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#  Nontransverse heterodimensional cycles: stabilisation and robust tangencies

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>Instituto de Matemáticas, Pontificia Universidad
Católica de Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso, Chile.
<EMAIL_ADDRESS>
###### Abstract.

We consider three-dimensional diffeomorphisms having simultaneously
heterodimensional cycles and heterodimensional tangencies associated to
saddle-foci. These cycles lead to a completely nondominated bifurcation
setting. For every $r{\geqslant}2$, we exhibit a class of such diffeomorphisms
whose heterodimensional cycles can be $C^{r}$ stabilised and (simultaneously)
approximated by diffeomorphisms with $C^{r}$ robust homoclinic tangencies. The
complexity of our nondominated setting with plenty of homoclinic and
heteroclinic intersections is used to overcome the difficulty of performing
$C^{r}$ perturbations, $r\geqslant 2$, which are remarkably more difficult
than $C^{1}$ ones. Our proof is reminiscent of the Palis-Takens’ approach to
get surface diffeomorphisms with infinitely many sinks (Newhouse phenomenon)
in the unfolding of homoclinic tangencies of surface diffeomorphisms. This
proof involves a scheme of renormalisation along nontransverse heteroclinic
orbits converging to a center-unstable Hénon-like family displaying blender-
horseshoes. A crucial step is the analysis of the embeddings of these blender-
horseshoes in a nondominated context.

###### Key words and phrases:

Blender-horseshoe, Center-unstable Hénon-like family, Heterodimensional cycle
and tangency, Homoclinic tangency, Renormalisation scheme, Stabilisation of a
cycle.

###### 2020 Mathematics Subject Classification:

Primary: 37C20. Secondary: 37C29, 37D20, 37D30

This paper is part of the PhD thesis of SP (PUC-Rio) supported by CNPq and
CAPES - Finance Code 001 (Brazil). LJD was partially supported by INCTMat-
Faperj and CNPq (Brazil). SP was supported by Programa Postdoctorado FONDECYT
3190174 (Chile) and partially supported by CMUP grant PTDC/CTM/BIO-4043-2014 –
project UID/MAT/00144/2019 (Portugal). The authors thank the hospitality of
CMUP, PUC (Chile), and USACH (Chile).

To Jacob Palis, in the occasion of his 80th birthday

##  1\. Introduction

Palis’ density conjecture [22] claims that bifurcations through _cycles_
(either homoclinic tangencies or heterodimensional cycles) associated to
_saddles_ (hyperbolic periodic points) are the main mechanisms for destroying
hyperbolic dynamics: any nonhyperbolic system can be approximated by
diffeomorphisms displaying one of those bifurcations. A homoclinic tangency
associated to a saddle occurs when the invariant (stable and unstable) sets of
a saddle have a nontransverse intersection. A heterodimensional cycle
associated with a pair of saddles of different _indices_ (dimension of the
unstable bundle) occurs when the invariant sets of these saddles intersects
cyclically. Note that heterodimensional cycles can only occur in dimension at
least three and that there are settings (as the one in this paper) where both
types of bifurcations occur simultaneously with overlapping effects.

As a consequence of the Kupka-Smale genericity theorem111Periodic points of
generic diffeomorphisms are hyperbolic and their invariant manifolds are in
general position (i.e., either they intersect transversely or they are
disjoint)., a cycle associated to saddles is a fragile configuration that can
be destroyed by small perturbations. However, these configurations can become
robust (indestructible by small perturbations) when these saddles are embedded
in some special type of horseshoes. Hence, it is natural to consider also
heterodimensional cycles and tangencies associated to (basic) hyperbolic sets
(for the precise definition see Section 1.2). One aims to understand when a
bifurcation through a fragile cycle associated to saddles can lead to such
robust cycles.

Bonatti222Formulated in Bonatti’s talk _The global dynamics of $C^{1}$ generic
diffeomorphisms or flows_, in the Second Latin American Congress of
Mathematicians, Cancún, México (2004). See also [6]. stated a stronger version
of Palis’ conjecture using robust cycles: the union of the $C^{r}$ open sets
of hyperbolic diffeomorphisms (satisfying the Axiom A and the no-cycles
properties) and of diffeomorphisms with $C^{r}$ robust cycles is dense in the
space of $C^{r}$ diffeomorphisms, see [8, Conjecture 1.10]. For results and
recent progress in the previous conjectures, see [26, 11, 12] for the Palis’
one and [21, 19, 1, 8] for Bonatti’s one. Some of these results will be
discussed below.

The latter conjecture has several motivations, one of them comes from the
study of global dynamics of diffeomorphisms when considering the decomposition
of the chain recurrence set into its chain of recurrence classes. Note first
that two saddles involved in a cycle are always in the same class of
recurrence. One aims to put these saddles robustly into the same class. If
such saddles are contained in a pair of transitive hyperbolic sets involved in
a robust cycle then the continuations of the hyperbolic sets (and hence the
ones of the initial saddles) are also in the same class of recurrence. This
gives a way to put saddles with different indices into prescribed recurrence
classes. This process is known as _stabilisation of a cycle_. More precisely,
a heterodimensional cycle of a $C^{r}$ diffeomorphism $f$ associated to
saddles $P$ and $Q$ can be _$C^{r}$ stabilised_ if there are diffeomorphisms
arbitrarily $C^{r}$ close to $f$ with a $C^{r}$ robust cycle associated to
transitive hyperbolic sets containing the continuations of $P$ and $Q$. The
stabilisation of a homoclinic tangency associated to a saddle is defined
analogously.

The stabilisation of cycles depends on the type of cycle, differentiability,
and dimension. To avoid technicalities, we will restrict our discussion to
dimensions two and three333This allows us to skip the technical discussion of
the so-called _coindex_ of a heterodimensional cycle, since in dimension three
the coindex is always one. For phenomena that may occur in higher dimensions,
as for instance robust tangencies of large codimension, we refer to [2, 4]..
We first consider homoclinic tangencies. For surface diffeomorphisms this
question is completely solved: there are no $C^{1}$ robust tangencies and
hence no homoclinic tangency can be $C^{1}$ stabilised, [19]. On the other
hand, if $r\geqslant 2$ then every such a tangency can be $C^{r}$ stabilised,
[20]. In dimension three, a combination of [20, 27, 25] and the theory of
normal hyperbolicity implies that, every $C^{r}$ homoclinic tangency can be
$C^{r}$ stabilised for $r\geqslant 2$. In the $C^{1}$ case, the stabilisation
of homoclinic tangencies involves geometrical constraints and, in general, it
is not known which tangencies can be stabilised (see also [10]). For instance,
combining normally hyperbolic surfaces and [19], one can get homoclinic
tangencies that cannot be $C^{1}$ stabilised, see also [6, Sections 4.3–6].

Consider now heterodimensional cycles. First, every three-dimensional
heterodimensional cycle leads to $C^{1}$ robust cycles [7], although these
cycles may be not related to the saddles in the initial cycle. In [8] there is
given a class of heterodimensional cycles that cannot be $C^{1}$ stabilised
(_twisted_ cycles). Finally, in [9] it is proved that every nontwisted cycle
can be $C^{1}$ stabilised. The techniques used in these works are genuinely
$C^{1}$. Due to the absence of suitable tools, the stabilisation problem in
higher differentiability is widely open.

To explain our results, we recall that, in dimension three, two saddles with
different indices have a _heterodimensional tangency_ if their two dimensional
invariant manifolds have some nontransverse intersection. These tangencies
were introduced in [15] as a source of robustly nondominated/wild dynamics,
see also [18, 3]. In this paper, we consider a class of three-dimensional
$C^{r}$ diffeomorphisms whose heterodimensional cycles involve
heterodimensional tangencies (see Figure 1). For every $r\geqslant 2$, we
state the $C^{r}$ stabilisation of such cycles and show that they also provide
$C^{r}$ robust homoclinic tangencies, see the Stabilisation and Robust
tangencies theorems below. Let us now provide further details of our
statements.

\begin{overpic}[scale={0.05}]{InicialCycle.jpg} \put(30.0,50.0){{\large$Q$}}
\put(112.0,70.0){{\large$P$}} \put(160.0,50.0){{\large$Q$}}
\put(244.0,70.0){{\large$P$}} \end{overpic} Figure 1. Heterodimensional cycles
with heterodimensional tangencies.

Let $M$ be a three-dimensional compact manifold. We consider a set
$\mathcal{H}^{r}_{{{\mathrm{BH}}}}(M)$ of $C^{r}$ diffeomorphisms of $M$
having a heterodimensional cycle with a heterodimensional tangency associated
to saddle-foci $P$ and $Q$ of indices two and one satisfying the following
conditions:

  * •

Linearising assumptions at $P$ and $Q$ and spectral conditions implying some
sort of locally dissipative behaviour (Section 2.1.1).

  * •

The one-dimensional invariant manifolds $W^{\mathrm{s}}(P,f)$ and
$W^{\mathrm{u}}(Q,f)$ have a quasi-transverse intersection along the orbit of
some point $X$ and the two dimensional invariant manifolds
$W^{\mathrm{u}}(P,f)$ and $W^{\mathrm{s}}(Q,f)$ have a heterodimensional
tangency along the orbit of some point $Y$. This tangency may be of hyperbolic
or elliptic type (Section 2.2). The type of tangency plays an important role
in the resulting dynamics.

  * •

Conditions on the “transitions” from $P$ to $Q$ and from $Q$ to $P$ along the
orbits of the heteroclinic points $X$ and $Y$ (Section 2.1.2).

The precise description of the set $\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$ is
given in Section 2. Our main results are the following, see Theorem 1.1 for
further details.

######  Stabilisation of cycles.

Let $r\geqslant 2$. Any cycle in $\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$ can be
$C^{r}$ stabilised.

The next result deals with diffeomorphisms in
$\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$ whose heterodimensional tangency is of
elliptic type (see the lefthand side of Figure 1). This leads to the
definition of the subset $\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{e}^{+}}(M)$
of $\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$, see Section 2.2 for the precise
definition and a discussion.

######  Robust tangencies.

Let $r\geqslant 2$. Every diffeomorphism in
$\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{e}^{+}}(M)$ can be $C^{r}$
approximated by diffeomorphisms with a $C^{r}$ robust homoclinic tangency
associated to a basic set containing the continuation of the saddle-focus of
index two.

###  1.1. Our approach: a renormalisation scheme leading to blender-horseshoes

To explain the strategy of the proof of our results let us first recall the
approach in [23, Chapter 6]444In [23] it is proved the generic coexistence of
infinitely many sinks, in this proof the occurrence of robust tangencies is a
key step. In this homoclinic case, these robust tangencies imply the
stabilisation of the tangency, defined similarly as in the case of a cycle. to
stabilise homoclinic tangencies of $C^{2}$ diffeomorphisms. The construction
in [23] has the following main ingredients: (a) a renormalisation scheme at a
homoclinic tangency, (b) convergence of the scheme to a quadratic one-
parameter family, (c) existence of parameters of the family corresponding to
_thick horseshoes_ (horseshoes with large “fractal-like dimension”), and (d)
control of the localisation of the thick horseshoe guaranteeing that it is
homoclinically related555Two hyperbolic sets with the same index are
homoclinically related if their invariant manifolds intersect cyclically and
transversally. to the continuation of the initial saddle. A key property in
this approach is that thick horseshoes are $C^{2}$ robust, thus their
existence for the limit map extends to nearby systems.

Our strategy to get the $C^{r}$ stabilisation of cycles in
$\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$ translates the ideas of [23] to a
heterodimensional setting following the approach started in [14]. In our
construction, the ingredients (a)–(d) above are replaced by: (a’) a
renormalisation scheme at a heterodimensional tangency, (b’) convergence of
the scheme to a center-unstable Hénon-like family, (c’) existence of
parameters corresponding to _blender-horseshoes,_ (d’) prove that the blender-
horseshoes are homoclinically related to the initial saddle of index two and
have a robust cycle with the initial saddle of index one. Let us observe that,
in very rough terms, blender-horseshoes are local hyperbolic plugs used to get
robust heterodimensional cycles, where they play a role similar to the one of
the thick horseshoes for homoclinic tangencies, see Section 5 for details. As
above, a key step is to analise how the blender-horseshoes are embedded in the
global dynamics.

Here we rely on preliminary results in [16, 17] towards the development of the
strategy (a’)–(d’). In our context, the “limit” family is the center-unstable
Hénon-like family given by

$G_{\varpi}(x,y,z)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(y,\mu+y^{2}+\eta_{1}\,y\,z+\eta_{2}\,z^{2},\xi\,z+y),\quad\varpi=(\xi,\mu,\eta_{1},\eta_{2}).$

For diffeomorphisms in $\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$, the
renormalisation scheme and its convergence to the family $G_{\varpi}$ (steps
(a’) and (b’)). were obtained in [16]. A crucial property (step (c’)) is that
there is an open set $\mathcal{O}_{\mathrm{BH}}$ of parameters $\varpi$ for
which the family $G_{\varpi}$ exhibits blender-horseshoes, see [17]. In the
final step (d’), we analyse how these blender-horseshoes are embedded in the
global dynamics (the blender-horseshoe is homoclinically related to $P$ and
has a robust cycle with $Q$). This is a major difficulty in our nondominated
setting. It turns out that the lack of domination is simultaneously a
difficulty and, in some sense, an advantage. First, the existence of nonreal
multipliers makes the renormalisation scheme and the “existence and
localisation” of blenders a difficult task. On the other hand, the dynamics at
the bifurcation is very rich and, in particular, enables us to find new
homoclinic and heteroclinic orbits close to the initial cycle. As a heuristic
principle, this richness allows us to overcome the difficulty of performing
$C^{r}$ perturbations, $r\geqslant 2$, which are notably more problematic than
$C^{1}$ ones.

The lack of domination also means that there are plenty of possibilities for
unfolding the cycles involving many parameters. For instance, comparing with
the setting of homoclinic tangencies where any transverse direction of
unfolding behaves in the same way, the lack of domination implies that any
direction of unfolding is different. Thus we have eight natural parameters:
six parameters corresponding to the unfolding of the nontransverse
intersections (three for the heterodimensional tangency and three for the
quasi-transverse heteroclinic intersection), and two parameters associated to
the arguments of the saddle-foci, see Section 6.1. We see that “unfoldings
following appropriate directions” lead to robust cycles. However, the
complexity of these cycles is huge and a complete description of the
bifurcations is beyond reach.

We now recall some definitions and state precisely our results.

###  1.2. Stabilisation of cycles and robust tangencies: precise statements

Let $M$ be a compact boundaryless manifold. Let $\mathrm{Diff}^{r}(M)$ be the
space of $C^{r}$ diffeomorphisms of $M$ endowed with the $C^{r}$ uniform
topology. Consider $f\in\mathrm{Diff}^{r}(M)$ and $\Lambda_{f}$ a hyperbolic
_transitive set_ (i.e. with a dense orbit) of $f$. Recall that there is a
$C^{r}$ neighbourhood $\mathcal{U}_{f}$ of $f$ such that every
$g\in\mathcal{U}_{f}$ has a hyperbolic set $\Lambda_{g}$ that is topologically
conjugate to $\Lambda_{f}$ called the _continuation_ of $\Lambda_{f}$. The
_index_ of $\Lambda_{f}$ is the dimension of its unstable bundle (by
transitivity, this number is well defined).

Consider $f\in\mathrm{Diff}^{r}(M)$ having a pair of transitive hyperbolic
sets $\Lambda_{f}$ and $\Upsilon_{f}$ with different indices. These sets form
a _heterodimensional cycle_ if their invariant stable and unstable sets
intersect cyclically, i.e., $W^{\mathrm{s}}(\Lambda_{f})\cap
W^{\mathrm{u}}(\Upsilon_{f})\neq\emptyset$ and
$W^{\mathrm{u}}(\Lambda_{f})\cap W^{\mathrm{s}}(\Upsilon_{f})\neq\emptyset$.
This cycle is _$C^{r}$ robust_ if there is a $C^{r}$ neighbourhood
$\mathcal{U}_{f}$ of $f$ consisting of diffeomorphisms $g$ such that the sets
$\Lambda_{g}$ and $\Upsilon_{g}$ have a heterodimensional cycle. The notion of
a _$C^{r}$ robust homoclinic tangency_ associated to $\Lambda_{f}$ is stated
similarly: there is a $C^{r}$ neighbourhood $\mathcal{U}_{f}$ of $f$ such that
for every $g\in\mathcal{U}_{f}$ the invariant stable and unstable sets of
$\Lambda_{g}$ have some nontransverse intersection. Recall that robust cycles
cannot be associated to _trivial_ hyperbolic sets (i.e., periodic orbits).

