter $\mu$. For notational simplicity, write

$\alpha_{k,P}=\alpha^{P}_{k}(\varphi_{P}),\quad\beta_{k,Q}=\beta^{Q}_{k}(\varphi_{Q}),\quad\bar{\upsilon}_{k}(\mu)=\bar{\upsilon}_{k}(\mu,\varphi_{P},\varphi_{Q}),$

and for $\bar{X}=(x,y,z)\in K$ let

(7.1)
$\Psi_{k}^{-1}\circ\mathcal{R}_{\bar{\upsilon}_{k}(\mu)}\circ\Psi_{k}(x,y,z)=\breve{X}_{k}(x,y,z)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(\breve{x}_{k},\breve{y}_{k},\breve{z}_{k}).$

The goal of this section is to determine the coordinates of $\breve{X}_{k}$,
this is done in equations (7.15)-(7.17). With these coordinates at hand, we
will obtain the convergence of the renormalisation scheme. The calculation of
these coordinates involves three intermediate steps, corresponding to the
compositions $\Psi_{k}(\bar{X})$,
$\mathcal{R}_{\bar{\upsilon}_{k}(\mu)}\circ\Psi_{k}(\bar{X})$, and
$\Psi_{k}^{-1}\circ\mathcal{R}_{\bar{\upsilon}_{k}(\mu)}\circ\Psi_{k}(\bar{X})$.

###  7.1. Coordinates of $\Psi_{k}(\bar{X})$

Write $\bar{X}_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\Psi_{k}(\bar{X})$ where

(7.2)
$\Psi_{k}(\bar{X})=(1+\sigma_{P}^{-m_{k}}{\sigma_{Q}}^{-n_{k}}\,x,\sigma_{Q}^{-n_{k}}+\sigma_{P}^{-2m_{k}}\sigma_{Q}^{-2n_{k}}\,y,1+\sigma_{P}^{-m_{k}}{\sigma_{Q}}^{-n_{k}}\,z).$

By the compactness of the set $K$, $\bar{X}_{k}\to\widetilde{Y}=(1,0,1)\in
U_{Q}$, as $k\to+\infty$. Note that this step does not depend on $\mu$.

###  7.2. Coordinates of
$\mathcal{R}_{\bar{\upsilon}_{k}(\mu)}\circ\Psi_{k}(\bar{X})$

Recall the definition of $f_{\bar{\upsilon}_{k}(\mu,\theta,\omega)}$ in (6.9).
The application of $\mathcal{R}_{\bar{\upsilon}_{k}(\mu)}$ involves the
following four “independent” intermediate steps (see Figure 4):

  * •

Step A: $n_{k}$ iterations nearby $Q$ given by
$f^{n_{k}}_{\bar{\upsilon}_{k}(\mu)}=f^{n_{k}}_{\beta_{k,Q}}$;

  * •

Step B: the transition from $Q$ to $P$ along the orbit of $X$ given by
$f^{N_{1}}_{\bar{\upsilon}_{k}(\mu)}=f_{\bar{\nu}_{k}}\circ f^{N_{1}-1}$;

  * •

Step C: $m_{k}$ iterations nearby $P$ given by
$f^{m_{k}}_{\bar{\upsilon}_{k}(\mu)}=f^{m_{k}}_{\alpha_{k,P}}$;

  * •

Step D: the transition from $P$ to $Q$ along the orbit of $Y$ given by
$f^{N_{2}}_{\bar{\upsilon}_{k}(\mu)}=f_{\bar{\mu}_{k}(\mu)}\circ f^{N_{2}-1}$.
This last step is the only one depending on $\mu$.

\begin{overpic}[scale={.13}]{pasos2.jpg} \put(65.0,90.0){{$\textbf{Step A}$}}
\put(125.0,140.0){\small{$\textbf{Step B}$}}
\put(190.0,203.0){\small{$\textbf{Step C}$}}
\put(210.0,90.0){\small{$\textbf{Step D}$}} \end{overpic} Figure 4. Steps A-D.

Step A: Recalling (6.9) and (7.2) , write

(7.3)
$f^{n_{k}}_{\beta_{k,Q}}(\bar{X}_{k})\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(x_{k},y_{k}+1,z_{k})=X_{k}\in U_{Q}.$

The following equalities follow from Remark 6.6,

(7.4)
$\begin{split}x_{k}&={\sigma_{P}}^{-m_{k}}\,\lambda_{Q}^{n_{k}}\,\sigma_{Q}^{-n_{k}}\,(\mathfrak{c}_{k}\,x-\mathfrak{s}_{k}\,z)+\lambda_{Q}^{n_{k}}\,(\mathfrak{c}_{k}-\mathfrak{s}_{k}),\\\
y_{k}&=\sigma_{P}^{-2m_{k}}\,{\sigma_{Q}}^{-n_{k}}\,y,\\\
z_{k}&={\sigma_{P}}^{-m_{k}}\,\lambda_{Q}^{n_{k}}\,\sigma_{Q}^{-n_{k}}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)+\lambda_{Q}^{n_{k}}(\mathfrak{c}_{k}+\mathfrak{s}_{k}).\end{split}$

Since $K$ is a compact set, we have that

(7.5) $x_{k}=O(\lambda_{Q}^{n_{k}}),\quad
y_{k}=O(\sigma_{P}^{-2m_{k}}\,{\sigma_{Q}}^{-n_{k}}),\quad
z_{k}=O(\lambda_{Q}^{n_{k}}),$

where $O(\cdot)$ denotes the symbol of Landau666Given two real valued
functions $f$ and $g$, one says that the symbol of Landau of $f(x)$ is
$O(g(x))$ (as $x\to+\infty$) if and only if there are $M>0$ and $x_{0}$ with
$|f(x)|\leq M|g(x)|$ for all $x>x_{0}$.. Therefore

$X_{k}=(x_{k},y_{k}+1,z_{k})\to X=(0,1,0)\in U_{X},\quad\mbox{as
$k\to+\infty$.}$ (7.6)
$\textbf{x}_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(x_{k},y_{k},z_{k}).$

Step B: The transition from a neighbourhood of $X$ to $U_{P}$ given by
$f^{N_{1}}_{\bar{\upsilon}_{k}(\mu)}=f_{\bar{\nu}_{k}}\circ f^{N_{1}-1}$.
Write

(7.7) $f_{\bar{\nu}_{k}}\circ
f^{N_{1}-1}(X_{k})\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(1+\tilde{x}_{k},\tilde{y}_{k},\tilde{z}_{k})=\tilde{X}_{k}\in
U_{P}.$

Using the definitions of $f^{N_{1}}$ in (3.12) (recall also (3.13)), of
$f_{\bar{\nu}_{k}}$ in (6.9), and of $X_{k}$ in (7.3), we get

$\begin{split}\tilde{x}_{k}&={\sigma_{P}}^{-m_{k}}\,\lambda_{Q}^{n_{k}}\,\sigma_{Q}^{-n_{k}}\,\alpha_{1}\,(\mathfrak{c}_{k}\,x-\mathfrak{s}_{k}\,z)+{\sigma_{P}}^{-2m_{k}}\,\sigma_{Q}^{-n_{k}}\,\alpha_{2}\,y+\\\
&\qquad+{\sigma_{P}}^{-m_{k}}\,\lambda_{Q}^{n_{k}}\,\sigma_{Q}^{-n_{k}}\,\alpha_{3}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)+\widetilde{H}_{1}\big{(}\textbf{x}_{k}\big{)},\\\
\tilde{y}_{k}&={\sigma_{P}}^{-2m_{k}}\,\sigma_{Q}^{-n_{k}}\,\beta_{2}\,y+\sigma_{P}^{-m_{k}}(\tilde{\mathfrak{c}}_{k}+\tilde{\mathfrak{s}}_{k})-\lambda_{Q}^{2n_{k}}\tilde{\rho}_{2,k}+\widetilde{H}_{2}\big{(}\textbf{x}_{k}\big{)},\\\
\tilde{z}_{k}&={\sigma_{P}}^{-m_{k}}\,\lambda_{Q}^{n_{k}}\,\sigma_{Q}^{-n_{k}}\,\gamma_{3}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)+\sigma_{P}^{-m_{k}}(\tilde{\mathfrak{c}}_{k}-\tilde{\mathfrak{s}}_{k})-\lambda_{Q}^{2n_{k}}\tilde{\rho}_{3,k}+\widetilde{H}_{3}\big{(}\textbf{x}_{k}\big{)}.\end{split}$