A heterodimensional cycle of a $C^{r}$ diffeomorphism $f$ associated to
saddles $P_{f}$ and $Q_{f}$ can be _$C^{r}$ stabilised_ if there are
diffeomorphisms $g\in\mathrm{Diff}^{r}(M)$ arbitrarily $C^{r}$ close $f$ with
a $C^{r}$ robust cycle associated to transitive hyperbolic sets $\Lambda_{g}$
and $\Upsilon_{g}$ containing the continuations $P_{g}$ and $Q_{g}$,
respectively.

Our main result is the following theorem.

######  Theorem 1.1.

Let $r\geqslant 2$ and $M$ be a compact boundaryless three-dimensional
manifold. Given $f\in\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$, with a cycle
associated to saddle-foci $P_{f}$ and $Q_{f}$ of indices two and one, there
are diffeomorphisms $g$ arbitrarily $C^{r}$ close to $f$ with a blender-
horseshoe $\Lambda_{g}$ of index two such that:

  * (i)

$\Lambda_{g}$ and $Q_{g}$ has a $C^{r}$ robust heterodimensional cycle and

  * (ii)

$\Lambda_{g}$ and $P_{g}$ are homoclinically related.

Moreover, if $f\in\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{e}^{+}}(M)$ then the
blender-horseshoe $\Lambda_{g}$ has a $C^{r}$ robust homoclinic tangency.

###  1.3. Steps of the proofs

We now explain the steps of the proof of Theorem 1.1. Consider
$f\in\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$ with a cycle associated to saddle-
foci $P_{f}$ and $Q_{f}$ as in the theorem. A preliminary step is to perturb
the original cycle to obtain a new diffeomorphism in
$\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$ (that continue denoting by $f$) having
transverse homoclinic points and new additional quasi-transverse heteroclinic
points associated to $P_{f}$ and $Q_{f}$ (see Proposition 4.1). We can now
apply the renormalisation scheme to this new cycle, getting diffeomorphisms
$g$ arbitrarily $C^{r}$ close to $f$ whose dynamics in a neighbourhood of the
cycle is close to a Hénon-like map $G_{\varpi}$ with
$\varpi\in\mathcal{O}_{\mathrm{BH}}$. By Proposition 7.2, each diffeomorphism
$g$ has a blender-horseshoe $\Lambda_{g}$ of index two. We will see that the
following holds:

(ia) The two-dimensional manifolds $W^{\mathrm{u}}(\Lambda_{g},g)$ and
$W^{{\mathrm{s}}}(Q_{g},g)$ intersect transversely, see Proposition 9.1. The
difficulty of this step is to control the size of the unstable manifold of
$\Lambda_{g}$, assuring that it is sufficiently “large” so that it is
connected to the stable manifold of $Q_{g}$. We overcome this difficulty with
an analysis motivated by the constructions in [23, Section 6.4] for homoclinic
tangencies of surface diffeomorphisms.

(ib) The one-dimensional manifolds $W^{\mathrm{s}}(\Lambda_{g},g)$ and
$W^{{\mathrm{u}}}(Q_{g},g)$ have nonempty intersection, see Proposition 10.1.
This step is inspired by [14, Theorem 1.4] (see Remark 1.1 for a discussion)
and involves quantitative aspects of the renormalisation scheme in [16]. In
this step the new quasi-transverse heteroclinic points above play an important
role.

(ii) The saddle $P_{g}$ and the blender-horseshoe $\Lambda_{g}$ are
homoclinically related, see Proposition 11.1. This step is a relatively simple
consequence of (ia) and (ib) where the existence of transverse homoclinic
points of $Q_{f}$ is used.

Conditions (ia) and (ii) are $C^{r}$ open, $r\geqslant 1$, while (ib) is not
(due to deficiency of the sum of the dimensions). The blender-horseshoe allows
us to make this nontransverse intersection $C^{r}$ robust, $r\geqslant 1$.
Thus conditions (ia) and (ib) give a $C^{r}$ robust cycle between $Q_{g}$ and
$\Lambda_{g}$. As $\Lambda_{g}$ and $P_{g}$ are homoclinically related, they
are contained in a larger hyperbolic set, implying the stabilisation of the
initial cycle.

In the second part of the theorem, about robust tangencies, we consider
diffeomorphisms with elliptic tangencies in
$\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{e}^{+}}(M)$ (lefthand side of Figure
1) and study the intersections between the two-dimensional manifolds of the
saddle-foci in the cycle. We see that these intersections generate “tubes
crossing the reference domain of the blender-horseshoe”, see Section 5.2.
These tubes will provide robust tangencies. This step involves the
constructions in [8] using folding manifolds.

######  Remark 1.1.

In [14] it is obtained a renormalisation scheme for $C^{r}$ diffeomorphisms
$f$, $r\geqslant 2$, with a configuration somewhat similar to the one here,
where the saddle-foci are replaced by a pair of saddles with real multipliers.
In [14] the intersection between the one-dimensional manifolds in (ib) is
obtained for $C^{1+\alpha}$ perturbations of $f$. Let us observe a perhaps
counterintuitive fact: the intersections between the “big” two dimensional
manifolds in (ia) are more difficult to obtain than the intersections between
the “small” one dimensional manifolds in (ib). Indeed, in [14] the
intersections (ia) and (ii) were not achieved.

### Organisation of the paper

The bifurcation setting is described in Section 2. In Section 3, we introduce
the perturbations used in our constructions. In Section 4, we prove that the
set of diffeomorphisms having additional “special” homoclinic and quasi-
transverse heteroclinic intersections is dense in
$\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$. These “special” homoclinic and
heteroclinic points will play an important role in our proof. Blender-
horseshoes and their occurrence in center-unstable Hénon-like families are
discussed in Section 5. In Section 6, we review some relevant ingredients
renormalisation scheme in [16] used in our constructions. In Section 7, we
study the interplay between the blender-horseshoes given by the
renormalisation scheme and the additional heteroclinic points. Section 8 deals
with orbits and itineraries associated to the renormalisation scheme. The
proof of Theorem 1.1 is completed in Sections 9–12. Section 9 deals with the
intersections between the two-dimensional invariant manifolds of $Q$ and of
the blender-horseshoe. In Section 10, we state the occurrence of robust
intersections between the one-dimensional invariant manifolds of $Q$ and of
the blender-horseshoe. In Section 11, we see that the saddle $P$ and the
blender-horseshoe are homoclinically related. Finally, in Section 12 we prove
the part of the theorem corresponding to robust tangencies. Section 13 is an
appendix collecting some explicit calculations of the renormalisation scheme
borrowed from [16].

##  2\. The bifurcation setting

In this section, we describe precisely the set
$\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$, see Definition 2.3. We close this
section with some comments on the geometry of the cycle. Throughout this
section we consider diffeomorphisms $f$ having a pair of saddle-foci of
different indices $P=P_{f}$ and $Q=Q_{f}$.

###  2.1. The set $\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$

We now explain the conditions in the definition of
$\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$: linearising dynamics and nontransverse
intersections and transition maps.

####  2.1.1. Linearisable local dynamics

(A) _Saddle-foci periodic points_ : Let $\pi(P)$ and $\pi(Q)$ be the periods
of $P$ and $Q$. We assume that $f^{\pi(P)}$ and $f^{\pi(Q)}$ are $C^{r}$
linearisable in small neighbourhoods $U_{P}$ of $P$ and $U_{Q}$ of $Q$. Denote
the eigenvalues of $Df^{\pi(P)}(P)$ and $Df^{\pi(Q)}(Q)$ by

(2.1) $\begin{split}&\big{(}\lambda_{P},\sigma_{P}\,e^{-2\pi
i\varphi_{P}},\sigma_{P}\,e^{2\pi i\varphi_{P}}\big{)}\mbox{ where\,\,
$0<|\lambda_{P}|<1<\sigma_{P},\,\varphi_{P}\in[0,1]$,}\\\
&\big{(}\lambda_{Q}\,e^{-2\pi i\varphi_{Q}},\lambda_{Q}\,e^{2\pi
i\varphi_{Q}},\sigma_{Q}\big{)}\mbox{ where\,\,
$0<\lambda_{Q}<1<|\sigma_{Q}|,\,\varphi_{Q}\in[0,1]$.}\end{split}$

We assume that

(2.2)
$0<\Big{|}\big{|}|\lambda_{P}|^{\frac{1}{2}}\,\sigma_{P}\big{|}^{\eta}\sigma_{Q}\Big{|}<1,\quad\mbox{where}\quad\eta=\dfrac{\log|\lambda_{Q}^{-1}|}{\log|\sigma_{P}|}.$

In what follows, we assume that in the linearising local coordinates the sets
$U_{P}$ and $U_{Q}$ are the “cubes” $[-a_{P},a_{P}]^{3}$ and
$[-a_{Q},a_{Q}]^{3}$, for some $a_{P},a_{Q}>0$. For simplicity, we also assume
that the periods $\pi(P)$ and $\pi(Q)$ are equal to one.

####  2.1.2. Nontransverse intersections and transition maps

(B) _Quasi-transverse intersection and its transition map_ : The one-
dimensional invariant manifolds of $P$ and $Q$ intersect quasi-transversely
along the orbit of a point $X=X_{f}$, that is $X\in W^{\mathrm{s}}(P,f)\cap
W^{\mathrm{u}}(Q,f)$ and

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

After replacing $X$ by some iterate, we can assume that $X\in U_{Q}$.
Associated to $X$ there is a transition map corresponding to some iterate of
$f$ going from $U_{Q}$ to $U_{P}$ defined as follows. There are
$N_{1}\in\mathbb{N}$ such that

$f^{N_{1}}(X)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\widetilde{X}\in U_{P}\quad\mbox{and}\quad f^{i}(X)\not\in
U_{P}\quad\mbox{for every}\quad 0\leqslant i<N_{1}$

and a small neighbourhood $U_{X}$ of $X$ contained in $U_{Q}$ such that

$f^{N_{1}}(U_{X})\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}U_{\widetilde{X}}\subset U_{P}.$

In the local coordinates at $P$ and $Q$, the restriction $\mathfrak{T}_{1}$ of
$f^{N_{1}}$ to $U_{X}$ is of the form:

(2.3)
$\mathfrak{T}_{1}(X+W)=f^{N_{1}}(X+W)=\widetilde{X}+A(W)+\widetilde{H}(W),$

where

(2.4) $A=\begin{pmatrix}\alpha_{1}&\alpha_{2}&\alpha_{3}\\\ 0&\beta_{2}&0\\\
0&0&\gamma_{3}\end{pmatrix},\qquad\alpha_{1}\beta_{2}\gamma_{3}\neq 0,$

and $\widetilde{H}:\mathbb{R}^{3}\to\mathbb{R}^{3}$ is such that
$\widetilde{H}(\textbf{0})=\textbf{0}$ and $D\widetilde{H}(\textbf{0})$ is the
null matrix. Note that $\alpha_{1}\beta_{2}\gamma_{3}\neq 0$ is not an
additional assumption since $f^{N_{1}}$ is a diffeomorphism.

(C) _Heterodimensional tangency and its transition map_ : The two-dimensional
invariant manifolds of $P$ and $Q$ intersect along the orbit of a point
$Y=Y_{f}$ that is a heterodimensional tangency, that is, the orbit of $Y$ is
contained in the set

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

As above, after replacing $Y$ by some iterate, we can assume that $Y\in
U_{P}$. Associated to $Y$ there is a transition map corresponding to some
iterate of $f$ going from $U_{P}$ to $U_{Q}$ defined as follows. There are
$N_{2}\in\mathbb{N}$ such that

$f^{N_{2}}(Y)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\widetilde{Y}\in U_{Q}\quad\mbox{and}\quad f^{i}(Y)\not\in
U_{Q}\quad\mbox{for every}\quad 0\leqslant i<N_{2}$

and a small neighbourhood $U_{Y}$ of $Y$ contained in $U_{P}$ such that

$f^{N_{2}}(U_{Y})\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}U_{\widetilde{Y}}\subset U_{Q}.$

In the local coordinates at $Q$ and $P$, the restriction $\mathfrak{T}_{2}$ of
$f^{N_{2}}$ to $U_{Y}$ is of the form:

(2.5) $\mathfrak{T}_{2}(Y+W)=f^{N_{2}}(Y+W)=\widetilde{Y}+B(W)+{H}(W),$

where $B$ is a quadratic map of the form

(2.6) $B\begin{pmatrix}x\\\ y\\\
z\end{pmatrix}=\begin{pmatrix}a_{1}x+a_{2}y+a_{3}z\\\
b_{1}x+b_{2}y^{2}+b_{3}z^{2}+b_{4}yz\\\
c_{1}x+c_{2}y+c_{3}z\end{pmatrix},\qquad b_{1}(a_{2}c_{3}-a_{3}c_{2})\neq 0.$

where ${H}\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$ is a map such that
${H}(\textbf{0})=\textbf{0}$, $D{H}(\textbf{0})$ is the null matrix, and

$\frac{\partial^{2}}{\partial
y^{2}}{H}_{2}(\textbf{0})=\frac{\partial^{2}}{\partial
z^{2}}{H}_{2}(\textbf{0})=\frac{\partial^{2}}{\partial y\partial
z}{H}_{2}(\textbf{0})=\textbf{0},$

here ${H}_{i}$ is the $i$-th component of $\widetilde{H}$. Note that
$b_{1}(a_{2}c_{3}-a_{3}c_{2})\neq 0$ is not an additional assumption since
$f^{N_{2}}$ is a diffeomorphism.

The constants $a_{1},...,c_{3}$ in the definition of $B$ satisfy the following
conditions

(2.7) $c_{2}=c_{3},\qquad\gamma_{3}(a_{3}-a_{2})>0$

that will guarantee the convergence of the renormalisation scheme.

######  Notation 2.1.

Given $f\in\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$ we say that $P,Q,X$, and $Y$
are the elements of the cycle of $f$ and that $N_{1}$ and $N_{2}$ are the
transition times of the cycle.

######  Notation 2.2 (Coordinates of the heteroclinic points).

In what follows, we will assume that, in our local coordinates, the
heteroclinic points above are of the form:

$\begin{split}&\widetilde{X}=(1,0,0),\quad Y=(0,1,1)\quad\mbox{(in the
neighbourhood $U_{P}$)},\\\
&{X}=(0,1,0),\quad\widetilde{Y}=(1,0,1)\quad\mbox{(in the neighbourhood
$U_{Q}$)}.\end{split}$

####  2.1.3. Parameters of the transition maps

To each diffeomorphism $f$ satisfying (A)-(C) and $\xi>1$ we associate the
following parameters

(2.8)
$\bar{\varsigma}=\bar{\varsigma}(\xi,f)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(\varsigma_{1},\varsigma_{2},\varsigma_{3},\varsigma_{4},\varsigma_{5})\in\mathbb{R}^{5},$

where

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

here $\beta_{2}$ is as in (2.4) and $a_{1},\dots,c_{3}$ are as in (2.6).

######  Definition 2.3 (The set $\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$).

The set $\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$ consists of the $C^{r}$
diffeomorphisms $f$ satisfying (A)-(C) such that

(2.10) $(a_{2}+a_{3})(b_{2}+b_{3}+b_{4})\neq 0$

and whose vector $\bar{\varsigma}(\xi,f)$ satisfies

$(\xi,\varsigma_{1}^{2}\varsigma_{3}\varsigma_{2}^{-1},\varsigma_{1}\varsigma_{4}\varsigma_{2}^{-1})\in(1.18,1.19)\times(-\varepsilon_{\mathrm{BH}},\varepsilon_{\mathrm{BH}})^{2},$

where $\varepsilon_{\mathrm{BH}}$ is a number fixed in Theorem 5.6.

######  Remark 2.4.

Equations (2.4) and (2.10) implies that
$\varsigma_{1}\,\varsigma_{2}\,\varsigma_{5}\neq 0$. These conditions are used
to get blender-horseshoes in the renormalisation scheme.

###  2.2. Geometry of the cycle: the sets
$\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{h}}(M)$,
$\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{e}}(M)$, and
$\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{e}^{+}}(M)$

For $R=P,Q$ consider

$W^{\ast}_{\mathrm{loc}}(R,f)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}C\big{(}R,W^{\ast}_{\mathrm{loc}}(R,f)\cap
U_{R}\big{)},\quad\ast={\mathrm{s}},{\mathrm{u}},$

here $C(x,A)$ is the connected component of the set $A$ containing the point
$x$.

The next definition classifies the two types of heterodimensional tangencies
that we will consider. Note that given any
$f\in\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$ the set $U_{P}\setminus
W^{\mathrm{u}}_{\mathrm{loc}}(P,f)$ has two connected components.

######  Definition 2.5 (Elliptic and hyperbolic tangencies).

The heterodimensional tangency at $Y$ is elliptic if there is a neighbourhood
$\operatorname{\mathcal{P}^{\mathrm{s}}_{Y}}$ of $Y$ in
$W^{\mathrm{s}}(Q,f)\cap U_{P}$ such that the set
$\operatorname{\mathcal{P}^{\mathrm{s}}_{Y}}\setminus\\{Y\\}$ is contained in
a connected component of $U_{P}\setminus W^{\mathrm{u}}_{\mathrm{loc}}(P,f)$.
The tangency is hyperbolic if every neighbourhood of $Y$ in
$W^{\mathrm{s}}(Q,f)$ contains points in both components of $U_{P}\setminus
W^{\mathrm{u}}_{\mathrm{loc}}(P,f)$.

In Figure 1, the heterodimensional tangency in the left-hand side is elliptic
while the one in the right-hand side is hyperbolic.

We observe that if $f\in\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$ then the
heterodimensional tangency at the point $Y$ is either hyperbolic or elliptic.
We split the set $\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$ in two parts,
$\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{h}}(M)$ and
$\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{e}}(M)$ consisting of hyperbolic and
elliptic heterodimensional tangencies, respectively.

For diffeomorphisms in $f\in\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{e}}(M)$ we
need to take in consideration the relative position of the tangency and the
quasi-transverse heteroclinic points. We consider the subset
$\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{e}^{+}}(M)$ of
$\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{e}}(M)$ such that (with the notation
above) the set $\operatorname{\mathcal{P}^{\mathrm{s}}_{Y}}\setminus\\{Y\\}$
and $\widetilde{X}$ are in the same connected component of $U_{P}\setminus
W^{\mathrm{u}}_{\mathrm{loc}}(P,f)$, see Figure 1. These geometrical
considerations have the same flavour of those in [24, Section 2].

##  3\. Translation and rotation-like perturbations

In this section, we describe the two types of $C^{r}$ perturbations used in
our constructions. We start by introducing a class of auxiliary bump
functions.

######  Caveat.

For simplicity, throughout this paper, we will use the term perturbation to
refer to arbitrarily small ones.

###  3.1. Auxiliary bump functions

Consider a family of $C^{r}$ bump functions $b^{\theta}$, $\theta>1$, such
that

(3.1) $b^{\theta}(x)=\begin{cases}0\,,&|x|\geqslant\theta,\\\ 0\leqslant
b^{\theta}(x)\leqslant 1\,,&1\leqslant|x|\leqslant\theta,\\\ 1\,,&|x|\leqslant
1.\end{cases}$

Associated to $b^{\theta}$ we consider the family of bump functions

$b^{\theta}_{\rho}(x)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}b^{\theta}\left(\frac{x}{\rho}\right),\quad\rho>0$

and the three-dimensional bump functions

(3.2)
$\Pi^{\theta}_{\rho}\colon\mathbb{R}^{3}\to[0,1],\quad\Pi^{\theta}_{\rho}(x,y,z)=b^{\theta}_{\rho}(x)\,b^{\theta}_{\rho}(y)\,b^{\theta}_{\rho}(z).$

Denote by $B(x,\tau)$ the open ball in $\mathbb{R}^{3}$ with center $x$ and
radius $\tau$ and by $\|\cdot\|_{r}$ the $C^{r}$ norm. Note that the support
of $\Pi^{\theta}_{\rho}$ is the closure of $B(\textbf{0},{\theta\rho})$ and
that

(3.3)
$\|\Pi^{\theta}_{\rho}\|_{r}\leqslant(\|b^{\theta}\|_{r})^{3}\,\rho^{-r}.$

In what follows, for simplicity, when $\theta=2$ we write
$b^{2}_{\rho}=b_{\rho}$ and $\Pi^{2}_{\rho}=\Pi_{\rho}$.

###  3.2. Translation-like perturbations

Given a point $Z_{0}\in\mathbb{R}^{3}$, a vector
$\widetilde{w}\in\mathbb{R}^{3}$, and small $\rho>0$, we consider the $C^{r}$
map $T_{Z_{0},\widetilde{w},\rho}:\mathbb{R}^{3}\to\mathbb{R}^{3}$ defined by

(3.4)
$T_{Z_{0},\widetilde{w},\rho}(Z)=\quad\left\\{\begin{split}&Z+\Pi_{\rho}(Z-Z_{0})\widetilde{w},\quad\mbox{if
$Z\in B(Z_{0},2\rho)$},\\\ &Z,\quad\mbox{if $Z\not\in
B(Z_{0},2\rho)$}.\end{split}\right.$

By construction and by (3.3), it holds

$\big{\|}T_{Z_{0},\widetilde{w},\rho}-\mathrm{id}\big{\|}_{r}\leqslant\big{\|}\Pi_{\rho}\big{\|}_{r}\,||\widetilde{w}||\leqslant(\|b\|_{r})^{3}\,\rho^{-r}\,||\widetilde{w}||.$

Therefore, for small $||\widetilde{w}||$, the map
$T_{Z_{0},\widetilde{w},\rho}$ is a $C^{r}$ perturbation of the identity
supported in $B(Z_{0},2{\rho})$. Finally observe that

$T_{Z_{0},\widetilde{w},\rho}\big{(}B(Z_{0},2\rho)\big{)}=B(Z_{0},2\rho).$

###  3.3. Rotation-like perturbations

We now consider maps
$I^{x}_{\omega},\,I^{y}_{\omega}\colon\mathbb{R}^{3}\to\mathbb{R}^{3},$
$\omega\in[-\pi,\pi]$, defined by

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

and for $\theta>1$ and $\kappa>0$ their associated $C^{\infty}$
diffeomorphisms

(3.5)
$R^{\ast}_{{\omega,\theta,\kappa}}\colon\mathbb{R}^{3}\to\mathbb{R}^{3},\quad
R^{\ast}_{\omega,\theta,\kappa}(W)=I^{\ast}_{b^{\theta}(\kappa||W||)\,\omega}(W^{T}),\quad\ast=x,y,$

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

Note that the restriction of $R^{\ast}_{{\omega,\theta,\kappa}}$ to the set
$[-\kappa^{-1},\kappa^{-1}]^{3}$ coincides with $I^{\ast}_{\omega}$ and
$R^{\ast}_{{\omega,\theta,\kappa}}$ is the identity map in the complement of
$[-\theta\kappa^{-1},\theta\kappa^{-1}]^{3}$. Note also

$R^{\ast}_{{\omega,\theta,\kappa}}\big{(}[-\theta\kappa^{-1},\theta\kappa^{-1}]^{3}\big{)}=[-\theta\kappa^{-1},\theta\kappa^{-1}]^{3},\quad\ast=x,y,$

and that there is a constant $C(\theta,\kappa)>0$ such that

$\big{\|}R^{\ast}_{{\omega,\theta,\kappa}}-\mathrm{id}\big{\|}_{C^{r}}<C(\theta,\kappa)|{\omega}|,\quad\ast=x,y.$

Thus, for every $\omega$ small enough, the map
$R^{\ast}_{{\omega},\theta,\kappa}$ is a $C^{r}$ perturbations of identity
supported in $[-\theta\kappa^{-1},\theta\kappa^{-1}]^{3}$.

##  4\. New heteroclinic and homoclinic intersections

Recall the definitions of the sets $\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$,
$\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{h}}(M)$, and
$\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{e}^{+}}(M)$ in Section 2.2. The main
result of this section is Proposition 4.1 claiming that for every $f$ in
$\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{h}}(M)$ (resp.
$\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{e}^{+}}(M)$) there are local $C^{r}$
perturbations $f_{\varepsilon}$ in
$\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{h}}(M)$ (resp.
$\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{e}^{+}}(M)$) of $f$ with pairs of
additional quasi-transverse heteroclinic points in
$W^{\mathrm{s}}(P,f_{\varepsilon})\cap W^{\mathrm{u}}(Q,f_{\varepsilon})$ and
additional transverse homoclinic points in
$W^{\mathrm{s}}(Q,f_{\varepsilon})\pitchfork
W^{\mathrm{u}}(Q,f_{\varepsilon})$. The proof of this proposition is done in
Section 4.2. To prove it, in Section 4.1, we state some preliminary results
about the invariant manifolds of the saddle-foci in the cycle. In Section 4.3,
we study the transitions associated to the new heteroclinic points. Finally,
in Section 4.4, we consider parameterisations of special unstable discs
throughout the new heteroclinic points contained the unstable manifold of $Q$.
The unfolding the cycle associated to these heteroclinic points will provide
unstable discs intersecting robustly the stable manifold of the blender-
horseshoes. We now go to the details.

Given $f\in\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$ with elements $P,Q,X,Y$ (recall
Notation 2.1) define the closed invariant set

(4.1) $\Gamma_{P,Q,X,Y}(f)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathrm{Orb}(X,f)\cup\mathrm{Orb}(Y,f)\cup\\{P,Q\\},$

where $\mathrm{Orb}(W,f)$ denotes the $f$-orbit of the point $W$.

Recall also the neighbourhoods $U_{X}$ and $U_{Y}$ of $X$ and $Y$ in Section
2.1.2.

In what follows, we use the notation $d_{r}(f,g)$ for the $C^{r}$ distance
between two maps $f,g\in\mathrm{Diff}^{r}(M)$.

######  Proposition 4.1.

Let $f\in\mathcal{H}^{r}_{{\mathrm{BH}},\ast}(M)$,
$\ast=\mathrm{h},\mathrm{e}^{+}$, with elements $P,Q,X,Y$ and transition times
$N_{1}$ and $N_{2}$. For every $\varepsilon,\delta>0$ there is
$f_{\varepsilon}\in\mathcal{H}^{r}_{{\mathrm{BH}},\ast}(M)$ with
$d_{r}(f,f_{\varepsilon})<\varepsilon$ such that:

  * (1)

$f_{\varepsilon}$ coincides with $f$ on the sets $\Gamma_{P,Q,X,Y}(f)$ and

$\bigcup_{i=0}^{N_{1}-1}f^{i}(U_{X})\,\cup\,\bigcup_{i=0}^{N_{2}-1}f^{i}(U_{Y,\varepsilon})$

where $U_{Y,\varepsilon}$ is a neighbourhood of $Y$ contained in $U_{Y}$
depending on $\varepsilon$.

  * (2)

$f_{\varepsilon}$ has two quasi-transverse heteroclinic points

$X_{1,\varepsilon},\,X_{2,\varepsilon}\in
f_{\varepsilon}^{-N_{1}}\big{(}W^{\mathrm{s}}_{{\mathrm{loc}}}(P,f_{\varepsilon})\big{)}\cap
W^{\mathrm{u}}(Q,f_{\varepsilon})\cap B(X,\delta)$

such that $\mathrm{Orb}(X_{1,\varepsilon},f_{\varepsilon})$,
$\mathrm{Orb}(X_{2,\varepsilon},f_{\varepsilon})$, and
$\mathrm{Orb}(X,f_{\varepsilon})$ are pairwise disjoint and
$X_{1,\varepsilon},X_{2,\varepsilon}\to X$ as $\varepsilon\to 0$.

  * (3)

$f_{\varepsilon}$ has two transverse intersection points

$Z^{\pm}_{\varepsilon}\in W^{\mathrm{s}}(Q,f_{\varepsilon})\pitchfork
W^{\mathrm{u}}_{{\mathrm{loc}}}(Q,f_{\varepsilon})$

such that in the local coordinates

$Z^{\pm}_{\varepsilon}=(0,1\pm\zeta^{\pm}_{\varepsilon},0),\quad
0<\zeta^{\pm}_{\varepsilon}<\delta.$

A preliminary step of the proof of this proposition is Lemma 4.1 in Section
4.1 claiming that the closure of the one dimensional invariant manifold of $P$
(resp. $Q$) contains the two dimensional invariant manifold of $Q$ (resp.
$P$).

###  4.1. Density properties of $W^{\mathrm{s}}(P,f)$ and
$W^{\mathrm{u}}(Q,f)$

Consider $f\in\operatorname{Diff}^{r}(M)$ with a saddle focus $R$ with
$f(R)=R$ such that the eigenvalues of $Df(R)$ are $\lambda\in\mathbb{R}$ and
$\sigma\,e^{\pm 2\,\pi i\varphi}\in\mathbb{C}$, where $0<|\lambda|<1<\sigma$
and $\varphi\in[0,1)$, and that is $C^{r}$ linearisable in a neighbourhood
$U_{R}$ of $R$. We identify $U_{R}$ with the Cartesian product of the local
invariant manifolds of $R$ (where the $x$\- and $yz$-spaces are the stable and
unstable eigenspaces of $Df(R)$, respectively.) We assume that there are (see
Figure 2):

  * •

A one-dimensional $C^{r}$ disc $L\subset U_{R}$ such that $L$ is quasi-
transverse to $W^{\mathrm{s}}_{\mathrm{loc}}(R,f)$ at some point $W$ in
interior of $L$. We let $L_{+}$ and $L_{-}$ the two connected components of
$L\setminus\\{W\\}$.

  * •

A two-dimensional $C^{r}$ disc $S\subset U_{R}$ intersecting transverselly
$W^{\mathrm{u}}_{\mathrm{loc}}(R,f)$ in a curve $\gamma$ which is not
contained in any radial direction of $W^{\mathrm{u}}_{\mathrm{loc}}(R,f)$
(i.e., a straight-line containing the origin). In this case, we say that the
curve $\gamma$ has a nontrivial radial projection.

\begin{overpic}[scale={0.09}]{Silla.jpg}
\scriptsize\put(112.0,168.0){\large{$L^{+}$}}
\put(79.0,167.0){\large{$L^{-}$}} \put(10.0,105.0){\large{$f^{j}(L)$}}
\put(150.0,150.0){\large{$S$}} \put(108.0,76.0){\Large{$R$}}
\put(150.0,81.0){\Large{$\gamma$}} \end{overpic} Figure 2. The discs
$L,L^{\pm}$ and $S$ and the curve $\gamma$

We need the following simple auxiliary lemma.

######  Lemma 4.1 (Accelerating angles).

Consider a diffeomorphism $f$, a saddle $R$, a disc $L$, a local surface $S$,
and a curve $\gamma\subset S\cap W^{\mathrm{u}}_{\mathrm{loc}}(R,f)$ as above.
Then there is $g$ arbitrarily $C^{r}$ close to $f$ such that
$W^{\mathrm{\ast}}_{\mathrm{loc}}(R,g)=W^{\mathrm{\ast}}_{\mathrm{loc}}(R,f),$
$\ast={\mathrm{s}},{\mathrm{u}}$, and

  * (a)

$W^{\mathrm{u}}_{\mathrm{loc}}(R,g)$ is simultaneously contained in the
closure of the sequences of discs $\big{(}g^{j}(L_{+})\big{)}$ and
$\big{(}g^{j}(L_{-})\big{)}$, $j\geqslant 1$, and

  * (b)

there are infinitely many $j_{\pm}\geqslant 1$ such that
$g^{j_{\pm}}(L_{\pm})$ meets transversely $S$ at points arbitrarily close to
$\gamma$.