For simplicity, in what follows, we write

(7.8)
$\displaystyle\widehat{H}_{i}\big{(}\textbf{x}_{k}\big{)}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\widetilde{H}_{i}\big{(}\textbf{x}_{k}\big{)}-\lambda_{Q}^{2n_{k}}\tilde{\rho}_{i,k},\quad
i=2,3.$

Step C: Recalling (6.9), Remark (6.6) and equation (7.7) writing

(7.9)
$f^{m_{k}}_{\alpha_{k,P}}(\tilde{X}_{k})\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(\hat{x}_{k},1+\hat{y}_{k},1+\hat{z}_{k})=\hat{X}_{k}\in U_{P},$

we have (after some straightforward simplifications and using that
$\mathfrak{s}_{k}^{2}+\mathfrak{c}_{k}^{2}=1$) the following equalities:

$\begin{split}\hat{x}_{k}&=\lambda_{P}^{m_{k}}+\lambda_{P}^{m_{k}}\,{\sigma_{P}}^{-m_{k}}\lambda_{Q}^{n_{k}}\,\sigma_{Q}^{-n_{k}}\,\big{(}\alpha_{1}\,(\mathfrak{c}_{k}\,x-\mathfrak{s}_{k}\,z)+\alpha_{3}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)\big{)}\\\
&\qquad+\lambda_{P}^{m_{k}}\sigma_{P}^{-2m_{k}}\,\sigma_{Q}^{-n_{k}}\,\alpha_{2}\,y+{\lambda_{P}}^{m_{k}}\,\widetilde{H}_{1}\big{(}\textbf{x}_{k}\big{)},\\\
\hat{y}_{k}&=\sigma_{P}^{-m_{k}}\,{\sigma_{Q}}^{-n_{k}}\tilde{\mathfrak{c}}_{k}\,\beta_{2}\,y-\lambda_{Q}^{n_{k}}\,\sigma_{Q}^{-n_{k}}\tilde{\mathfrak{s}}_{k}\,\gamma_{3}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)\\\
&\qquad+\sigma_{P}^{m_{k}}\,\Big{(}\tilde{\mathfrak{c}}_{k}\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}-\tilde{\mathfrak{s}}_{k}\widehat{H}_{3}\big{(}\textbf{x}_{k}\big{)}\Big{)},\\\
\hat{z}_{k}&=\sigma_{P}^{-m_{k}}\,{\sigma_{Q}}^{-n_{k}}\tilde{\mathfrak{s}}_{k}\,\beta_{2}\,y+\lambda_{Q}^{n_{k}}\,\sigma_{Q}^{-n_{k}}\tilde{\mathfrak{c}}_{k}\,\gamma_{3}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)\\\
&\qquad+{\sigma_{P}}^{m_{k}}\,\Big{(}\tilde{\mathfrak{s}}_{k}\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}+\tilde{\mathfrak{c}}_{k}\widehat{H}_{3}\big{(}\textbf{x}_{k}\big{)}\Big{)}.\end{split}$

######  Lemma 7.1.

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

###### Proof.

Note that $\hat{y}_{k}$ and $\hat{z}_{k}$ have the same symbol of Landau. The
conditions in (3.12) and (3.14) and the definition of $\textbf{x}_{k}$ in
(7.6) imply that the higher order terms $\widetilde{H}_{i}$, $i=1,2,3$, are
dominated by quadratic terms. Thus, for $i=1,2,3$, it holds

(7.10)
$\begin{split}\widetilde{H}_{i}\big{(}\textbf{x}_{k}\big{)}&=\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}\widetilde{H}_{i}(\textbf{0})\,x_{k}^{2}+\frac{\partial^{2}}{\partial
x\partial
y}\widetilde{H}_{i}(\textbf{0})\,x_{k}\,y_{k}+\frac{\partial^{2}}{\partial
x\partial z}\widetilde{H}_{i}(\textbf{0})\,x_{k}\,z_{k}+\\\
&\quad+\frac{1}{2}\frac{\partial^{2}}{\partial
y^{2}}\widetilde{H}_{i}(\textbf{0})\,y_{k}^{2}+\frac{\partial^{2}}{\partial
y\partial
z}\widetilde{H}_{i}(\textbf{0})\,y_{k}\,z_{k}+\frac{1}{2}\frac{\partial^{2}}{\partial
z^{2}}\widetilde{H}_{i}(\textbf{0})\,z_{k}^{2}+\mathrm{h.o.t.}.\end{split}$

Since $x_{k}$ and $z_{k}$ have the same Landau symbol
$O(\lambda_{Q}^{2n_{k}})$ and $y_{k}$ has Landau symbol
$O(\sigma_{P}^{-2m_{k}}\,{\sigma_{Q}}^{-n_{k}})$ (see (7.5)), it follows that

(7.11)
$\widetilde{H}_{i}\big{(}\textbf{x}_{k}\big{)}=O(\lambda_{Q}^{2n_{k}})+O({\sigma_{P}}^{-2m_{k}}\lambda_{Q}^{n_{k}}\sigma_{Q}^{-n_{k}})+O({\sigma_{P}}^{-4m_{k}}\,\sigma_{Q}^{-2n_{k}}),\quad
i=1,2,3.$

Finally, as the set $K$ is compact, it follows from (7.11) that
$\hat{x}_{k}=O({\lambda_{P}}^{m_{k}})$.

We now determine the symbol of $\hat{y}_{k}$ (and hence the one of
$\hat{z}_{k}$). For this, its is enough to estimate the terms
${\sigma_{P}}^{m_{k}}\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}$ and
${\sigma_{P}}^{m_{k}}\widehat{H}_{3}\big{(}\textbf{x}_{k}\big{)}$ (as the
other terms in the expression of $\hat{y}_{k}$ are bigger). Using the Taylor
formula in (7.10), the definition of the coordinates (7.4), and the definition
of $\tilde{\rho}_{i,k}$ in (6.6), it follows from (7.5) and (7.8) that

$\begin{split}\widehat{H}_{i}\big{(}\textbf{x}_{k}\big{)}&=\widetilde{H}_{i}\big{(}\textbf{x}_{k}\big{)}-\lambda_{Q}^{2n_{k}}\tilde{\rho}_{i,k}=O({\sigma_{P}}^{-2m_{k}}\,\lambda_{Q}^{2n_{k}}\,\sigma_{Q}^{-2n_{k}})+O({\sigma_{P}}^{-m_{k}}\,\lambda_{Q}^{2n_{k}}\,\sigma_{Q}^{-n_{k}})+\\\
&+O({\sigma_{P}}^{-2m_{k}}\,\lambda_{Q}^{n_{k}}\,\sigma_{Q}^{-n_{k}})+O(\sigma_{P}^{-4m_{k}}\,{\sigma_{Q}}^{-2n_{k}}),\quad
i=2,3.\end{split}$