###### Proof.

The result is obvious if the argument $\varphi$ of $Df(R)$ is irrational, in
that case we can take $g=f$. Otherwise, it is enough to consider a sequence
$(\alpha_{j})\to 0$ such that $\varphi+{\alpha_{j}}$ is irrational, rotations
$I^{x}_{\alpha_{j}}$ (defined on $U_{R}$) with argument $\alpha_{j}$ (recall
the definition in Section 3.3), and local perturbations $g=f_{j}$ of $f$ of
the form $f_{j}=I^{x}_{\alpha_{j}}\circ f$ in the set $U_{R}$. These
perturbations can be chosen supported on an small neighbourhood of the closure
of $U_{R}$. ∎

######  Remark 4.2.

There are the corresponding version of Lemma 4.1 for saddle foci with index
one.

######  Remark 4.3.

Changing the surface “$S$” by a one-dimensional disc and “transversality” by
“quasi-transversality”, the part (b) of Lemma 4.1 can be stated as follows:
there is arbitrarily large $j_{\pm}$ such that $g^{j_{\pm}}(L_{\pm})$ meets
quasi-transversely $S$ at some point arbitrarily close to $\gamma$.

######  Remark 4.4.

Assume that $f\in\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$ and that $g$ is obtained
perturbing $f$ using Lemma 4.1. Then $g$ can be taken such that
$g\in\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$ and
$\Gamma_{P,Q,X,Y}(g)=\Gamma_{P,Q,X,Y}(f)$, recall (4.1).

###  4.2. Proof of Proposition 4.1

We first consider hyperbolic tangencies, that is, we assume that
$f\in\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{h}}(M)$. Consider the points $Y$
and $\widetilde{Y}=f^{N_{2}}(Y)$ corresponding to the heterodimensional
tangency in condition (C) in Section 2.1.2 and their neighbourhoods $U_{Y}$
and $U_{\widetilde{Y}}$ in the definition of the transition map
$\mathfrak{T}_{2}$ in (2.5).

Using the notation in Definition 2.5, we can select small two-discs

$\mathcal{P}_{Y}^{\mathrm{s}}\subset W^{\mathrm{s}}(Q,f)\cap
U_{Y}\quad\mbox{and}\quad\mathcal{P}^{\mathrm{u}}_{\widetilde{Y}}\subset
W^{\mathrm{u}}(P,f)\cap U_{\widetilde{Y}}$

containing $Y$ and $\widetilde{Y}$ (respectively) in their interiors and
assume that $\mathcal{P}_{Y}^{\mathrm{s}}$ contains a pair of disjoint
surfaces $S_{+}$ and $S_{-}$ intersecting transversely
$W^{\mathrm{u}}_{\mathrm{loc}}(P,f)$ throughout curves $\gamma_{+}$ and
$\gamma_{-}$ with nontrivial radial projection, see Figure 3.

Consider now the quasi-transverse heteroclinic points $X$ and
$\widetilde{X}=f^{N_{1}}(X)$ in condition (B) in Section 2.1.2. Fix small
$\delta>0$ such that $B(X,\delta)$ is contained in $U_{X}$. Consider small
curves

$L^{\mathrm{u}}\subset U_{P}\cap
f^{N_{1}}\big{(}W_{{\mathrm{loc}}}^{\mathrm{u}}(Q,f)\cap
B(X,\delta)\big{)},\quad L^{\mathrm{s}}\subset W^{\mathrm{s}}(P,f)\cap U_{Q}$

containing $\widetilde{X}$ and $X$ in their interiors, respectively. See
Figure 3. As above, we let $L^{\mathrm{u}}_{\pm}$ the connected components of
$L^{\mathrm{u}}\setminus\\{\widetilde{X}\\}$.

Fix small $\varepsilon>0$. Applying item (b) of Lemma 4.1 to $P$, the surfaces
$S_{\pm}$, and the disc $L^{\mathrm{u}}$, we get a diffeomorphism
$\widetilde{f}_{\varepsilon}$ with
$d_{r}(f,\widetilde{f}_{\varepsilon})<\frac{\varepsilon}{2}$ and arbitrarily
large numbers $i_{+},i_{-}\geqslant 0$ such that
$\widetilde{f}_{\varepsilon}^{i_{\pm}}(L^{\mathrm{u}}_{\pm})$ transversely
intersects $S_{\pm}$ at some point $Z_{i_{\pm}}$. The points $Z_{i_{\pm}}$ can
be chosen converging to some point of $\gamma_{\pm}$. Item (3) of the
proposition follows taking
$Z_{\varepsilon}^{\pm}=\widetilde{f}_{\varepsilon}^{-(N_{1}+i_{\pm})}(Z_{i_{\ell}})$.
Note that, arguing as before, we can assume that the argument of the complex
eigenvalue of $D\widetilde{f}_{\varepsilon}(Q)$ is irrational.

Note that by Remark 4.4, the transitions of $\widetilde{f}_{\varepsilon}$ and
$f$ are the same, therefore
$\widetilde{f}_{\varepsilon}\in\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{h}}(M)$
and $\Gamma_{P,Q,X,Y}(\widetilde{f}_{\varepsilon})=\Gamma_{P,Q,X,Y}(f)$.
Moreover, we observe that all perturbations that we will perform in what
follows will keep this property.

To prove item (2), consider small disjoint closed subdiscs
$\widetilde{L}^{\mathrm{u}}_{\pm}=\widetilde{L}^{\mathrm{u}}_{\pm}(\varepsilon)\subset
L^{{\mathrm{u}}}_{\pm}$ as follows (see Figure 3): Let $\ell\in\\{+,-\\}$,
there are numbers $i_{\ell}=i_{\ell}(\varepsilon)$ with

  * •

$\widetilde{f}^{j}_{\varepsilon}(\widetilde{L}^{\mathrm{u}}_{\ell})\subset
U_{P}$ for all $j\in\\{0,\dots,i_{\ell}\\}$,

  * •

$\widetilde{f}^{j}_{\varepsilon}(\widetilde{L}^{\mathrm{u}}_{\ell})\cap
U_{Y}=\emptyset$ for all $j\in\\{0,\dots,i_{\ell}-1\\}$ and
$\widetilde{f}_{\varepsilon}^{i_{\ell}}(\widetilde{L}^{\mathrm{u}}_{\ell})\subset
U_{Y}$,

  * •

$\widetilde{f}_{\varepsilon}^{i_{\ell}}(\widetilde{L}^{\mathrm{u}}_{\ell})\pitchfork
S_{\ell}$ at the point ${Z}_{i_{\ell}}$,

  * •

the family

$\mathcal{L}^{\mathrm{u}}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{\\{}\widetilde{f}_{\varepsilon}^{j}(\widetilde{L}_{\ell}^{\mathrm{u}})\colon
j\in\\{0,\dots,i_{\ell}+N_{2}\\},\,\,\ell\in\\{+,-\\}\big{\\}}$

consists of pairwise disjoint sets,

  * •

$\widetilde{f}_{\varepsilon}^{i_{\ell}+N_{2}}(\widetilde{L}^{\mathrm{u}}_{\ell})\subset
U_{\widetilde{Y}}$, $\ell=+,-$.

By the definition of the transition $\mathfrak{T}_{2}$ and the choice of the
neighbourhoods $U_{Q},U_{Y},U_{\widetilde{Y}}$, the subfamily of
$\mathcal{L}^{\mathrm{u}}$ given by

$\mathcal{L}^{\mathrm{u}}_{0}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathcal{L}^{\mathrm{u}}\setminus\\{\widetilde{f}_{\varepsilon}^{i_{\ell}+N_{2}}(\widetilde{L}_{\ell}^{\mathrm{u}})\\}$

is disjoint from $U_{P}$.

\begin{overpic}[scale={0.053}]{bora.jpg}
\scriptsize\put(80.0,50.0){\Large{$P$}}
\put(83.0,126.0){\large{$\widetilde{X}$}}
\put(51.0,150.0){\large{$\widetilde{L}^{\mathrm{u}}_{+}$}}
\put(84.0,152.0){\large{$\widetilde{L}^{\mathrm{u}}_{-}$}}
\put(110.0,142.0){\large{$L^{\mathrm{u}}$}}
\put(114.0,58.0){\large{$\gamma_{+}$}} \put(165.0,70.0){\large{$\gamma_{-}$}}
\put(117.0,80.0){\large{$Z_{i_{+}}$}} \put(170.0,90.0){\large{$Z_{i_{-}}$}}
\put(110.0,110.0){\large{$S_{+}$}} \put(158.0,110.0){\large{$S_{-}$}}
\put(255.0,50.0){\large{$Q$}} \put(273.0,142.0){\large{$L^{\mathrm{s}}$}}
\put(219.0,150.0){\large{${L}^{\mathrm{s}}_{+}$}}
\put(253.0,152.0){\large{${L}^{\mathrm{s}}_{-}$}}
\put(249.0,128.0){\large{${X}$}} \put(293.0,45.0){\large{$\widehat{Z}_{+}$}}
\par\end{overpic} Figure 3. The discs $L^{\mathrm{u}}_{\pm}$ in
$W^{\mathrm{u}}(Q,f)$ and $L^{\mathrm{s}}_{\pm}$ in $W^{\mathrm{s}}(P,f)$.

To prove item (2) of the proposition, consider the homoclinic points of $Q$

$\widehat{Z}_{\ell}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\widetilde{f}_{\varepsilon}^{N_{2}}(Z_{i_{\ell}})\in
W^{\mathrm{s}}_{\mathrm{loc}}(Q,\widetilde{f}_{\varepsilon})\pitchfork
W^{\mathrm{u}}(Q,\widetilde{f}_{\varepsilon}),\quad\ell=+,-$

and take arbitrarily small

(4.2) $0<\rho<\frac{\varepsilon}{2(\|f\|_{r}+\varepsilon)}$

such that

$\begin{split}&B(\,\widetilde{Y},2\rho\,)\cup B(\,\widehat{Z}_{+},2\rho\,)\cup
B(\,\widehat{Z}_{-},2\rho\,)\subset U_{\widetilde{Y}}\\\
&\mbox{$B(\,\widehat{Z}_{+},2\rho\,),\,B(\,\widehat{Z}_{-},2\rho\,),\,B(\,\widetilde{Y},2\rho\,)$
are pairwise disjoint.}\end{split}$

Arguing as above, considering $\widetilde{f}_{\varepsilon}^{-1}$ and applying
item (a) of Lemma 4.1 to $Q$, the disc $L^{\mathrm{s}}$, and the points
$\widehat{Z}_{\pm}$ we get disjoint closed subdiscs
$L^{\mathrm{s}}_{\pm}=L^{\mathrm{s}}_{\pm}(\rho)$ of $L^{\mathrm{s}}$
satisfying the following conditions (see Figure 3): Let $K$ be the $C^{r}$
norm of the map $b_{\rho}$ in (3.1), there are numbers $k_{\pm}(\rho)=k_{\pm}$
such that:

  * •

the family of sets

$\\{\widetilde{f}^{-i}_{\varepsilon}(L^{\mathrm{s}}_{+})\colon
i=0,\dots,k_{+}\\}\cup\\{\widetilde{f}_{\varepsilon}^{-j}(L^{\mathrm{s}}_{-})\colon
j=0,\dots,k_{-}\\}$

is pairwise disjoint,

  * •

$\widetilde{f}^{-i}_{\varepsilon}(L^{\mathrm{s}}_{\pm})\subset\Big{(}U_{Q}\setminus
B\big{(}\,\widehat{Z}_{\pm},\frac{\rho^{r+1}}{K^{3}}\,\big{)}\Big{)}$ for
every $i\in\\{0,\dots,k_{\pm}-1\\}$,

  * •

$\widetilde{f}_{\varepsilon}^{-k_{\pm}}(L^{\mathrm{s}}_{\pm})\subset
B\big{(}\,\widehat{Z}_{\pm},\frac{\rho^{r+1}}{K^{3}}\,\big{)}$.

Let $X_{\pm}$ be the closest point of $\widehat{Z}_{\pm}$ in
$\widetilde{f}_{\varepsilon}^{-k_{\pm}}(L^{\mathrm{s}}_{\pm})$ and define the
vector

(4.3) $\mathbf{w}_{\pm}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}X_{\pm}-\widehat{Z}_{\pm},\qquad\|\mathbf{w}_{\pm}\|\leqslant\frac{\rho^{r+1}}{K^{3}}.$

Using the function $\Pi_{\rho}$ in (3.2), consider the perturbation of the
identity given by

$\vartheta_{\pm,\rho}(\widehat{Z}_{\pm}+W)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\widehat{Z}_{\pm}+W+\Pi_{\rho}(W)\,\mathbf{w}_{\pm},\quad\mbox{if}\quad\widehat{Z}_{\pm}+W\in
B(\widehat{Z}_{\pm},2\rho)$

and the identity otherwise. Since
$\|\mathbf{w}_{\pm}\|\leqslant\frac{\rho^{r+1}}{K^{3}}$ and
$\|\Pi^{\theta}_{\rho}\|_{r}\leqslant K^{3}\,\rho^{-r}$ (recall (3.3)), it
holds

$\|\vartheta_{\pm,\rho}-\mathrm{id}\|_{r}\leqslant\|\Pi_{\rho}\|_{r}\cdot\|\mathbf{w}_{\pm}\|<\rho.$

Finally, consider the perturbation of $\widetilde{f}_{\varepsilon}$ defined by

$f_{\varepsilon}=f_{\varepsilon,\rho}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\vartheta_{\pm,\rho}\circ\widetilde{f}_{\varepsilon}.$

Recalling the choice of $\rho$ in (4.2), we get

$d(f,f_{\varepsilon})_{r}\leqslant
d(f,\widetilde{f}_{\varepsilon})_{r}+d(\widetilde{f}_{\varepsilon},f_{\varepsilon})_{r}\leqslant\frac{\varepsilon}{2}+\rho\|\widetilde{f}_{\varepsilon}\|_{r}<\frac{\varepsilon}{2}+\rho(\|f\|_{r}+\varepsilon)<\varepsilon.$

By construction, $f_{\varepsilon,\rho}$ coincides with
$\widetilde{f}_{\varepsilon}$ outside
$\widetilde{f}^{-1}_{\varepsilon}\big{(}B(\widehat{Z}_{+},2\rho)\cup
B(\widehat{Z}_{-},2\rho)\big{)}$ and by (4.3) we have that the points $X$,
${X}_{1,\varepsilon}$, and ${X}_{2,\varepsilon}$ with

${X}_{1,\varepsilon}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}f_{\varepsilon}^{k_{+}}(X_{+}),\quad{X}_{2,\varepsilon}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}f_{\varepsilon}^{k_{-}}({X}_{-})\in
f_{\varepsilon,\rho}^{-N_{1}}\big{(}W^{\mathrm{s}}_{{\mathrm{loc}}}(P,f_{\varepsilon,\rho})\big{)}\cap
W^{\mathrm{u}}(Q,f_{\varepsilon,\rho})$

are quasi-transverse heteroclinic points of $f_{\varepsilon,\rho}$ with
different orbits.

Thus, $f_{\varepsilon,\rho}\in\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{h}}(M)$
and satisfies items (1)-(3) in the proposition.

We now study the elliptic case when
$f\in\mathcal{H}^{r}_{{\mathrm{BH}},\mathrm{e}^{+}}(M)$. We apply the
variation of Lemma 4.1 in Remark 4.3 and observe that an arbitrarily small
modification of the angle provides two transverse intersections (for the same
iterate). The rest of the proof is identical to the hyperbolic case.

Finally, note that in our construction the transitions are preserved, thus if
$f\in\mathcal{H}^{r}_{{\mathrm{BH}},\ast}(M)$ then
$f_{\varepsilon,\rho}\in\mathcal{H}^{r}_{{\mathrm{BH}},\ast}(M)$,
$\ast=\mathrm{h},\mathrm{e}^{+}$. The proof of the proposition is now
complete. ∎

###  4.3. Transitions for the new heteroclinic points

Given $f\in\mathcal{H}_{{\mathrm{BH}},\ast}^{r}(M)$,
$\ast=\mathrm{h},\mathrm{e}^{+}$, and small $\varepsilon>0$, consider its
perturbation $f_{\varepsilon}\in\mathcal{H}_{{\mathrm{BH}},\ast}^{r}(M)$ and
the heteroclinic points $X_{1,\varepsilon}$ and $X_{2,\varepsilon}$ given by
Proposition 4.1. Take disjoint neighbourhoods $U_{1,\varepsilon}$ and
$U_{2,\varepsilon}$ of $X_{1,\varepsilon}$ and $X_{2,\varepsilon}$ contained
in $U_{X}$ where the transition map $\mathfrak{T}_{1}$ in (2.3) is defined. By
shrinking these neighbourhoods, we can take a small neighbourhood
$U_{0,\varepsilon}\subset U_{X}$ of $X_{0,\varepsilon}=X$ disjoint from
$U_{1,\varepsilon}$ and $U_{2,\varepsilon}$. We write

(4.4) $X_{i,\varepsilon}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}X+Z_{i,\varepsilon},\quad
Z_{i,\varepsilon}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(x_{i,\varepsilon},y_{i,\varepsilon},z_{i,\varepsilon}),\quad
i=0,1,2.$

Note that $Z_{1,\varepsilon},Z_{2,\varepsilon}\to\mathbf{0}$ as
$\varepsilon\to 0$.

Denote by $\mathfrak{T}_{1,i,\varepsilon}$ the restriction of
$\mathfrak{T}_{1}$ to $U_{i,\varepsilon}$,

(4.5) $\mathfrak{T}_{1,i,\varepsilon}:U_{i,\varepsilon}\to
U_{\widetilde{X}},\quad\mathfrak{T}_{1,i,\varepsilon}(Z)=f^{N_{1}}_{\varepsilon}(Z)=f^{N_{1}}(Z),\quad
i=0,1,2.$

Hence, recalling (2.3),

$\widetilde{X}_{i,\varepsilon}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathfrak{T}_{1,i,\varepsilon}({X}_{i,\varepsilon})=\widetilde{X}+A(Z_{i,\varepsilon})+\widetilde{H}(Z_{i,\varepsilon}),$

Using equation (4.4) and that ${X}_{i,\varepsilon}\in
f_{\varepsilon}^{-N_{1}}\big{(}W^{\mathrm{s}}_{{\mathrm{loc}}}(P,f_{\varepsilon})\big{)}\cap
U_{X}$ we get

(4.6)
$\widetilde{X}_{i,\varepsilon}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\widetilde{X}+\widetilde{Z}_{i,\varepsilon}\in
W^{\mathrm{s}}_{\mathrm{loc}}(P,f_{\varepsilon}),\quad\mbox{where}\quad\widetilde{Z}_{i,\varepsilon}=A(Z_{i,\varepsilon})+\widetilde{H}(Z_{i,\varepsilon}).$

Note that $\widetilde{X},\widetilde{X}_{i,\varepsilon}\in
W^{\mathrm{s}}_{\mathrm{loc}}(P,f_{\varepsilon})$ and that (in local
coordinates) $W^{\mathrm{s}}_{\mathrm{loc}}(P,f_{\varepsilon})$ is contained
in $\\{(x,0,0)\\}\subset U_{P}$. Hence

(4.7) $\widetilde{Z}_{i,\varepsilon}=(\tilde{x}_{i,\varepsilon},0,0).$

Then the map $\mathfrak{T}_{1,i,\varepsilon}$ can be written as follows: for
$X_{i,\varepsilon}+W\in U_{i,\varepsilon}$ we have

(4.8)
$\mathfrak{T}_{1,i,\varepsilon}(X_{i,\varepsilon}+W)=\widetilde{X}_{i,\varepsilon}+A(W)+\widetilde{H}_{\varepsilon}^{i}(W),$

where $\widetilde{H}^{i}_{\varepsilon}\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$
is defined by

(4.9)
$\widetilde{H}^{i}_{\varepsilon}(W)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\widetilde{H}(Z_{i,\varepsilon}+W)-\widetilde{H}(Z_{i,\varepsilon}).$

######  Remark 4.5.

Let
$\widetilde{H}^{i}_{\varepsilon}=(\widetilde{H}^{i}_{\varepsilon,1},\widetilde{H}^{i}_{\varepsilon,2},\widetilde{H}^{i}_{\varepsilon,3})$.
Then

$\widetilde{H}^{i}_{\varepsilon,1}(\textbf{0})=\widetilde{H}^{i}_{\varepsilon,2}(\textbf{0})=\widetilde{H}^{i}_{\varepsilon,3}(\textbf{0})=0.$

Although the maps $\widetilde{H}^{i}_{\varepsilon,j}$ do not satisfy the same
“flat conditions” at $\mathbf{0}$ satisfied by the terms $\widetilde{H}_{i}$
of $\mathfrak{T}_{1}$, see (2.3), the following convergence property holds:

$\frac{\partial}{\partial x}\widetilde{H}^{i}_{\varepsilon,k}(\textbf{0})\to
0,\quad\frac{\partial}{\partial
y}\widetilde{H}^{i}_{\varepsilon,k}(\textbf{0})\to
0,\quad\frac{\partial}{\partial
z}\widetilde{H}^{i}_{\varepsilon,k}(\textbf{0})\to 0,\quad\varepsilon\to
0,\quad k=1,2,3.$

###  4.4. Parameterisations of unstable discs throughout the points
$X_{i,\varepsilon}$

Take an unitary vector $\mathbf{v}_{i,\varepsilon}\in
T_{X_{i,\varepsilon}}W^{\mathrm{u}}(Q,f_{\varepsilon})$. For small $\delta>0$,
consider the parameterised segment of the local unstable manifold of $Q$
containing $X_{i,\varepsilon}$ in $U_{i,\varepsilon}$ obtained considering its
Taylor expansion,

(4.10)
$L_{i,\varepsilon}^{\mathrm{u}}(\delta)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{\\{}X_{i,\varepsilon}+t\,\mathbf{v}_{i,\varepsilon}+\widetilde{\rho}_{i,\varepsilon}(t):|t|<\delta\big{\\}}\subset
W^{\mathrm{u}}(Q,f_{\varepsilon}),$

here $\widetilde{\rho}_{i,\varepsilon}$ is an $C^{r}$ map satisfying

(4.11)
$\widetilde{\rho}_{i,\varepsilon}(0)=\frac{d}{dt}\widetilde{\rho}_{i,\varepsilon}(0)=\textbf{0}.$

######  Remark 4.6.

By the $\lambda$-lemma we have that
$\mathbf{v}_{i,\varepsilon}\to\textbf{e}_{2}=(0,1,0)$ and
$\|\widetilde{\rho}_{i,\varepsilon}\|_{r}\to 0$ as $\varepsilon\to 0$. By a
$C^{r}$ perturbation of $f_{\varepsilon}$, we can assume that
$\mathbf{v}_{i,\varepsilon}=\textbf{e}_{2}$. To see this, let
$\pi_{i,\varepsilon}$ be the plane generated by $\mathbf{v}_{i,\varepsilon}$
and $\textbf{e}_{2}$ in $T_{X_{i,\varepsilon}}M$ and $\alpha_{i,\varepsilon}$
the smallest angle (modulus $2\pi$) of the rotation map taking
$\mathbf{v}_{i,\varepsilon}$ into $\textbf{e}_{2}$. Note that
$\alpha_{i,\varepsilon}\to 0$ as $\varepsilon\to 0$. Performing a rotation-
like $C^{r}$ perturbation as in Section 3.3 at $X_{i,\varepsilon}$ around of
the orthogonal direction to $\Pi_{i,\varepsilon}$, we get a diffeomorphism
$O(\alpha_{i,\varepsilon})$ $C^{r}$ close to $f_{\varepsilon}$ (that we
continue to call $f_{\varepsilon}$) such that

(4.12)
$L_{i,\varepsilon}^{\mathrm{u}}(\delta)=\big{\\{}X_{i,\varepsilon}+t\,\textbf{e}_{2}+\widetilde{\rho}_{i,\varepsilon}(t):|t|<\delta\big{\\}}\subset
W^{\mathrm{u}}(Q,f_{\varepsilon}),\quad i=1,2.$

With a slight abuse of notation, the higher order terms
$\widetilde{\rho}_{i,\varepsilon}$ in (4.12) are denoted as the ones in
(4.10). As the latter are obtained as “small rotations” of the terms in
(4.10), they satisfy the flat conditions in (4.11).

##  5\. Blender-horseshoes and center-unstable Hénon-like families

In this section, we introduce blender-horseshoes and their main properties
(Section 5.1) and explain how they may lead to robust tangencies (Section
5.2). We also state their occurrence in center-unstable Hénon-like families
(Section 5.3). Finally, we study the geometry of the unstable manifolds of
these blenders (Lemma 5.5). All blenders considered in this paper are blender-
horseshoes, thus if there is no misunderstanding in some cases we will refer
to them simply as blenders.

###  5.1. Blender-horseshoes

We refrain to give a precise definition of a blender-horseshoe (for details
see [8, 13]), instead we will focus on their relevant properties. We also
restrict our discussion to our three dimensional context. A blender-horseshoe
is a locally maximal hyperbolic set $\Gamma_{f}$ of a diffeomorphism $f:M\to
M$ that is conjugate to the complete full shift on two symbols and satisfies a
geometrical condition stated in Lemmas 5.1 and 5.2. We now go to the details.

There is an open neighbourhood $\Delta$ of $\Gamma$ such that

$\Gamma_{f}=\bigcap_{i\in\mathbb{Z}}f^{i}(\,\overline{\Delta}\,)\subset\Delta$

The set $\Gamma_{f}$ is also partially hyperbolic: there is a dominated
splitting with one-dimensional bundles $E^{\mathrm{s}}\oplus
E^{\mathrm{cu}}\oplus E^{{\mathrm{u}}{\mathrm{u}}}$ of $T_{\Gamma_{f}}M$ such
that $E^{\mathrm{u}}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}E^{\mathrm{cu}}\oplus E^{{\mathrm{u}}{\mathrm{u}}}$ and
$E^{\mathrm{s}}$ are the unstable and stable bundles of $\Gamma$,
respectively. The bundle $E^{{\mathrm{u}}{\mathrm{u}}}$ is the strong unstable
direction. We consider a $Df$-invariant cone fields
$\mathcal{C}^{{\mathrm{u}}{\mathrm{u}}}$ and $C^{\mathrm{u}}$ around
$E^{{\mathrm{u}}{\mathrm{u}}}$ and $E^{\mathrm{u}}$, a $Df^{-1}$-invariant
cone field $\mathcal{C}^{{\mathrm{s}}{\mathrm{s}}}$ around
$E^{{\mathrm{ss}}}$, and a center unstable cone field $C^{\mathrm{cu}}$ around
$E^{\mathrm{cu}}$. The latter is not $Df$ invariant, but the norm of the
vectors in $C^{\mathrm{cu}}$ are uniformly expanded by $Df$.

As a hyperbolic set, the blender-horseshoe $\Gamma_{f}$ has a continuation
$\Gamma_{g}$ for every $g$ sufficiently close. The important fact is that for
diffeomorphisms nearby these continuations are also blender-horseshoes.
Blender-horseshoes can be also defined for endomorphisms, see [17, Definition
2.7], with the following reformulation of the continuation property: every map
(diffeomorphism or endomorphism) close to an endomorphism with a blender-
horseshoe also has a blender-horseshoe with the same reference domain.

A key ingredient of a blender-horseshoe is its superposition region. To
describe it define the local stable manifold of $\Gamma_{f}$ by

(5.1)
$W^{\mathrm{s}}_{\mathrm{loc}}(\Gamma_{f},f)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{\\{}\,x\in
M\,\colon\,f^{i}(x)\in\overline{\Delta}\,\,\,\,\mbox{for every $\,\,i\geqslant
0$}\,\big{\\}}$

and observe that the set $\Gamma_{f}$ has two fixed points $P^{+}$ and
$P^{-}$, called the _reference fixed points of the blender_. One defines
“large” one-dimensional ${\mathrm{uu}}$-discs (contained in $U$) tangent to
$\mathcal{C}^{{\mathrm{u}}{\mathrm{u}}}$ at the right and at the left of
$W^{\mathrm{s}}_{\mathrm{loc}}(P^{\pm},f)$. The set of such discs at the right
of $W^{\mathrm{s}}_{\mathrm{loc}}(P^{-},f)$ and at the left of
$W^{\mathrm{s}}_{\mathrm{loc}}(P^{+},f)$ form the superposition region denoted
by $\mathcal{D}_{f}$. We say that these ${\mathrm{uu}}$-discs are in-between.

The next two lemmas (see for instance [5, Lemma 3.13] and [17, Lemma 2.5])
state an intersection property for discs in the superposition region of the
blender that will play a key role to get robust heterodimensional cycles.

######  Lemma 5.1 (The superposition region).

Let $\Gamma_{f}$ be a blender-horseshoe of $f$ and $D$ compact disc whose
interior contains a disc in the superposition region of $\Gamma_{f}$. Then for
every $C^{1}$-neighbourhood $\mathcal{U}$ of $D$ there is a
$C^{1}$-neighbourhood666Let $r>1$ and $f\in\mathrm{Diff}^{r}(M)$. Any
$C^{1}$-neighbourhood of $f$ contains a $C^{r}$ neighbourhood of $f$.
$\mathcal{V}$ of $f$ such that every compact disc in $\mathcal{U}$ contains a
disc in the superposition region of $\Gamma_{g}$ for every $g\in\mathcal{V}$.

For further explanation of this lemma see Remark 5.4. See also Lemma 5.4 for a
“quantitative version” of this result.

######  Lemma 5.2.

Let $\Gamma_{f}$ be a blender-horseshoe of $f$ and $D$ a compact disc
containing a disc the superposition region of $\Gamma_{f}$. Then
$W^{\mathrm{s}}_{\mathrm{loc}}(\Gamma_{f},f)\cap D\neq\emptyset$ and this
intersection is quasi-transverse.

This lemma follows from the fact that the image of any disc $D$ in the
superposition region contains a disc in the superposition region. Arguing
inductively, it follows that any disc in the superposition region contains a
point whose forward orbit is contained in $\Delta$. The lemma now follows from
the characterisation of $W^{\mathrm{s}}_{\mathrm{loc}}(\Gamma_{f},f)$ in
(5.1). This construction also guarantees that the obtained intersection
between $D$ and $W^{\mathrm{s}}_{\mathrm{loc}}(\Gamma_{f},f)$ is quasi-
transverse: the disks in the superposition region are tangent to a strong
unstable cone field, $W^{\mathrm{s}}_{\mathrm{loc}}(\Gamma_{f},f)$ is tangent
to a stable cone field, and these cone fields have no common directions.

###  5.2. Blender-horseshoes: tubes and folding manifolds

We now analise when the local stable manifold
$W^{{\mathrm{s}}}_{\mathrm{loc}}(\Gamma_{f},f)$ of a blender-horseshoe
$\Gamma_{f}$ has a (robust) tangency with a surface $S$ “passing throughout
its domain $\Delta$”. In [8] it is proved that occurrence when $S$ is a
folding manifold. Motivated by this fact, we introduce the notion of
${\mathrm{u}}$-tubes and prove that they generate folding manifolds after
iterations. In this way, Proposition 5.2 provides a mechanism guaranteeing the
robust tangencies in Theorem 1.1. We now go into the details of this
construction that follows the ideas in [8, Section 4].

####  5.2.1. Folding manifolds and tubes

Throughout this section, we consider a diffeomorphism $f$ with a blender-
horseshoe $\Gamma_{f}$ with domain $\Delta$ and reference fixed points $P^{-}$
and $P^{+}$. Next definition is an extension of [8, Definition 4.2].

######  Definition 5.1 (Strips, tubes, and folding manifolds).

Consider a surface with boundary $S$ of the form

$S=\bigcup_{t\in[0,1]}D_{t}\subset\Delta,$

where $(D_{t})_{t\in[0,1]}$ is a family of ${\mathrm{uu}}$-discs depending
continuously on $t$. We say that $S$ is a

  * •

${\mathrm{u}}$-strip if the family of discs $(D_{t})_{t\in[0,1]}$ is pairwise
disjoint and $S$ is tangent to the unstable cone field
$\mathrm{C}^{\mathrm{u}}$,

  * •

${\mathrm{u}}{\mathrm{u}}$-tube if the family of discs $(D_{t})_{t\in[0,1)}$
is pairwise disjoint and $D_{0}=D_{1}$.