For the next estimate, recall that by (6.2) the sequence
$\lambda_{Q}^{n_{k}}\,\sigma_{P}^{m_{k}}$ converges to some number different
from $0$. Hence $\lambda_{Q}^{n_{k}}\,\sigma_{P}^{m_{k}}$ and
$\lambda_{Q}^{-n_{k}}\,\sigma_{P}^{-m_{k}}$ are bounded from above by some
$K_{0}>0$. This bound and $\sigma_{P},\sigma_{Q}>1$ provide the following
estimates,

  * •

${\sigma_{P}}^{-2m_{k}}\,\lambda_{Q}^{2n_{k}}\,\sigma_{Q}^{-2n_{k}}<{\sigma_{P}}^{-m_{k}}\,\lambda_{Q}^{2n_{k}}\,\sigma_{Q}^{-n_{k}}$,

  * •

${\sigma_{P}}^{-2m_{k}}\,\lambda_{Q}^{n_{k}}\,\sigma_{Q}^{-n_{k}}=(\lambda_{Q}^{-n_{k}}\,\sigma_{P}^{-m_{k}})\,{\sigma_{P}}^{-m_{k}}\,\lambda_{Q}^{2n_{k}}\,\sigma_{Q}^{-n_{k}}<K_{0}\,{\sigma_{P}}^{-m_{k}}\,\lambda_{Q}^{2n_{k}}\,\sigma_{Q}^{-n_{k}}$,

  * •

$\sigma_{P}^{-4m_{k}}\,{\sigma_{Q}}^{-2n_{k}}=(\lambda_{Q}^{-2n_{k}}\,\sigma_{P}^{-2m_{k}})(\sigma_{P}^{-m_{k}}\,\sigma_{Q}^{-n_{k}})\,{\sigma_{P}}^{-m_{k}}\,\lambda_{Q}^{2n_{k}}\,\sigma_{Q}^{-n_{k}}$

  * $<K_{0}^{2}\,{\sigma_{P}}^{-m_{k}}\,\lambda_{Q}^{2n_{k}}\,\sigma_{Q}^{-n_{k}}$.

These inequalities imply that

(7.12)
$\widehat{H}_{i}\big{(}\textbf{x}_{k}\big{)}=\widetilde{H}_{i}\big{(}\textbf{x}_{k}\big{)}-\lambda_{Q}^{2n_{k}}\tilde{\rho}_{i,k}=O({\sigma_{P}}^{-m_{k}}\,\lambda_{Q}^{2n_{k}}\,\sigma_{Q}^{-n_{k}}),\quad
i=2,3.$

Therefore

${\sigma_{P}}^{m_{k}}\widehat{H}_{i}\big{(}\textbf{x}_{k}\big{)}=O(\lambda_{Q}^{2n_{k}}\sigma_{Q}^{-n_{k}}),\quad
i=2,3.$

Finally, observing that

$\lambda_{Q}^{2n_{k}}\,\sigma_{Q}^{-n_{k}}=(\sigma_{P}^{m_{k}}\,\lambda_{Q}^{n_{k}})\lambda_{Q}^{n_{k}}\,\sigma_{P}^{-m_{k}}\,{\sigma_{Q}}^{-n_{k}}<K_{0}\,\sigma_{P}^{-m_{k}}\,{\sigma_{Q}}^{-n_{k}}$

we get $\hat{y}_{k}=O(\sigma_{P}^{-m_{k}}\,{\sigma_{Q}}^{-n_{k}})$, proving
the lemma. ∎

Lemma 7.1 implies that

$\hat{X}_{k}=(\hat{x}_{k},1+\hat{y}_{k},1+\hat{z}_{k})\to Y=(0,1,1)\in
V_{Y},\quad k\to+\infty.$

For notational simplicity we write

(7.13) $\hat{\textbf{x}}_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(\hat{x}_{k},\hat{y}_{k},\hat{z}_{k}).$

Step D: The transition from a neighbourhood of $Y$ to $U_{Q}$ is given by
$f^{N_{2}}_{\bar{\upsilon}_{k}(\mu)}=f_{\bar{\mu}_{k}(\mu)}\circ f^{N_{2}-1}$.
Write

(7.14) $f_{\bar{\mu}_{k}(\mu)}\circ
f^{N_{2}-1}(\hat{X}_{k})\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(1+\ddot{x}_{k},\ddot{y}_{k},1+\ddot{z}_{k})=\ddot{X_{k}}\in U_{Q}$

Using equation (3.22), the expression of $f_{\bar{\mu}_{k}(\mu)}\circ
f^{N_{2}-1}$ (see Remark 5.1), equation (6.9), the definition $\hat{X}_{k}$ in
(7.9), and the choice of $\bar{\mu}_{k}(\mu)$ in (6.4) we have

$\begin{split}\ddot{x}_{k}&=a_{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+a_{1}\,\lambda_{P}^{m_{k}}\,\sigma_{P}^{-2m_{k}}\,{\sigma_{Q}}^{-n_{k}}\,\alpha_{2}\,y+\big{(}\tilde{\mathfrak{c}}_{k}\,a_{2}+\tilde{\mathfrak{s}}_{k}\,a_{3}\big{)}\,\sigma_{P}^{-m_{k}}\,{\sigma_{Q}}^{-n_{k}}\,\beta_{2}\,y+\\\
&\quad+\lambda_{Q}^{n_{k}}\,\sigma_{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)+\mathrm{h.o.t.}^{*}_{k},\\\
\ddot{y}_{k}&=\sigma_{Q}^{-n_{k}}+\sigma_{P}^{-2m_{k}}\,{\sigma_{Q}}^{-2n_{k}}\mu+b_{1}\,\lambda_{P}^{m_{k}}\,\sigma_{P}^{-2m_{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+\sigma_{P}^{-2m_{k}}\,{\sigma_{Q}}^{-2n_{k}}\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}^{-m_{k}}\,\lambda_{Q}^{n_{k}}\,\sigma_{Q}^{-2n_{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+\lambda_{Q}^{2n_{k}}\sigma_{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}+\mathrm{h.o.t.}^{**}_{k},\\\
\ddot{z}_{k}&=c_{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+c_{1}\,\lambda_{P}^{m_{k}}\,\sigma_{P}^{-2m_{k}}\,{\sigma_{Q}}^{-n_{k}}\,\alpha_{2}\,y+\big{(}\tilde{\mathfrak{c}}_{k}\,c_{2}+\tilde{\mathfrak{s}}_{k}\,c_{3}\big{)}\,\sigma_{P}^{-m_{k}}\,{\sigma_{Q}}^{-n_{k}}\,\beta_{2}\,y+\\\
&\quad+\lambda_{Q}^{n_{k}}\,\sigma_{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)+\mathrm{h.o.t.}^{***}_{k},\end{split}$