  * •

a folding surface if the family of discs $(D_{t})_{t\in[0,1]}$ is pairwise
disjoint, $D_{0}$ and $D_{1}$ both intersect
$W^{{\mathrm{s}}}_{{\mathrm{loc}}}(P^{-},f)$ (or both intersect
$W^{{\mathrm{s}}}_{{\mathrm{loc}}}(P^{+},f)$), and $D_{t}$ is in-between
$P^{-}$ and $P^{+}$ for every $t\in(0,1)$.

A strip or a tube $S$ is in-between if every $D_{t}$ is in-between $P^{+}$ and
$P^{-}$. Note that a folding surface cannot be in-between.

In what follows, we will use the letters $S$, $T$, and $F$ to denote strips,
tubes, and folding manifolds, respectively. The main result of this section is
the following:

######  Proposition 5.2.

Let $\Gamma_{f}$ be a blender-horseshoe of a diffeomorphism $f$ and $T$ a
${\mathrm{uu}}$-tube in-between. Then
$W^{{\mathrm{s}}}_{{\mathrm{loc}}}(\Gamma_{f},f)$ and $T$ have a tangency
point.

An immediate consequence of this proposition is the following version of [8,
Corollary 4.11] where folding manifolds are replaced by ${\mathrm{uu}}$-tubes:

######  Corollary 5.1.

Let $\Gamma_{f}$ be a blender-horseshoe of $f$ and $T$ a ${\mathrm{uu}}$-tube
in-between contained in the unstable manifold of a saddle $R_{f}$ of index
two. Then there is a $C^{r}$ neighbourhood $\mathcal{V}_{f}$ of $f$ such that
$W^{{\mathrm{s}}}_{{\mathrm{loc}}}(\Gamma_{g},g)$ and
$W^{{\mathrm{u}}}(R_{g},g)$ have a tangency point for every
$g\in\mathcal{V}_{f}$.

Proposition 5.2 will follow from [8, Proposition 4.4]: any folding manifold
has a tangency with $W^{{\mathrm{s}}}_{{\mathrm{loc}}}(\Gamma_{f},f)$. The
main ingredient of the proof in [8] is the fact that the image of any folding
manifold contains a folding manifold, see [8, Lemma 4.5]. We reformulate that
lemma in our context. For that we need to analise the central width of
iterations of ${\mathrm{u}}$-tubes and ${\mathrm{u}}$-strips. We refer to see
[13, Section 2] for a detailed analysis of iterations of strips. We now go
into the details.

First, a central curve is a curve tangent to the center unstable cone field
$\mathcal{C}^{\mathrm{cu}}$. The central width of a ${\mathrm{u}}$-strip
$S=\cup_{t\in[0,1]}D_{t}$ is defined by

$\mathrm{w}(S)=\inf\\{\mathrm{length}(\ell)\colon\mbox{$\ell\subset S$ is a
central curve joining $D_{0}$ and $D_{1}$}\\}.$

Note that there is $\kappa=\kappa(\Gamma_{f})>0$ such that
$\mathrm{w}(S){\leqslant}\kappa$ for every ${\mathrm{u}}$-strip in-between.

To define the central width of a ${\mathrm{uu}}$-tube $T$ in-between (denoted
with a slight abuse of notation also by $\mathrm{w}(T)$), we note that
$(\Delta\setminus T)$ has two connected components, one of them is disjoint
from $P^{-}$ and $P^{+}$. We denote this component by $\Delta_{T}$ and let

$\mathrm{w}(T)=\sup\\{\mathrm{w}(S)\colon\mbox{$S$ is a ${\mathrm{u}}$-strip
contained in $\Delta_{T}$}\\}.$

Note that the width of any ${\mathrm{uu}}$-tube in-between is bounded by the
constant $\kappa$ above.

######  Lemma 5.3.

Let $\Gamma_{f}$ be a blender-horseshoe. Then there is $\lambda>1$ such that
for every ${\mathrm{uu}}$-tube $T$ in-between one of the following
possibilities holds true:

  * (a)

$f(T)$ has tangency with either $W^{{\mathrm{s}}}_{\mathrm{loc}}(P^{-},f)$ or
with $W^{{\mathrm{s}}}_{\mathrm{loc}}(P^{+},f)$,

  * (b)

$f(T)$ contains a folding manifold,

  * (c)

$f(T)$ contains a ${\mathrm{uu}}$-tube $T^{\prime}$ in-between with
$\mathrm{w}(T^{\prime}){\geqslant}\lambda\,\mathrm{w}(T)$.

Proposition 5.2 easily follows from this lemma. Observe that, by the comments
above, in cases (a) and (b) we get a tangency between $T$ and
$W^{{\mathrm{s}}}_{{\mathrm{loc}}}(\Gamma_{f},f)$ and we are done. Otherwise,
we let $T_{0}=T$ and get a new tube $T_{1}$ in-between such that $T_{1}\subset
f(T_{0})$ and $\mathrm{w}(T_{1}){\geqslant}\lambda\,\mathrm{w}(T_{0})$. We can
now apply Lemma 5.3 to $T_{1}$ and argue recurrently. But case (c) cannot
occur infinitely many consecutive times: in such a case we get a sequence of
tubes $(T_{n})$ in-between with

$T_{n}\subset
f(T_{n-1})\quad\mbox{and}\quad\mathrm{w}(T_{n}){\geqslant}\lambda\,\mathrm{w}(T_{n-1}){\geqslant}\lambda^{n}\,\mathrm{w}(T_{0}).$

Since the widths of the tubes $T_{n}$ are bounded by $\kappa$ there is a first
step of the recurrent construction when we fall in cases (a) or (b). We now
prove the lemma.

####  5.2.2. Proof of Lemma 5.3

We have the following version of Lemma 5.3 for ${\mathrm{u}}$-strips imported
from [8, Lemma 4.5] (see also the proof of [13, Proposition 2.3]).

######  Lemma 5.4.

Consider a blender-horseshoe $\Gamma_{f}$ of $f$. Then there is $\lambda>1$
such that for every ${\mathrm{u}}$-strip $S$ in-between then one of the two
possibilities holds:

  * (i)

$f(S)$ intersects transversely either
$W^{{\mathrm{s}}}_{\mathrm{loc}}(P^{-},f)$ or
$W^{{\mathrm{s}}}_{\mathrm{loc}}(P^{+},f)$,

  * (ii)

$f(S)$ contains a ${\mathrm{u}}$-strip $S^{\prime}$ in-between with
$\mathrm{w}(S^{\prime}){\geqslant}\lambda\,\mathrm{w}(S)$.

The main ingredient of the proof of this lemma is in Remark 5.4.

######  Remark 5.3 (Iterations of ${\mathrm{u}}$-strips in-between).

Lemma 5.4 implies that the orbit of any ${\mathrm{u}}$-strip in-between
transversely meets either $W^{s}_{{\mathrm{loc}}}(P^{-},f)$ or
$W^{s}_{{\mathrm{loc}}}(P^{+},f)$. More precisely, let $S_{0}=S$ and assume
that $f(S_{0})$ intersects transversely neither
$W^{{\mathrm{s}}}_{\mathrm{loc}}(P^{-},f)$ nor
$W^{{\mathrm{s}}}_{\mathrm{loc}}(P^{+},f)$. In that case, we get a new strip
$S_{1}$ in-between such that $S_{1}\subset f(S_{0})$ and
$\mathrm{w}(S_{1}){\geqslant}\lambda\,\mathrm{w}(S_{0})$. We can now apply
Lemma 5.4 to $S_{1}$ and argue recurrently. As above, this possibility cannot
occurs infinitely many consecutive times.

We are now ready to prove the lemma. Given any $\varepsilon>0$, associated to
the ${\mathrm{uu}}$-tube $T_{0}=T$ there is an internal strip
$S_{0}\subset\Delta_{T_{0}}$, with
$\mathrm{w}(S_{0}){\geqslant}\mathrm{w}(T_{0})-\varepsilon$. Note that this
strip is in-between. Consider now $f(T_{0})$ and assume that cases (a) and (b)
in Lemma 5.3 do not hold. This implies that the image of $f(S_{0})$ satisfies
(ii) in Lemma 5.4. We now see that and in that case there is
${\mathrm{u}}$-tube $T_{1}\subset f(T_{0})$ such that
$S_{1}\subset\Delta_{T_{1}}$. Since this holds for every $\varepsilon>0$ item
(c) in the lemma follows.

We now explain how the tube $T_{1}$ is obtained. We will refer to [13, Section
2] for details. A blender-horseshoe has a Markov partition associated to two
disjoint “subrectangles”
$\Delta_{\mathbb{A}},\Delta_{\mathbb{B}}\subset\Delta$ such that

$\Gamma_{f}=\bigcap_{i\in\mathbb{Z}}f^{i}(\Delta_{\mathbb{A}}\cup\Delta_{\mathbb{B}}).$

Moreover, the map that associates to each $x\in\Gamma_{f}$ the sequence
$(\xi_{i}(x))_{\in\mathbb{Z}}\in\\{\mathbb{A},\mathbb{B}\\}^{\mathbb{Z}}$
defined by $f^{i}(x)\in\Delta_{\xi_{i}(x)}$ is a conjugation between
$\Gamma_{f}$ and the complete shift on the symbols $\mathbb{A},\,\mathbb{B}$.
In particular, the reference fixed points of the blender satisfy
$P^{-}\in\Delta_{\mathbb{A}}$ and $P^{+}\in\Delta_{\mathbb{B}}$. In what
follows, we denote by $f_{\mathbb{E}}$ the restriction of $f$ to
${\mathbb{E}}$, $\mathbb{E}\in\\{\mathbb{A},\mathbb{B}\\}$. Given a set $X$ we
let $X_{\mathbb{E}}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}X\cap\Delta_{\mathbb{E}}$.

Next remark explains the mechanism guaranteeing Lemma 5.1 and is a consequence
of the definition of a blender-horseshoe.

######  Remark 5.4 (Iterations of ${\mathrm{uu}}$-discs ans
${\mathrm{u}}$-strip).

There is $\lambda>1$ with the following property: Consider a
${\mathrm{uu}}$-disc $D$ and a ${\mathrm{u}}$-strip $S$ in-between.

  * •

Then either $f(D_{\mathbb{A}})$ or $f(D_{\mathbb{B}})$ contains a
${\mathrm{uu}}$-disc in-between.

  * •

Suppose that $f(S_{\mathbb{E}})$ is a ${\mathrm{u}}$-strip in-between, then
$\mathrm{w}f(S_{\mathbb{E}}){\geqslant}\lambda\mathrm{w}(S_{\mathbb{E}})$.

By Remark 5.4, for a given $t\in[0,1]$ either $f(D_{t,\mathbb{A}})$ or
$f(D_{t,\mathbb{B}})$ is a ${\mathrm{uu}}$-disc in-between. We have the
following three cases:

  * (A)

$f(D_{t,\mathbb{A}})$ is a ${\mathrm{uu}}$-disc in-between for every
$t\in[0,1]$,

  * (B)

$f(D_{t,\mathbb{B}})$ is a ${\mathrm{uu}}$-disv in-between for every
$t\in[0,1]$,

  * (C)

Cases (A) and (B) do not hold.

In case (A), by Remark 5.4 we have that $T_{1}=f(T_{t,\mathbb{A}})$ is a
${\mathrm{uu}}$-tube such that
$\mathrm{w}(T_{1}){\geqslant}\lambda\mathrm{w}(T)$. Analogously, in case (B)
$T_{1}=f(T_{t,\mathbb{B}})$ is a ${\mathrm{uu}}$-tube satisfying
$\mathrm{w}(T_{1}){\geqslant}\lambda\mathrm{w}(T)$. In both cases, item (3) in
the lemma holds. Thus it remains to consider case (iii).

In case (C), there is $c\in[0,1)$ such that $f(D_{c,\mathbb{A}})$ is a
${\mathrm{uu}}$-disc in-between but $f(D_{c,\mathbb{B}})$ is not a
${\mathrm{uu}}$-disc in-between (this possibility includes the case where
$f(D_{c,\mathbb{B}})$ is not a ${\mathrm{uu}}$-disc). After changing the
parameterisation of the tube we can take $c=0$. Recall that $D_{0}=D_{1}$. Let

$\begin{split}t_{1}&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\sup\\{t\in[0,1]\colon\mbox{$f(D_{s,\mathbb{A}})$ is in-between for
all $s\in[0,t]$}\\},\\\ t_{2}&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\inf\\{t\in[0,1]\colon\mbox{$f(D_{s,\mathbb{A}})$ is in-between for
all $s\in[t,1]$}\\}.\end{split}$

Note that $t_{1}=1$ if and only if $t_{2}=0$. Moreover, if this does not hold
then $t_{1}{\leqslant}t_{2}$. Thus, a priory, there are the following cases,
(i) $t_{1}=1$ and $t_{2}=0$, (ii) $t_{1}=t_{2}\in(0,1)$, and (iii)
$0<t_{1}<t_{2}<1$. Case (i) implies that we are in case (A) above, a
contradiction. Thus it can be discarded.

Note that, by continuity, if $t_{1}<1$ then $f(D_{t_{1},\mathbb{A}})\cap
W^{s}_{\mathrm{loc}}(P^{-},f)\neq\emptyset$. Similarly, if $t_{2}>0$ then
$f(D_{t_{2},\mathbb{A}})\cap W^{s}_{\mathrm{loc}}(P^{-},f)\neq\emptyset$. Note
that the local stable manifolds are tangent to the stable cone field and the
${\mathrm{uu}}$-discs are tangent to the strong unstable cone field, hence the
previous intersections are quasi-transverse.

In case (iii), consider the interval $[t_{2}-1,t_{1}]$ and let
$\widehat{D}_{t}=D_{t}$ if $t\in[0,t_{1}]$ and $\widehat{D}_{t}=D_{t+1}$ if
$t\in[t_{2}-1,1]$. By construction,

$S\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\bigcup_{t\in[t_{2}-1,t_{1}]}f(\widehat{D}_{t,\mathbb{A}})$

is a folding manifold, thus item (b) in the lemma holds.

Finally, in case (ii) we have that
$T_{1}=\bigcup_{t\in[0,1]}f(D_{t,\mathbb{A}})$ is tangent to
$W^{s}_{\mathrm{loc}}(P^{-},f)$ and we are in case (a) in the lemma. This
completes the proof of the lemma. ∎

###  5.3. Blender-horseshoes in the center-unstable Hénon-like family

Let us start by defining the Hénon-like families of endomorphisms that we will
consider.

####  5.3.1. Center-unstable Hénon-like families of endomorphisms

We consider the parameterised families of Hénon-like endomorphisms
$E_{\xi,\mu,\bar{\varsigma}},\,G_{\xi,\mu,\bar{\eta}}\colon\mathbb{R}^{3}\to\mathbb{R}^{3},$
defined by

(5.2)
$\begin{split}E_{\xi,\mu,\bar{\varsigma}}(x,y,z)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(\xi\,x+\varsigma_{1}\,y,\,\mu+\varsigma_{2}\,y^{2}+\varsigma_{3}\,x^{2}+\varsigma_{4}\,x\,y,\,\varsigma_{5}\,y),\\\
G_{\xi,\mu,\bar{\eta}}(x,y,z)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(y,\mu+y^{2}+\eta_{1}\,y\,z+\eta_{2}\,z^{2},\xi\,z+y),\end{split}$

where $\xi>1,\mu\in\mathbb{R}$,
$\bar{\varsigma}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(\varsigma_{1},\varsigma_{2},\varsigma_{3},\varsigma_{4},\varsigma_{5})\in\mathbb{R}^{5}$,
and $\bar{\eta}=(\eta_{1},\eta_{2})\in\mathbb{R}^{2}$. These families are
called center-unstable Hénon-like.