where

$\begin{split}\mathrm{h.o.t.}^{*}_{k}&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}a_{1}\,{\lambda_{P}}^{m_{k}}\,\widetilde{H}_{1}\big{(}\textbf{x}_{k}\big{)}+a_{2}\,\sigma_{P}^{m_{k}}\,\Big{(}\tilde{\mathfrak{c}}_{k}\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}-\tilde{\mathfrak{s}}_{k}\widehat{H}_{3}\big{(}\textbf{x}_{k}\big{)}\Big{)}+\\\
&\quad+a_{3}\,\sigma_{P}^{m_{k}}\,\Big{(}\tilde{\mathfrak{s}}_{k}\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}+\tilde{\mathfrak{c}}_{k}\widehat{H}_{3}\big{(}\textbf{x}_{k}\big{)}\Big{)}+H_{1}\big{(}\hat{\textbf{x}}_{k}\big{)},\\\
\mathrm{h.o.t.}^{**}_{k}&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}b_{1}\,{\lambda_{P}}^{m_{k}}\widetilde{H}_{1}\big{(}\textbf{x}_{k}\big{)}+b_{2}\,\Big{[}\sigma_{P}^{2m_{k}}\,\Big{(}\tilde{\mathfrak{c}}_{k}\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}-\tilde{\mathfrak{s}}_{k}\widehat{H}_{3}\big{(}\textbf{x}_{k}\big{)}\Big{)}^{2}+\\\
&\quad+2\sigma_{P}^{m_{k}}\,\Big{(}\tilde{\mathfrak{c}}_{k}\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}-\tilde{\mathfrak{s}}_{k}\widehat{H}_{3}\big{(}\textbf{x}_{k}\big{)}\Big{)}\Big{(}\sigma_{P}^{-m_{k}}\,{\sigma_{Q}}^{-n_{k}}\tilde{\mathfrak{c}}_{k}\,\beta_{2}\,y-\lambda_{Q}^{n_{k}}\,\sigma_{Q}^{-n_{k}}\tilde{\mathfrak{s}}_{k}\,\gamma_{3}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)\Big{)}\Big{]}+\\\
&\quad+b_{3}\,\Big{[}\sigma_{P}^{2m_{k}}\,\Big{(}\tilde{\mathfrak{s}}_{k}\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}+\tilde{\mathfrak{c}}_{k}\widehat{H}_{3}\big{(}\textbf{x}_{k}\big{)}\Big{)}^{2}+\\\
&\quad+2\sigma_{P}^{m_{k}}\,\Big{(}\tilde{\mathfrak{s}}_{k}\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}+\tilde{\mathfrak{c}}_{k}\widehat{H}_{3}\big{(}\textbf{x}_{k}\big{)}\Big{)}\Big{(}\sigma_{P}^{-m_{k}}\,{\sigma_{Q}}^{-n_{k}}\tilde{\mathfrak{s}}_{k}\,\beta_{2}\,y+\lambda_{Q}^{n_{k}}\,\sigma_{Q}^{-n_{k}}\tilde{\mathfrak{c}}_{k}\,\gamma_{3}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)\Big{)}\Big{]}\\\
&\quad+b_{4}\,\Big{[}\sigma_{P}^{2m_{k}}\,\Big{(}\tilde{\mathfrak{c}}_{k}\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}-\tilde{\mathfrak{s}}_{k}\widehat{H}_{3}\big{(}\textbf{x}_{k}\big{)}\Big{)}\Big{(}\tilde{\mathfrak{s}}_{k}\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}+\tilde{\mathfrak{c}}_{k}\widehat{H}_{3}\big{(}\textbf{x}_{k}\big{)}\Big{)}+\\\
&\quad+\sigma_{P}^{m_{k}}\,\Big{(}\tilde{\mathfrak{c}}_{k}\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}-\tilde{\mathfrak{s}}_{k}\widehat{H}_{3}\big{(}\textbf{x}_{k}\big{)}\Big{)}\Big{(}\sigma_{P}^{-m_{k}}\,{\sigma_{Q}}^{-n_{k}}\tilde{\mathfrak{s}}_{k}\,\beta_{2}\,y+\lambda_{Q}^{n_{k}}\,\sigma_{Q}^{-n_{k}}\tilde{\mathfrak{c}}_{k}\,\gamma_{3}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)\Big{)}+\\\
&\quad+\sigma_{P}^{m_{k}}\,\Big{(}\tilde{\mathfrak{s}}_{k}\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}+\tilde{\mathfrak{c}}_{k}\widehat{H}_{3}\big{(}\textbf{x}_{k}\big{)}\Big{)}\Big{(}\sigma_{P}^{-m_{k}}\,{\sigma_{Q}}^{-n_{k}}\tilde{\mathfrak{c}}_{k}\,\beta_{2}\,y-\lambda_{Q}^{n_{k}}\,\sigma_{Q}^{-n_{k}}\tilde{\mathfrak{s}}_{k}\,\gamma_{3}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)\Big{)}\Big{]}+\\\
&\quad+H_{2}\big{(}\hat{\textbf{x}}_{k}\big{)},\\\
\mathrm{h.o.t.}^{**\ast}_{k}&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}c_{1}\,{\lambda_{P}}^{m_{k}}\,\widetilde{H}_{1}\big{(}\textbf{x}_{k}\big{)}+c_{2}\,\sigma_{P}^{m_{k}}\,\Big{(}\tilde{\mathfrak{c}}_{k}\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}-\tilde{\mathfrak{s}}_{k}\widehat{H}_{3}\big{(}\textbf{x}_{k}\big{)}\Big{)}+\\\
&\quad+c_{3}\,\sigma_{P}^{m_{k}}\,\Big{(}\tilde{\mathfrak{s}}_{k}\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}+\tilde{\mathfrak{c}}_{k}\widehat{H}_{3}\big{(}\textbf{x}_{k}\big{)}\Big{)}+H_{3}\big{(}\hat{\textbf{x}}_{k}\big{)}.\end{split}$

This concludes the computations in Steps A -D.

###  7.3. Coordinates of
$\Psi_{k}^{-1}\circ\mathcal{R}_{\bar{\upsilon}_{k}(\mu)}\circ\Psi_{k}(\bar{X})$

To conclude the renormalisation calculations, it remains to apply
$\Psi_{k}^{-1}$ to the points $\ddot{X}_{k}$ in (7.14). Note that, by the
definition of $\breve{X}_{k}$ in (7.1) and by construction,

$\Psi^{-1}_{k}(\ddot{X}_{k})=(\breve{x}_{k},\breve{y}_{k},\breve{z}_{k})=\breve{X}_{k}.$

Recalling the expression of $\Psi_{k}$ in (7.2) we get

${\Psi^{-1}_{k}}(1+\ddot{x_{k}},\ddot{y_{k}},1+\ddot{z_{k}})=(\sigma_{P}^{m_{k}}{\sigma_{Q}}^{n_{k}}\ddot{x},\sigma_{P}^{2m_{k}}\,{\sigma_{Q}}^{2n_{k}}(\ddot{y}-\sigma_{Q}^{-n_{k}}),\sigma_{P}^{m_{k}}\,{\sigma_{Q}}^{n_{k}}\ddot{z}).$

Applying the corresponding substitutions for $\ddot{x}_{k}$, $\ddot{y}_{k}$,
and $\ddot{z}_{k}$, we have:

(7.15)
$\begin{split}&\breve{x}_{k}=a_{1}\,\lambda_{P}^{m_{k}}\,{\lambda_{Q}}^{n_{k}}\,\big{(}\alpha_{1}\,(\mathfrak{c}_{k}\,x-\mathfrak{s}_{k}\,z)+\alpha_{3}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)\big{)}+\\\
&\qquad+a_{1}\,\lambda_{P}^{m_{k}}\,\sigma_{P}^{-m_{k}}\,\alpha_{2}\,y+\big{(}\tilde{\mathfrak{c}}_{k}\,a_{2}+\tilde{\mathfrak{s}}_{k}\,a_{3}\big{)}\,\beta_{2}\,y+\\\
&\qquad+\sigma_{P}^{m_{k}}\,\lambda_{Q}^{n_{k}}\,\gamma_{3}\,\big{(}\tilde{\mathfrak{c}}_{k}\,a_{3}-\tilde{\mathfrak{s}}_{k}\,a_{2}\big{)}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)+\\\
&\qquad+{\sigma_{P}}^{m_{k}}\,\sigma_{Q}^{n_{k}}\,\mathrm{h.o.t.}^{*}_{k},\end{split}$
(7.16)
$\begin{split}&\breve{y}_{k}=\mu+b_{1}\,\lambda_{P}^{m_{k}}\,{\sigma_{Q}}^{n_{k}}\,\alpha_{2}\,y+\\\
&\qquad+b_{1}\,{\lambda_{P}}^{m_{k}}\,{\sigma_{P}}^{m_{k}}\,\lambda_{Q}^{n_{k}}\,\sigma_{Q}^{n_{k}}\,\big{(}\alpha_{1}\,(\mathfrak{c}_{k}\,x-\mathfrak{s}_{k}\,z)+\alpha_{3}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)\big{)}+\\\
&\qquad+\Big{(}\tilde{\mathfrak{c}}_{k}^{2}\,b_{2}+\tilde{\mathfrak{s}}_{k}^{2}\,b_{3}+\tilde{\mathfrak{c}}_{k}\,\tilde{\mathfrak{s}}_{k}\,b_{4}\Big{)}\,\beta_{2}^{2}\,y^{2}+\\\
&\qquad+\sigma_{P}^{2m_{k}}\,\lambda_{Q}^{2n_{k}}\Big{(}\tilde{\mathfrak{s}}_{k}^{2}\,b_{2}+\tilde{\mathfrak{c}}_{k}^{2}\,b_{3}-\tilde{\mathfrak{c}}_{k}\,\tilde{\mathfrak{s}}_{k}\,b_{4}\Big{)}\,\gamma_{3}^{2}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)^{2}+\\\
&\qquad+\sigma_{P}^{m_{k}}\,\lambda_{Q}^{n_{k}}\,\Big{(}2\tilde{\mathfrak{c}}_{k}\,\tilde{\mathfrak{s}}_{k}\,(b_{3}-b_{2})+(\tilde{\mathfrak{c}}_{k}^{2}\,-\tilde{\mathfrak{s}}_{k}^{2})\,b_{4}\Big{)}\,\beta_{2}\,\gamma_{3}\,(\mathfrak{s}_{k}\,x\,y+\mathfrak{c}_{k}\,y\,z)+\\\
&\qquad+{\sigma_{P}}^{2m_{k}}\,\sigma_{Q}^{2n_{k}}\,\mathrm{h.o.t.}^{**}_{k},\end{split}$
(7.17)
$\begin{split}&\breve{z}_{k}=c_{1}\,\lambda_{P}^{m_{k}}\,{\lambda_{Q}}^{n_{k}}\,\big{(}\alpha_{1}\,(\mathfrak{c}_{k}\,x-\mathfrak{s}_{k}\,z)+\alpha_{3}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)\big{)}+\\\
&\qquad+c_{1}\,\lambda_{P}^{m_{k}}\,\sigma_{P}^{-m_{k}}\,\alpha_{2}\,y+\big{(}\tilde{\mathfrak{c}}_{k}\,c_{2}+\tilde{\mathfrak{s}}_{k}\,c_{3}\big{)}\,\beta_{2}\,y+\\\
&\qquad+\sigma_{P}^{m_{k}}\,\lambda_{Q}^{n_{k}}\,\gamma_{3}\,\big{(}\tilde{\mathfrak{c}}_{k}\,c_{3}-\tilde{\mathfrak{s}}_{k}\,c_{2}\big{)}\,(\mathfrak{s}_{k}\,x+\mathfrak{c}_{k}\,z)+\\\
&\qquad+{\sigma_{P}}^{m_{k}}\,\sigma_{Q}^{n_{k}}\,\mathrm{h.o.t.}^{***}_{k}.\end{split}$

This completes the calculations of
$\Psi_{k}^{-1}\circ\mathcal{R}_{\bar{\upsilon}_{k}(\mu)}\circ\Psi_{k}(\bar{X})$.

##  8\. Convergence of the renormalised sequence: end of the proof of Theorem
1.

Recall the choices of $\xi>0$ and
$\tau=\dfrac{\gamma_{3}(a_{3}-a_{2})}{\sqrt{2}}>0$ in Remark 6.2 satisfying

$\sigma_{P}^{m_{k}}\,\lambda_{Q}^{n_{k}}\rightarrow\tau^{-1}\xi,\quad
k\to+\infty.$

Consider the vector
$\bar{\varsigma}=\bar{\varsigma}(\xi,f)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(\varsigma_{1},\varsigma_{2},\varsigma_{3},\varsigma_{4},\varsigma_{5})\in\mathbb{R}^{5}$
where

(8.1) $\begin{split}\varsigma_{1}&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\frac{\beta_{2}(a_{2}+a_{3})}{\sqrt{2}},\quad\varsigma_{2}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\frac{\beta_{2}^{2}(b_{2}+b_{3}+b_{4})}{2},\quad\varsigma_{3}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\xi^{2}\left(\frac{b_{2}+b_{3}-b_{4}}{(a_{3}-a_{2})^{2}}\right),\\\
\varsigma_{4}&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\xi\sqrt{2}\left(\frac{\beta_{2}(b_{3}-b_{2})}{a_{3}-a_{2}}\right),\quad\varsigma_{5}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\frac{\beta_{2}(c_{2}+c_{3})}{\sqrt{2}}.\end{split}$

Note that the coordinates of $\bar{\varsigma}$ depend on $\xi>0$ and the
numbers $a_{1},\dots,c_{3}$ in the definition of $f^{N_{2}}$ (see (3.22)).

The next proposition implies Theorem 1.

######  Proposition 8.1.

Consider any compact set $K\subset\mathbb{R}^{3}$. Then

$\lim_{k\to+\infty}\Big{\|}\big{(}\Psi_{k}^{-1}\circ\mathcal{R}_{\bar{\upsilon}_{k}(\mu)}\circ\Psi_{k}-E_{(\xi,\mu,\bar{\varsigma}(\xi,f))}\big{)}|_{K}\Big{\|}_{C^{r}}=0,$

where $E_{(\xi,\mu,\bar{\varsigma})}$ is the endomorphism in (2.2).

###### Proof.

To make more transparent our calculations, let us first consider the leading
terms (low order terms) of the coordinates
$(\breve{x}_{k},\breve{y}_{k},\breve{z}_{k})$ in (7.15)-(7.17). Write

$\breve{X}_{k}^{\prime}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(\breve{x}_{k}^{\prime},\breve{y}_{k}^{\prime},\breve{z}_{k}^{\prime}),\qquad\left\\{\begin{split}\breve{x}_{k}^{\prime}=&\breve{x}_{k}-\sigma_{P}^{m_{k}}\,{\sigma_{Q}}^{n_{k}}\mathrm{h.o.t.}^{*}_{k},\\\
\breve{y}_{k}^{\prime}=&\breve{y}_{k}-\sigma_{P}^{2m_{k}}\,{\sigma_{Q}}^{2n_{k}}\mathrm{h.o.t.}^{**}_{k},\\\
\breve{z}_{k}^{\prime}=&\breve{z}_{k}-\sigma_{P}^{m_{k}}\,{\sigma_{Q}}^{n_{k}}\mathrm{h.o.t.}^{***}_{k}.\end{split}\right.$

To prove the proposition it is enough to see that

$\breve{X}_{k}^{\prime}\to E_{(\xi,\mu,\bar{\varsigma})},$

and that

$\sigma_{P}^{m_{k}}\,{\sigma_{Q}}^{n_{k}}\mathrm{h.o.t.}^{*}_{k}\to
0,\quad\sigma_{P}^{2m_{k}}\,{\sigma_{Q}}^{2n_{k}}\mathrm{h.o.t.}^{**}_{k}\to
0,\quad\sigma_{P}^{m_{k}}\,{\sigma_{Q}}^{n_{k}}\mathrm{h.o.t.}^{***}_{k}\to
0,$

where the converge occurs in the $C^{r}$-topology. This is done in the next
two lemmas.