These families are conjugate, this allows us to translate properties from one
family to the other. More precisely:

######  Remark 5.5.

Consider the families of endomorphisms

$\widehat{E}_{\xi,\bar{\varsigma}},\,\widehat{G}_{\xi,\bar{\eta}}\colon\mathbb{R}^{4}\to\mathbb{R}^{4}$

defined by

(5.3)
$\begin{split}(\mu,x,y,z)&\mapsto\widehat{E}_{\xi,\bar{\varsigma}}(\mu,x,y,x)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{(}\mu,E_{\xi,\mu,\bar{\varsigma}}(x,y,z)\big{)},\\\
(\mu,x,y,z)&\mapsto\widehat{G}_{\xi,\bar{\eta}}(\mu,x,y,z)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{(}\mu,G_{\xi,\mu,\bar{\eta}}(x,y,z)\big{)}.\end{split}$

Consider the map

(5.4)
$\bar{\eta}(\bar{\varsigma})=(\eta_{1}(\bar{\varsigma}),\eta_{2}(\bar{\varsigma}))=(\varsigma_{1}^{2}\varsigma_{3}\varsigma_{2}^{-1},\varsigma_{1}\varsigma_{4}\varsigma_{2}^{-1}).$

Suppose that $\bar{\varsigma}$ is such that

$\varsigma_{1}\varsigma_{2}\varsigma_{5}\neq 0,$

then $\widehat{E}_{\xi,\bar{\varsigma}}$ and
$\widehat{G}_{\xi,\bar{\eta}(\bar{\varsigma})}$ are conjugate:

$\widehat{\Theta}^{-1}_{\bar{\varsigma}}\circ\widehat{E}_{\xi,\bar{\varsigma}}\circ\widehat{\Theta}_{\bar{\varsigma}}=\widehat{G}_{\xi,\bar{\eta}(\bar{\varsigma})}.$

where

(5.5)
$\begin{split}&\widehat{\Theta}_{\bar{\varsigma}}\colon\mathbb{R}^{4}\to\mathbb{R}^{4},\quad\widehat{\Theta}_{\bar{\varsigma}}(\mu,x,y,z)=\big{(}\varsigma_{2}^{-1}\mu,\Theta_{\bar{\varsigma}}(x,y,x)\big{)},\\\
&\Theta_{\bar{\varsigma}}\colon\mathbb{R}^{3}\to\mathbb{R}^{3},\quad\Theta_{\bar{\varsigma}}(x,y,z)=(\varsigma_{2}^{-1}\varsigma_{1}z,\varsigma_{2}^{-1}y,\varsigma_{2}^{-1}\varsigma_{5}x).\end{split}$

####  5.3.2. Occurrence of blender-horseshoes

Consider the set

(5.6) $\Delta\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}[-4,4]^{2}\times[-40,22].$

######  Theorem 5.6 (Theorem 1 in [17]).

There is $\varepsilon_{\mathrm{BH}}>0$ such that for every

$(\xi,\mu,\bar{\eta})\in\mathcal{O}_{\mathrm{BH}}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(1.18,1.19)\times(-10,-9)\times(-\varepsilon_{\mathrm{BH}},\varepsilon_{\mathrm{BH}})^{2}$

the endomorphism $G_{\xi,\mu,\bar{\eta}}$ has a blender-horseshoe
$\Lambda_{\xi,\mu,\bar{\eta}}$ with domain of reference $\Delta$ such that

$\Lambda_{\xi,\mu,\bar{\eta}}=\bigcap_{i\in\mathbf{Z}}G^{i}_{\xi,\mu,\bar{\eta}}(\Delta)\subset\mathrm{interior}(\Delta).$

As a consequence, every diffeomorphism or endomorphism sufficiently $C^{1}$
close to $G_{\xi,\mu,\bar{\eta}}$ has a blender-horseshoe in $\Delta$.

######  Remark 5.7.

By Remark 5.5, the map $E_{\xi,\mu,\bar{\varsigma}}$, with
$\bar{\varsigma}=(\varsigma_{1},\dots,\varsigma_{5})$, has blender-horseshoes
if $\varsigma_{1}\varsigma_{2}\varsigma_{5}\neq 0$ and

$(\xi,\mu,\varsigma_{1}^{2}\varsigma_{3}\varsigma_{2}^{-1},\varsigma_{1}\varsigma_{4}\varsigma_{2}^{-1})\in(1.18,1.19)\times(-10,-9)\times(-\varepsilon_{\mathrm{BH}},\varepsilon_{\mathrm{BH}})^{2}.$

Let us say a few words about the blenders in Theorem 5.6. Let
$P^{\pm}_{\xi,\mu,\bar{\eta}}$ be the reference fixed points of the blender
$\Lambda_{\xi,\mu,\bar{\eta}}$ of $G_{\xi,\mu,\bar{\eta}}$ in in $\Delta$. The
fixed points of $G_{\xi,\mu}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}G_{\xi,\mu,\bar{0}}$ 777With the notation in [17], say
$P^{-}_{\xi,\mu}=P_{\xi,\mu}$ and $P^{+}_{\xi,\mu}=Q_{\xi,\mu}$ in $\Delta$,
satisfy

(5.7)
$P^{-}_{\xi,\mu}=(p^{-}_{\xi,\mu},p^{-}_{\xi,\mu},\widetilde{p}^{-}_{\xi,\mu}),\qquad
P^{+}_{\xi,\mu}=(p^{+}_{\xi,\mu},p^{+}_{\xi,\mu},\widetilde{p}^{+}_{\xi,\mu})$

with

$\begin{split}-2.7&<p^{-}_{\xi,\mu}<-2.5,\quad
13<\widetilde{p}^{-}_{\xi,\mu}<15,\\\
3.5&<p^{+}_{\xi,\mu}<3.71,\quad-20.6<\widetilde{p}^{+}_{\xi,\mu}<-18.4.\end{split}$

For the map $G_{\xi,\mu}$, the strong unstable cone field is given by

(5.8)
$\mathcal{C}^{\mathrm{uu}}(Z)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{\\{}\,(u,v,w)\in\mathbb{R}^{3}\,:\,\sqrt{u^{2}+w^{2}}<\tfrac{1}{2}|v|\,\big{\\}},$

see [17, Lemma 3.10]. We also have that

$W^{\mathrm{s}}_{\mathrm{loc}}(P^{\pm}_{\xi,\mu},G_{\xi,\mu})=\big{\\{}\,(p^{\pm}_{\xi,\mu}+t,p^{\pm}_{\xi,\mu},\widetilde{p}^{\pm}_{\xi,\mu})\,\colon\,-4-p^{\pm}_{\xi,\mu}\leqslant
t\leqslant 4-p^{\pm}_{\xi,\mu}\,\big{\\}}.$

The superposition region of the blender
$\Lambda_{\xi,\mu}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\Lambda_{\xi,\mu,\bar{0}}$ consists of (large) discs tangent to the
cone field $\mathcal{C}^{{\mathrm{u}}{\mathrm{u}}}$ which are at the right of
$W^{\mathrm{s}}_{\mathrm{loc}}(P^{-}_{\xi,\mu},G_{\xi,\mu})$ and at the left
of $W^{\mathrm{s}}_{\mathrm{loc}}(P^{+}_{\xi,\mu},G_{\xi,\mu})$. These
observations imply the following:

######  Remark 5.8 (A disc in the superposition region).

Consider the disc

(5.9) $L\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{\\{}\,(0,y,0):|y|<4\,\big{\\}}\subset\Delta$

in the superposition region the blender $\Lambda_{\xi,\mu,\bar{\eta}}$ of
$G_{\xi,\mu,\bar{\eta}}$ for every
$(\xi,\mu,\bar{\eta})\in\mathcal{O}_{{\mathrm{BH}}}$. Then, by Lemma 5.1, for
every diffeomorphism $F$ close enough to $G_{\xi,\mu,\bar{\eta}}$ every disc
sufficiently close to $L$ is in the superposition region of the blender
$\Lambda_{F}$.

######  Lemma 5.5.

The unstable manifold
$W^{\mathrm{u}}\big{(}P^{+}_{\xi,\mu,\bar{\eta}},G_{\xi,\mu,\bar{\eta}}\big{)}$
is unbounded in the $y$\- and $z$-directions.

###### Proof.

We will show that $W^{\mathrm{u}}\big{(}P^{+}_{\xi,\mu},G_{\xi,\mu}\big{)}$
contains the set

$G_{\xi,\mu}(\Pi^{+}_{\xi,\mu}),\quad\mbox{where}\quad\Pi^{+}_{\xi,\mu}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{\\{}(\,x,y,z)\colon y\geqslant
p^{+}_{\xi,\mu}\,\big{\\}}\subset\mathbb{R}^{3}.$

The general case follows studying 2-dimensional projections of
$W^{\mathrm{u}}\big{(}P^{+}_{\xi,\mu,\bar{\eta}},G_{\xi,\mu,\bar{\eta}}\big{)}$.

Consider the plane $\Pi_{\xi,\mu}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\\{\,y=p^{+}_{\xi,\mu}\,\\}\subset\mathbb{R}^{3}$ and note that

$G_{\xi,\mu}(\Pi_{\xi,\mu})=\mathcal{L}_{\xi,\mu}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{\\{}\,(p^{+}_{\xi,\mu},p^{+}_{\xi,\mu},\widetilde{p}^{+}_{\xi,\mu}+t):t\in\mathbb{R}\,\big{\\}}\subset\Pi_{\xi,\mu}.$

It is easy to see that $\mathcal{L}_{\xi,\mu}$ is $G_{\xi.\mu}$-invariant and
that the restriction of $G_{\xi.\mu}$ to $\mathcal{L}_{\xi,\mu}$ is an
expanding linear map (with expansion factor $\xi>1$). Consider the cone field

$\mathcal{C}^{\mathrm{u}}(Z)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{\\{}(\,u,v,w)\in\mathbb{R}^{3}:|u|<\tfrac{1}{2}\sqrt{v^{2}+w^{2}}\,\big{\\}}.$

By [17, Claims 3.12 and 3.14], this cone field is $DG_{\xi,\mu}$-invariant and
uniformly expanding for every $Z=(x,y,z)$ with $|y|>\sqrt{5}$. A simple
calculation implies that for every $Z\in G_{\xi,\mu}(\Pi^{+}_{\xi,\mu})$ the
following holds

$G_{\xi,\mu}(\Pi^{+}_{\xi,\mu})\subset\Pi^{+}_{\xi,\mu}\qquad\mbox{and}\qquad
T_{Z}\big{(}G_{\xi,\mu}(\Pi^{+}_{\xi,\mu})\big{)}\subset\mathcal{C}^{\mathrm{u}}(Z).$

These properties and the expanding property of $\mathcal{C}^{\mathrm{u}}$
imply the lemma. ∎

######  Notation.

We will consider blenders in $\mathbb{R}^{3}$ and in the ambient manifold $M$.
We will denote the first ones by $\Lambda$ and the second ones by $\Upsilon$.

##  6\. The renormalisation scheme

In this section, we outline the renormalisation scheme in [16], see
Proposition 6.5. For that we embed any
$f\in\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$ in a bifurcating eight-parameter
family

$\mathbb{R}^{8}\ni\bar{\upsilon}\,\to\,f_{\bar{\upsilon}}\in\mathrm{Diff}^{r}(M)\quad\mbox{with}\quad
f_{\textbf{0}}=f$

and construct a renormalisation scheme for $f$ consisting of:

  * •

a sequence of local charts $\Psi_{k}$ from $\mathbb{R}^{3}$ to $U_{Q}$,

  * •

a sequence of reparameterisations
$\mathbb{R}\ni\mu\mapsto\bar{\upsilon}^{\xi}_{k}(\mu)\in\mathbb{R}^{8}$ with
$\xi>1$ and $\bar{\upsilon}^{\xi}_{k}(\mu)\to\textbf{0}$ on compact sets,

  * •

sequences $n_{k},\,m_{k}\in\mathbb{N}$ such that the “return maps”
$f^{N_{2}+m_{k}+N_{1}+n_{k}}_{\bar{\upsilon}_{k}^{\xi}(\mu)}$, defined on a
neighbourhood of the heterodimensional tangency $\widetilde{Y}$, satisfies

$\Psi_{k}^{-1}\circ
f^{N_{2}+m_{k}+N_{1}+n_{k}}_{\bar{\upsilon}_{k}^{\xi}(\mu)}\circ\Psi_{k}\to
E_{\xi,\mu,\bar{\varsigma}}$

where the convergence is $C^{r}$, $E_{\xi,\mu,\bar{\varsigma}}$ is defined
(5.2), and $\bar{\varsigma}=\bar{\varsigma}(\xi,f)$ satisfies (2.9).

This section is organised as follows. We define the unfolding family
$(f_{\bar{\upsilon}})_{\bar{\upsilon}\in\mathbb{R}^{8}}$ in Section 6.1 and
review the renormalisation scheme in Section 6.2. The convergence of the
scheme is stated in Section 6.3.

###  6.1. The unfolding family

Given $f\in\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$ and

$\bar{\upsilon}=(\bar{\mu},\bar{\nu},\alpha,\beta)\in\mathbb{R}^{3}\times\mathbb{R}^{3}\times\mathbb{R}\times\mathbb{R}=\mathbb{R}^{8},$

we consider a (smooth) family
$(f_{\bar{\upsilon}})_{\bar{\upsilon}\in\mathbb{R}^{8}}$ in
$\mathrm{Diff}^{r}(M)$ with $f_{\textbf{0}}=f$, $f_{\bar{\upsilon}}(Q)=Q$, and
$f_{\bar{\upsilon}}(P)=P$ such that

  * •

the parameter $\bar{\mu}$ unfolds the heterodimensional tangency
$\widetilde{Y}$,

  * •

the parameter $\bar{\nu}$ unfolds the quasi-transverse intersection
$\widetilde{X}$, and

  * •

the parameters $\alpha$ and $\beta$ modify the arguments of $Df(P)$ and
$Df(Q)$.

Recall the translation-like perturbations $T_{W,\bar{\omega},\varrho}$ and
rotation-like perturbations $R^{\ast}_{\alpha,\theta,\kappa}$ in (3.4) and
(3.5), respectively. For $\bar{\upsilon}=(\bar{\mu},\bar{\nu},{\alpha},\beta)$
and small $\rho>0$ consider the perturbation of the identity defined by

$\Omega_{\bar{\upsilon},\rho}(Z)=\Omega_{(\bar{\mu},\bar{\nu},{\alpha},\beta),\rho}(Z)=\left\\{\begin{split}&R^{x}_{\alpha,\theta_{1},\kappa_{1}}(Z),\quad\mbox{if
$Z\in U_{P}$},\\\ &T_{\widetilde{X},\bar{\nu},\rho}(Z),\quad\mbox{if $Z\in
V_{\widetilde{X}}$},\\\ &R^{y}_{\beta,\theta_{2},\kappa_{2}}(Z),\quad\mbox{if
$Z\in U_{Q}$},\\\ &T_{\widetilde{Y},\bar{\mu},\rho}(Z),\quad\mbox{if $Z\in
V_{\widetilde{Y}}$},\\\ &\mathrm{id}(Z),\quad\mbox{\,\,\,if $Z\not\in
V_{P}\cup V_{Q}\cup V_{\widetilde{X}}\cup
V_{\widetilde{Y}}$},\end{split}\right.$

where $V_{P},V_{Q},V_{\widetilde{X}}$, and $V_{\widetilde{Y}}$ are small
neighbourhoods of $P,Q,\widetilde{X},$ and $\widetilde{Y}$ contained in
$U_{P},U_{Q},U_{\widetilde{X}}$, and $U_{\widetilde{Y}}$, respectively, recall
Section 2. The numbers $\theta_{1},\,\theta_{2}>1$,
$\kappa_{1},\,\kappa_{2}>0$ are chosen such that

$\begin{split}&\widetilde{X},Y\in[-\kappa_{1}^{-1},\kappa_{1}^{-1}]^{3},\qquad\widetilde{Y},X\in[-\kappa_{2}^{-1},\kappa_{2}^{-1}]^{3},\\\
&[-\theta_{1}\kappa_{1}^{-1},\theta_{1}\kappa_{1}^{-1}]^{3}\subset
U_{P},\qquad[-\theta_{2}\kappa_{2}^{-1},\theta_{2}\kappa_{2}^{-1}]^{3}\subset
U_{Q}.\end{split}$

Finally, we let

(6.1) $f_{\bar{\upsilon},\rho}=\Omega_{\bar{\upsilon},\rho}\circ f.$

######  Remark 6.1 (The parameter $\rho$).

Above we emphasise the role of the parameter $\rho$ related to the size of the
translation-like perturbation of $f$.