######  Lemma 8.2.

$\lim_{k\to+\infty}\Big{\|}\big{(}\breve{X}_{k}^{\prime}-E_{(\xi,\mu,\bar{\varsigma}(\xi,f))}\big{)}|_{K}\Big{\|}_{C^{r}}=0.$

###### Proof.

This follows directly from
$\tilde{\mathfrak{c}}_{k},\tilde{\mathfrak{s}}_{k}\to\frac{1}{\sqrt{2}}$,
$\mathfrak{c}_{k}\to 0$, $\mathfrak{s}_{k}\to 1$ (see (6.8)), and
$c_{2}=c_{3}$ (see (3.23)). ∎

######  Lemma 8.3.

The terms $\sigma_{P}^{m_{k}}\,{\sigma_{Q}}^{n_{k}}\mathrm{h.o.t.}^{*}_{k}$,
$\sigma_{P}^{2m_{k}}\,{\sigma_{Q}}^{2n_{k}}\mathrm{h.o.t.}^{**}_{k}$, and
$\sigma_{P}^{m_{k}}\,{\sigma_{Q}}^{n_{k}}\mathrm{h.o.t.}^{***}_{k}$ converge
to zero in the $C^{r}$-topology on compact sets.

###### Proof.

We begin with some preliminary observations. Since
$\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}$ and
$\widehat{H}_{3}\big{(}\textbf{x}_{k}\big{)}$ have the same symbol of Landau
$O(\sigma_{P}^{-m_{k}}\,\lambda_{Q}^{2n_{k}}\,\sigma_{Q}^{-n_{k}})$, see
(7.12), and
$\tilde{\mathfrak{c}}_{k},\tilde{\mathfrak{s}}_{k}\to\tfrac{1}{\sqrt{2}}$, see
(6.8), the terms

$\tilde{\mathfrak{c}}_{k}\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}-\tilde{\mathfrak{s}}_{k}\widehat{H}_{3}\big{(}\textbf{x}_{k}\big{)},\quad\tilde{\mathfrak{s}}_{k}\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}+\tilde{\mathfrak{c}}_{k}\widehat{H}_{3}\big{(}\textbf{x}_{k}\big{)}$

have both symbol of Landau777Here we use the following property: if
$g_{1}=O(f_{1})$ and $g_{2}=O(f_{2})$ then $g_{1}+g_{2}=O(|f_{1}|+|f_{2}|)$.
equal to $O(\sigma_{P}^{-m_{k}}\,\lambda_{Q}^{2n_{k}}\,\sigma_{Q}^{-n_{k}})$.
With this in mind and using the multiplicative property of symbols of
Landau888i.e., $O(f\cdot g)=O(f)\cdot O(g)$., we get that

  * (i)

${O}(\mathrm{h.o.t.}^{*}_{k})={O}\big{(}{\lambda_{P}}^{m_{k}}\,\widetilde{H}_{1}\big{(}\textbf{x}_{k}\big{)}\big{)}+{O}(\sigma_{P}^{m_{k}}\,\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)})+{O}(H_{1}\big{(}\hat{\textbf{x}}_{k}\big{)}).$

  * (ii)

$O(\mathrm{h.o.t.}^{**}_{k})=O({\lambda_{P}}^{m_{k}})+O\big{(}\widetilde{H}_{1}\big{(}\textbf{x}_{k}\big{)}\big{)}+O\big{(}\big{(}\sigma_{P}^{m_{k}}\,\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}\big{)}^{2}\big{)}+O\big{(}\sigma_{Q}^{-n_{k}}\,\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}\big{)}+O\big{(}H_{2}\big{(}\hat{\textbf{x}}_{k}\big{)}\big{)}$.
In this estimate we use that $\sigma_{P}^{m_{k}}\,\lambda_{Q}^{n_{k}}$ is
convergent.

  * (iii)

$O(\mathrm{h.o.t.}^{***}_{k})=O\big{(}{\lambda_{P}}^{m_{k}}\big{(}\widetilde{H}_{1}\big{(}\textbf{x}_{k}\big{)}\big{)}\big{)}+O\big{(}\sigma_{P}^{m_{k}}\,\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}\big{)}+O\big{(}H_{3}\big{(}\hat{\textbf{x}}_{k}\big{)}\big{)}.$

Observing that
$O\big{(}H_{1}\big{(}\hat{\textbf{x}}_{k}\big{)}\big{)}=O\big{(}H_{3}\big{(}\hat{\textbf{x}}_{k}\big{)}\big{)}$
have the same symbol of Landau (see (3.22)), the proof of lemma is reduced to
the following claim.

######  Claim 8.4.

The following terms

  * (a)

${\lambda_{P}}^{m_{k}}\,\sigma_{P}^{2m_{k}}\,{\sigma_{Q}}^{2n_{k}}\,\widetilde{H}_{1}\big{(}\textbf{x}_{k}\big{)},$

  * (b)

$\sigma_{P}^{2m_{k}}\,{\sigma_{Q}}^{n_{k}}\,\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)},$

  * (c)

$\sigma_{P}^{m_{k}}\,{\sigma_{Q}}^{n_{k}}\,H_{1}\big{(}\hat{\textbf{x}}_{k}\big{)}$,

  * (d)

$\sigma_{P}^{2m_{k}}\,{\sigma_{Q}}^{2n_{k}}\,H_{2}\big{(}\hat{\textbf{x}}_{k}\big{)}$,

converge to zero in the $C^{r}$-topology on compact sets.

###### Proof.

We begin with some preliminary estimates. Recalling the symbols of Landau of
$\widetilde{H}_{i}\big{(}\textbf{x}_{k}\big{)}$ and
$\widehat{H}_{i}(\textbf{x}_{k})$ in (7.11) and (7.12), respectively, the
definition and properties of $H_{i}$ in (3.24), and the definition of
$\hat{\textbf{x}}_{k}$ in (7.13) whose coordinates satisfy Lemma 7.1, it
follows that

(8.2)
$\begin{split}\widetilde{H}_{1}\big{(}\textbf{x}_{k}\big{)}&=O(\lambda_{Q}^{2n_{k}})+O({\sigma_{P}}^{-2m_{k}}\,\lambda_{Q}^{n_{k}}\sigma_{Q}^{-n_{k}})+O(\sigma_{P}^{-4m_{k}}\,{\sigma_{Q}}^{-2n_{k}}),\\\
\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}&=O({\sigma_{P}}^{-m_{k}}\,\lambda_{Q}^{2n_{k}}\,\sigma_{Q}^{-n_{k}}),\\\
H_{1}\big{(}\hat{\textbf{x}}_{k}\big{)}&=O({\lambda_{P}}^{2m_{k}})+O({\lambda_{P}}^{m_{k}}\sigma_{P}^{-m_{k}}{\sigma_{Q}}^{-n_{k}})+O(\sigma_{P}^{-2m_{k}}\,{\sigma_{Q}}^{-2n_{k}}),\\\
H_{2}\big{(}\hat{\textbf{x}}_{k}\big{)}&=O({\lambda_{P}}^{2m_{k}})+O({\lambda_{P}}^{m_{k}}\sigma_{P}^{-m_{k}}{\sigma_{Q}}^{-n_{k}}).\end{split}$

To prove (a), note that

(8.3) ${\lambda_{P}}^{m_{k}}\sigma_{P}^{2m_{k}}{\sigma_{Q}}^{2n_{k}}\to
0,\quad k\to+\infty.$

Lemma 6.1 provides the constant $C\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}({\lambda_{P}}^{\frac{1}{2}}{\sigma_{P}})^{2(1+\tilde{\eta})}>0$
such that

${\lambda_{P}}^{m_{k}}\sigma_{P}^{2m_{k}}{\sigma_{Q}}^{2n_{k}}=\big{(}{\lambda_{P}}^{\frac{m_{k}}{2}}{\sigma_{P}}^{m_{k}}\sigma_{Q}^{n_{k}}\big{)}^{2}<C\,\Big{(}\big{(}({\lambda_{P}}^{\frac{1}{2}}{\sigma_{P}})^{\eta}\sigma_{Q}\big{)}^{n_{k}}\Big{)}^{2},$

where

$\eta=\dfrac{\log\lambda_{Q}^{-1}}{\log{\sigma_{P}}}\quad\mbox{and}\quad\tilde{\eta}=\dfrac{\log{(\tau\,\xi^{-1})}}{\log{\sigma_{P}}}.$

The convergence
${\lambda_{P}}^{m_{k}}\sigma_{P}^{2m_{k}}{\sigma_{Q}}^{2n_{k}}\to 0$ as
$k\to+\infty$ follows from
$0<({\lambda_{P}}^{\frac{1}{2}}\,{\sigma_{P}})^{\eta}\sigma_{Q}<1$ (see
(3.3)). Thus, on any compact set $K$ of $\mathbb{R}^{3}$, we have the
convergence

${\lambda_{P}}^{m_{k}}\,\sigma_{P}^{2m_{k}}\,{\sigma_{Q}}^{2n_{k}}\,\widetilde{H}_{1}\big{(}\textbf{x}_{k}\big{)}\to
0,\quad k\to+\infty.$

It is easy to see that this convergence also holds for the derivatives of
order $1\leq k\leq r$. Therefore

$\lim_{k\to+\infty}\Big{\|}{\lambda_{P}}^{m_{k}}\,\sigma_{P}^{2m_{k}}\,{\sigma_{Q}}^{2n_{k}}\,\widetilde{H}_{1}\big{(}\textbf{x}_{k}\big{)}|_{K}\Big{\|}_{C^{r}}=0.$

To prove (b), observe that (8.2) and (6.2) imply that

$\sigma_{P}^{2m_{k}}\,\sigma_{Q}^{n_{k}}\,\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}=O(\sigma_{P}^{m_{k}}\,\lambda_{Q}^{2n_{k}})=O(\lambda_{Q}^{n_{k}})\to
0,\quad k\to+\infty.$

Arguing as in (a) we get

$\lim_{k\to+\infty}\Big{\|}\sigma_{P}^{2m_{k}}\,\sigma_{Q}^{n_{k}}\,\widehat{H}_{2}\big{(}\textbf{x}_{k}\big{)}|_{K}\Big{\|}_{C^{r}}=0.$

To prove (c), observe that (8.2) and (8.3) imply that

$\begin{split}\sigma_{P}^{m_{k}}\,\sigma_{Q}^{n_{k}}\,H_{1}\big{(}\hat{\textbf{x}}_{k}\big{)}&=O({\lambda_{P}}^{2m_{k}}\,\sigma_{P}^{m_{k}}\,\sigma_{Q}^{n_{k}})+O({\lambda_{P}}^{m_{k}})+O(\sigma_{P}^{-m_{k}}\,{\sigma_{Q}}^{-n_{k}})\\\
&\leq O({\lambda_{P}}^{m_{k}})\cdot
O({\lambda_{P}}^{m_{k}}\,\sigma_{P}^{2m_{k}}\,\sigma_{Q}^{2n_{k}})\\\
&+O({\lambda_{P}}^{m_{k}})+O(\sigma_{P}^{-m_{k}}\,{\sigma_{Q}}^{-n_{k}})\to
0,\quad k\to+\infty.\end{split}$

Thus, arguing as in the previous cases,

$\lim_{k\to+\infty}\Big{\|}\sigma_{P}^{m_{k}}\,{\sigma_{Q}}^{n_{k}}\,H_{1}\big{(}\hat{\textbf{x}}_{k}\big{)}|_{K}\Big{\|}_{C^{r}}=0.$

Finally, to prove (d), observe that (8.2) and (8.3) imply that

$\begin{split}\sigma_{P}^{2m_{k}}\,\sigma_{Q}^{2n_{k}}\,H_{2}\big{(}\hat{\textbf{x}}_{k}\big{)}&=O({\lambda_{P}}^{2m_{k}}\,\sigma_{P}^{2m_{k}}\,\sigma_{Q}^{2n_{k}})+O({\lambda_{P}}^{m_{k}}\sigma_{P}^{m_{k}}{\sigma_{Q}}^{n_{k}})\\\
&\leq O({\lambda_{P}}^{m_{k}})\cdot
O({\lambda_{P}}^{m_{k}}\,\sigma_{P}^{2m_{k}}\,\sigma_{Q}^{2n_{k}})\\\
&+O({\lambda_{P}}^{m_{k}}\sigma_{P}^{2m_{k}}{\sigma_{Q}}^{2n_{k}})\to
0.\end{split}$

Thus, as in the previous cases,

$\lim_{k\to+\infty}\Big{\|}\sigma_{P}^{2m_{k}}\,{\sigma_{Q}}^{2n_{k}}\,H_{2}\big{(}\hat{\textbf{x}}_{k}\big{)}|_{K}\Big{\|}_{C^{r}}=0.$

This proves the claim. ∎

The proof or the lemma is now complete. ∎

The proof of the proposition is now complete. ∎

This completes the proof of Proposition 8.1.

##  9\. Non-transverse cycles leading blender-horseshoes

We discuss here the existence of diffeomorphisms $f$ satisfying the hypotheses
of Corollary 1.

######  Lemma 9.1.

There are diffeomorphisms $f$ satisfying conditions
$\mathbf{(A)}$-$\mathbf{(C)}$ whose vector
$\bar{\varsigma}=\bar{\varsigma}(f,\xi)$ satisfies equation (2.3) for every
$\xi\in(1.18,1.19)$.

###### Proof.

Consider the vector

$v=(a_{1},a_{2},a_{3},,b_{1},b_{2},b_{3},b_{4},c_{1},c_{2},c_{3})\in\mathbb{R}^{10},$

where $a_{1},\dots,c_{3}$ are the constants defining the transition map from
$P$ to $Q$ in (3.22). The precise formula for the vector $\bar{\varsigma}$ is
given in equation (8.1) and is of the form (with a slight abuse of notation)
$\bar{\sigma}(f,\xi)=\bar{\varsigma}(v,\xi)=(\bar{\varsigma}_{1}(v,\xi),\dots,\bar{\varsigma}_{5}(v,\xi))$,
where we interpret the coeficients $\bar{\varsigma}_{i}$ as maps depending on
$(v,\xi)$.

Consider the family of maps
$\gamma_{\xi}\colon\mathbb{R}^{11}\to\mathbb{R}^{2}$ defined by

$\gamma_{\xi}:\mathbb{R}^{11}\to(-\varepsilon,\varepsilon)^{2},\quad\gamma_{\xi}(v)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{(}\kappa(v,\xi),\eta(v,\xi)\big{)},$

where $\kappa,\eta\colon\mathbb{R}^{11}\to\mathbb{R}$ are given by

$\kappa(v,\xi)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\frac{\varsigma_{1}^{2}(v,\xi)\varsigma_{3}(v,\xi)}{\varsigma_{2}(v,\xi)},\qquad\eta(v,\xi)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\frac{\varsigma_{1}(v,\xi)\varsigma_{5}(v,\xi)}{\varsigma_{2}(v,\xi)}.$

It is immediate to verify that given any $\xi\in(1.18,1.19)$ then every
$(\kappa_{0},\eta_{0})\in(-\varepsilon,\varepsilon)^{2}$ is a regular value of
$\gamma_{\xi}$. This implies the lemma. ∎

## References

  * [1] P. G. Barrientos and A. Raibekas, Robustly non-hyperbolic transitive symplectic dynamics, arXiv:1707.06473, (2017).
  * [2] M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math. (2), 133 (1991), pp. 73–169.
  * [3] C. Bonatti, S. Crovisier, L. J. Díaz, and A. Wilkinson, What is$\ldots$ a blender?, Notices Amer. Math. Soc., 63 (2016), pp. 1175–1178.
  * [4] C. Bonatti and L. Díaz, Connexions hétéroclines et généricité d’une infinité de puits et de sources, Ann. Sci. École Norm. Sup. (4), 32 (1999), pp. 135–150.
  * [5] , On maximal transitive sets of generic diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., (2002), pp. 171–197 (2003).
  * [6] , Robust heterodimensional cycles and $C^{1}$-generic dynamics, J. Inst. Math. Jussieu, 7 (2008), pp. 469–525.
  * [7] C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math. (2), 143 (1996), pp. 357–396.
  * [8] , Abundance of $C^{1}$-robust homoclinic tangencies, Trans. Amer. Math. Soc., 364 (2012), pp. 5111–5148.
  * [9] C. Bonatti and L. J. Díaz, Fragile cycles, J. Differential Equations, 252 (2012), pp. 4176–4199.
  * [10] C. Bonatti, L. J. Díaz, and M. Viana, Discontinuity of the Hausdorff dimension of hyperbolic sets, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), pp. 713–718.
  * [11] C. Bonatti, L. J. Díaz, and M. Viana, Dynamics beyond uniform hyperbolicity: A global geometric and probabilistic perspective, vol. 102, Springer Science & Business Media, 2006.
  * [12] L. J. Díaz, Persistence of cycles and nonhyperbolic dynamics at heteroclinic bifurcations, Nonlinearity, 8 (1995), pp. 693–713.
  * [13] , Robust nonhyperbolic dynamics and heterodimensional cycles, Ergodic Theory Dynam. Systems, 15 (1995), pp. 291–315.
  * [14] L. J. Díaz, S. Kiriki, and K. Shinohara, Blenders in centre unstable Hénon-like families: with an application to heterodimensional bifurcations, Nonlinearity, 27 (2014), pp. 353–378.
  * [15] L. J. Díaz, A. Nogueira, and E. R. Pujals, Heterodimensional tangencies, Nonlinearity, 19 (2006), pp. 2543–2566.
  * [16] L. J. Díaz and S. A. Pérez, Blender-horseshoes in center-unstable Hénon-like families, arXiv:1804.02577v1, (2018).
  * [17] L. J. Díaz and J. Rocha, Non-critical saddle-node cycles and robust non-hyperbolic dynamics, Dynam. Stability Systems, 12 (1997), pp. 109–135.
  * [18] , Partially hyperbolic and transitive dynamics generated by heteroclinic cycles, Ergodic Theory Dynam. Systems, 21 (2001), pp. 25–76.
  * [19] L. J. Díaz and J. Rocha, Heterodimensional cycles, partial hyperbolicity and limit dynamics, Fund. Math., 174 (2002), pp. 127–186.
  * [20] S. V. Gonchenko, V. S. Gonchenko, and J. C. Tatjer, Three-dimensional dissipative diffeomorphisms with codimension two homoclinic tangencies and generalized Hénon maps, in Progress in nonlinear science, Vol. 1 (Nizhny Novgorod, 2001), RAS, Inst. Appl. Phys., Nizhniy Novgorod, 2002, pp. 63–79.
  * [21] , Bifurcations of three-dimensional diffeomorphisms with non-simple quadratic homoclinic tangencies and generalized Hénon maps, Regul. Chaotic Dyn., 12 (2007), pp. 233–266.
  * [22] S. V. Gonchenko, J. D. Meiss, and I. I. Ovsyannikov, Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation, Regul. Chaotic Dyn., 11 (2006), pp. 191–212.
  * [23] , Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation, Regul. Chaotic Dyn., 11 (2006), pp. 191–212.
  * [24] S. V. Gonchenko, I. I. Ovsyannikov, C. Simó, and D. Turaev, Three-dimensional Hénon-like maps and wild Lorenz-like attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), pp. 3493–3508.
  * [25] S. V. Gonchenko, L. P. Shilnikov, and O. V. Stenkin, On Newhouse regions with infinitely many stable and unstable invariant tori, in Progress in nonlinear science, Vol. 1 (Nizhny Novgorod, 2001), RAS, Inst. Appl. Phys., Nizhniy Novgorod, 2002, pp. 80–102.
  * [26] S. V. Gonchenko, L. P. Shilnikov, and D. V. Turaev, On dynamical properties of multidimensional diffeomorphisms from Newhouse regions. I, Nonlinearity, 21 (2008), pp. 923–972.
  * [27] , On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors, Regul. Chaotic Dyn., 14 (2009), pp. 137–147.
  * [28] S. V. Gonchenko, O. V. Sten’kin, and L. P. Shilnikov, On the existence of infinitely many stable and unstable invariant tori for systems from Newhouse regions with heteroclinic tangencies, Nelineinaya Dinamika [Russian Journal of Nonlinear Dynamics], 2 (2006), pp. 3–25.
  * [29] S. Hittmeyer, B. Krauskopf, H. Osinga, and K. Shinohara, The geometry of blenders in a three-dimensional Hénon-like family, preprint, (2017).
  * [30] S. Kiriki, Y. Nishizawa, and T. Soma, Heterodimensional tangencies on cycles leading to strange attractors, Discrete Contin. Dyn. Syst., 27 (2010), pp. 285–300.
  * [31] S. Kiriki and T. Soma, $C^{2}$-robust heterodimensional tangencies, Nonlinearity, 25 (2012), pp. 3277–3299.
  * [32] L. Mora and M. Viana, Abundance of strange attractors, Acta Math., 171 (1993), pp. 1–71.
  * [33] C. G. Moreira, There are no $C^{1}$-stable intersections of regular Cantor sets, Acta Math., 206 (2011), pp. 311–323.
  * [34] S. E. Newhouse, Nondensity of axiom ${\rm A}({\rm a})$ on $S^{2}$, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), (1970), pp. 191–202.
  * [35] , Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), pp. 9–18.
  * [36] , The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., (1979), pp. 101–151.
  * [37] J. Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors, Astérisque, (2000), pp. xiii–xiv, 335–347.  Géométrie complexe et systèmes dynamiques (Orsay, 1995).
  * [38] J. Palis and F. Takens, Cycles and measure of bifurcation sets for two-dimensional diffeomorphisms, Invent. Math., 82 (1985), pp. 397–422.
  * [39] , Hyperbolicity and the creation of homoclinic orbits, Ann. of Math. (2), 125 (1987), pp. 337–374.
  * [40] , Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations: Fractal Dimensions and Infinitely Many Attractors in Dynamics, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1995.
  * [41] S. A. Pérez, $C^{r}$-stabilisation of non-transverse heterodimensional cycles, PhD Thesis, Pontifícia Universidade Católica do Rio de Janeiro, Brazil, (2016).
  * [42] J. C. Tatjer, Three-dimensional dissipative diffeomorphisms with homoclinic tangencies, Ergodic Theory Dynam. Systems, 21 (2001), pp. 249–302.
  * [43] L. Tedeschini-Lalli and J. A. Yorke, How often do simple dynamical processes have infinitely many coexisting sinks?, Comm. Math. Phys., 106 (1986), pp. 635–657.
  * [44] D. V. Turaev and L. P. Shilnikov, An example of a wild strange attractor, Mat. Sb., 189 (1998), pp. 137–160.