######  Remark 6.2 (Support of the rotation-like part of
$\Omega_{\bar{\upsilon},\rho}$).

Consider the linear maps

$\begin{split}&f_{\lambda,\sigma,\varphi}=\begin{pmatrix}\lambda&0&\\\
0&\sigma\sin 2\pi\varphi&\sigma\cos 2\pi\varphi\\\ 0&-\sigma\cos
2\pi\varphi&\sigma\sin 2\pi\varphi\end{pmatrix},\\\
&\widetilde{f}_{\lambda,\sigma,\varphi}=\begin{pmatrix}\sigma\sin
2\pi\varphi&0&\sigma\cos 2\pi\varphi\\\ 0&\lambda&0\\\ -\sigma\cos
2\pi\varphi&0&\sigma\sin 2\pi\varphi\end{pmatrix}.\end{split}$

With this notation, the restriction of $f$ to $U_{P}$ is the map
$f_{\lambda_{P},\sigma_{P},\varphi_{P}}$, recall Section 2.1.1. Note that if
$Z\in
f^{-1}\big{(}[-\kappa_{1},\kappa_{1}]^{3}\big{)}\cap[-\kappa_{1},\kappa_{1}]^{3}$
then

$\Omega_{\bar{\upsilon},\rho}\circ
f(Z)=R^{x}_{\alpha,\theta_{1},\kappa_{1}}\circ
f(Z)=R^{x}_{\alpha,\theta_{1},\kappa_{1}}\circ
f_{\lambda_{P},\sigma_{P},\varphi_{P}}=f_{\lambda_{P},\sigma_{P},\varphi_{P}+\alpha}(Z).$

If $Z\in
f^{-1}\big{(}[-\theta_{1}\kappa_{1}^{-1},\theta_{1}\kappa_{1}^{-1}]^{3}\big{)}^{c}\cap
U_{P}$ then

$\Omega_{\bar{\upsilon},\rho}\circ
f(Z)=R^{x}_{\alpha,\theta_{1},\kappa_{1}}\circ f(Z)=f(Z).$

Similarly, the restriction of $f$ to $U_{Q}$ is
$\widetilde{f}_{\lambda_{Q},\sigma_{Q},\varphi_{Q}}$ and analogous conditions
hold.

######  Remark 6.3 (Support of the translation-like part of
$\Omega_{\bar{\upsilon},\rho}$).

Note that

  * •

$\Omega_{\bar{\upsilon},\rho}\circ f(Z)=T_{\widetilde{X},\bar{\nu},\rho}\circ
f(Z)=f(Z)$ for every $Z\in M\setminus
f^{-1}\big{(}B(\widetilde{X},2\rho)\big{)}$,

  * •

$\Omega_{\bar{\upsilon},\rho}\circ f(Z)=T_{\widetilde{X},\bar{\nu},\rho}\circ
f(f^{-1}(\widetilde{X}))=\widetilde{X}+\bar{\nu}$.

Analogously, we have that

  * •

$\Omega_{\bar{\upsilon},\rho}\circ f(Z)=T_{\widetilde{y},\bar{\mu},\rho}\circ
f(Z)=f(Z)$ for every $Z\in M\setminus
f^{-1}\big{(}B(\widetilde{Y},2\rho)\big{)}$,

  * •

$\Omega_{\bar{\upsilon},\rho}\circ f(Z)=T_{\widetilde{Y},\bar{\mu},\rho}\circ
f(f^{-1}(\widetilde{Y}))=\widetilde{Y}+\bar{\mu}$.

###  6.2. The renormalisation scheme

We now summarise the ingredients of the renormalisation scheme: Sojourn times
and adjusting arguments (Section 6.2.1), reparameterisations (Section 6.2.2),
and changes of coordinates (Section 6.2.3).

####  6.2.1. Sojourn times and adjusting arguments

Fix $\xi>1$ and consider

$\tau\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\frac{\gamma_{3}(a_{3}-a_{2})}{\sqrt{2}},\quad\mbox{where
$\gamma_{3}$ is as in \eqref{e.Aisoftheform} and $a_{2},a_{3}$ are as in
\eqref{e.Bisoftheform}}.$

Note that $\tau>0$, see (2.7). By [16, Lemma 6.1], associated to
$\tau^{-1}\xi$, there is a residual subset
$\mathcal{R}=\mathcal{R}_{\tau^{-1}\xi}$ of $(0,1)\times(1,\infty)$ consisting
of pairs $(\sigma,\lambda)$ having a sequence of sojourn times
$\mathbf{s}_{k}=(m_{k},n_{k})$ in $\mathbb{N}^{2}$,  adapted to $\tau^{-1}\xi$
satisfying

(6.2) $\lim_{k\to\infty}\sigma^{m_{k}}\lambda^{n_{k}}=\tau^{-1}\xi$

where $m_{k}$ and $n_{k}$ are related by the inequality

$m_{k}<\eta\,n_{k}+\tilde{\eta}+1,\quad\eta\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\frac{\log(\lambda^{-1})}{\log(\sigma)}\quad\mbox{and}\quad\tilde{\eta}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\frac{\log(\tau\,\xi^{-1})}{\log(\sigma)}.$

Our hypotheses allow us to consider $(\sigma_{P},\lambda_{Q})\in\mathcal{R}$
having a sequence of sojourn times ${\mathbf{s}}_{k}=(m_{k},n_{k})$ adapted to
$\tau^{-1}\,\xi$, see [14, Lemma 5.1] and [16, Lemma 6.1]. The spectral
condition in (2.2) provides a constant $C>0$ such that

(6.3)
${\lambda_{P}}^{\frac{m_{k}}{2}}\sigma_{P}^{m_{k}}{{\sigma_{Q}}}^{n_{k}}<C\,\big{(}({\lambda_{P}}^{\frac{1}{2}}\sigma_{P})^{\eta}{{\sigma_{Q}}}\big{)}^{n_{k}}\to
0$

Associated to the sequence $({\mathbf{s}}_{k})$ there is a sequence
$\Theta_{k}=(\zeta_{m_{k}},\vartheta_{n_{k}})\in\mathbb{R}^{2}$, with
$\Theta_{k}\to(0,0)$, of adjusting arguments leading to the following argument
maps

(6.4)
$\begin{split}&\alpha_{k}(\theta)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\frac{1}{2\pi m_{k}}\left(\frac{\pi}{4}-2\pi
m_{k}\theta+2\pi[m_{k}\theta]+\zeta_{m_{k}}\right);\\\
&\beta_{k}(\omega)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\frac{1}{2\pi n_{k}}\left(\frac{\pi}{2}-2\pi
n_{k}\omega+2\pi[n_{k}\omega]+\vartheta_{n_{k}}\right),\end{split}$

where $[x]$ denotes the integer part of $x\in\mathbb{R}$. The sequence
$(\vartheta_{n_{k}})$ is chosen such that $\varphi_{Q}+{\beta}_{k}$ is
irrational (here $\varphi_{Q}$ is the argument in item (A) in Section 2.1.1).
There are no further restrictions on the definition of $(\zeta_{m_{k}})$.

####  6.2.2. Reparameterisations

Associated to the sequences $({\mathbf{s}}_{k})$ and $(\Theta_{k})$ we define
the sequence of reparameterisations $\bar{\upsilon}=\bar{\upsilon}^{\xi}_{k}$
of the family $f_{\bar{\upsilon},\rho}$ in (6.1) by:

$\bar{\upsilon}^{\xi}_{k}\colon\mathbb{R}\to\mathbb{R}^{8},\quad\bar{\upsilon}^{\xi}_{k}(\mu)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{(}\bar{\mu}^{\xi}_{k}(\mu),\bar{\nu}^{\xi}_{k},\alpha^{\xi}_{k}({\varphi_{P}}),\beta^{\xi}_{k}(\varphi_{Q})\big{)}\in\mathbb{R}^{3}\times\mathbb{R}^{3}\times\mathbb{R}\times\mathbb{R},$

where (for simplicity, in what follows we eliminate the dependence888This
dependence is given by the choice in (6.2). of the coordinate maps of
$\bar{\upsilon}^{\xi}_{k}$ on $\xi$):

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

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

where $a_{1},b_{1},c_{1}$ are as in (2.6). Note that
$\bar{\mu}_{k}(\mu)\to(0,0,0)$ as $k\to\infty$.

$\quad\bullet$ To define $\bar{\nu}_{k}\in\mathbb{R}^{3}$ consider first

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

and the sequences

(6.6)
$\begin{split}\tilde{\mathfrak{c}}_{k}&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\cos\big{(}2\pi
m_{k}(\varphi_{P,k})\big{)},\quad\tilde{\mathfrak{s}}_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\sin\big{(}2\pi m_{k}(\varphi_{P,k})\big{)},\\\
\mathfrak{c}_{k}&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\cos\big{(}2\pi
n_{k}(\varphi_{Q,k})\big{)},\quad\mathfrak{s}_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\sin\big{(}2\pi n_{k}(\varphi_{Q,k})\big{)}.\end{split}$

######  Remark 6.4.

By the definition of $\alpha_{k}(\varphi_{P})$ and $\beta_{k}(\varphi_{Q})$ in
(LABEL:e.alphakbetak), it follows that ${\mathfrak{c}}_{k}\to 0$,
${\mathfrak{s}}_{k}\to 1$, and
$\tilde{\mathfrak{c}}_{k},\tilde{\mathfrak{s}}_{k}\to 1/\sqrt{2}$.

Recalling the coordinated maps $\widetilde{H}_{2}$ and $\widetilde{H}_{3}$ of
$\widetilde{H}$ in (2.3), we let

(6.7)
$\begin{split}\tilde{\rho}_{2,k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}\widetilde{H}_{2}(\textbf{0})(\mathfrak{c}_{k}-\mathfrak{s}_{k})^{2}+\frac{1}{2}\frac{\partial^{2}}{\partial
z^{2}}\widetilde{H}_{2}(\textbf{0})(\mathfrak{s}_{k}+\mathfrak{c}_{k})^{2},\\\
\tilde{\rho}_{3,k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}\widetilde{H}_{3}(\textbf{0})(\mathfrak{s}_{k}-\mathfrak{c}_{k})^{2}+\frac{1}{2}\frac{\partial^{2}}{\partial
z^{2}}\widetilde{H}_{3}(\textbf{0})(\mathfrak{s}_{k}+\mathfrak{c}_{k})^{2}.\end{split}$

Finally, we let

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

where $\alpha_{1},\alpha_{3},\gamma_{3}$ are as in (2.4). Note that
$\bar{\nu}_{k}\to\mathbf{0}$ as $k\to\infty$.

####  6.2.3. Change of coordinates

Using the local coordinates in $U_{Q}$, we consider the sequence of maps
$\Psi_{k}\colon U_{k}\to U_{Q}=[-a_{Q},a_{Q}]^{3}$ defined by

(6.9)
$\begin{split}\Psi_{k}(x,y,z)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(1+\sigma_{P}^{-m_{k}}\sigma_{Q}^{-n_{k}}\,x,\\\
&\quad\,\,\,\,\,\,\,\sigma_{Q}^{-n_{k}}+\sigma_{P}^{-2m_{k}}{\sigma_{Q}}^{-2n_{k}}\,y,1+\sigma_{P}^{-m_{k}}{\sigma_{Q}}^{-n_{k}}\,z),\end{split}$

where $U_{k}$ is the “cube” of $\mathbb{R}^{3}$ such that
$\Psi_{k}(U_{k})=U_{Q}$. Recall that $\widetilde{Y}=(1,0,1)$ and note that for
any compact set $K\subset\mathbb{R}^{3}$ it holds
$\Psi_{k}(K)\to\\{\widetilde{Y}\\}$ as $k\to\infty$.

###  6.3. Convergence of the renormalisation scheme

Fixed $\xi>1$, small $\rho>0$, and $f\in\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$,
consider the renormalisation scheme above and the sequence of one-parameter
family of maps

(6.10)
$\begin{split}&\mathbb{R}\ni\mu\,\to\,\mathcal{R}_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho}(f)\in\mathrm{Diff}^{r}(M)\\\
&\mathcal{R}_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho}(f)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}f_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho}^{N_{2}}\circ
f_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho}^{m_{k}}\circ
f_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho}^{N_{1}}\circ
f_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho}^{n_{k}},\end{split}$

called renormalised sequence of $f$. Here we are emphasising the roles of
$\xi$ and $\rho$.

######  Proposition 6.5 (Theorem 1,[16]).

Fix $\xi>1$, small $\rho>0$, and $\mu\in\mathbb{R}$. Given any
$f\in\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$ the sequence of maps

$\Psi_{k}^{-1}\circ\mathcal{R}_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho}(f)\circ\Psi_{k}\colon
U_{k}\to\mathbb{R}^{3},\quad k\in\mathbb{N},$

converges, on compact sets of $\mathbb{R}^{3}$ and in the $C^{r}$ topology, to
the endomorphism $E_{\xi,\mu,\bar{\varsigma}}$ in (5.2), where
$\bar{\varsigma}=\bar{\varsigma}(\xi,f)$ is as in (2.8).

######  Notation 6.6 (The parameters $\rho$ and $\xi$).

When the role of $\rho$ is not relevant it will omitted, writing
$f_{\bar{\upsilon}}$ and $\mathcal{R}_{\bar{\upsilon}^{\xi}_{k}(\mu)}$ instead
of $f_{\bar{\upsilon},\rho}$ and
$\mathcal{R}_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho}$. Similarly with $\xi>1$.

######  Remark 6.7.

Recall the perturbation $f_{\varepsilon}$ of $f$ in Proposition 4.1. The
renormalisation scheme of $f$ associated to $X$ and $Y$ can be applied to
$f_{\varepsilon}$ and it is preserved. In this way, we get the one-parameter
family of diffeomorphisms $f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho}$.

##  7\. Interplay between blenders and heteroclinic points

In Section 7.1, see Proposition 7.2, we state the occurrence of blender-
horseshoes in the renormalisation scheme for diffeomorphisms
$f\in\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$ and their perturbations
$f_{\varepsilon}\in\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$ given by Proposition
4.1.

Note that $f_{\varepsilon}$ has additional heteroclinic points
$X_{1,\varepsilon},X_{2,\varepsilon}$. In Section 7.2, we see how these
intersections are unfolded without modifying the blenders given by the
renormalisation scheme.

Before going into the details, recall the definitions of the transitions
$\mathfrak{T}_{1,i,\varepsilon}$ associated to $X_{i,\varepsilon}$ and their
domains $U_{i,\varepsilon}$ in (4.5) and consider the neighbourhoods

$\widetilde{U}_{i,\varepsilon}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathfrak{T}_{1,i,\varepsilon}(U_{i,\varepsilon})\quad\mbox{of}\quad\widetilde{X}_{i,\varepsilon}=\mathfrak{T}_{1,i,\varepsilon}({X}_{i,\varepsilon}),\quad
i=0,1,2.$

Take sufficiently small $\rho=\rho(\varepsilon)>0$ such that

(7.1) $B(\widetilde{X},2\,\rho)\subset\widetilde{U}_{0,\varepsilon},\quad
B(\widetilde{X}_{1,\varepsilon},2\,\rho)\subset\widetilde{U}_{1,\varepsilon},\quad
B(\widetilde{X}_{2,\varepsilon},2\,\rho)\subset\widetilde{U}_{2,\varepsilon}.$

In particular, these three balls are disjoint. We can now consider the
renormalisation scheme
$\mathcal{R}_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho}(f_{\varepsilon})$ and
observe that by the choice of $\rho$ the renormalisation preserves the
heteroclinic points $X_{1,\varepsilon}$ and $X_{2,\varepsilon}$. Note that as
the transitions of $f$ are not modified, recalling (2.8), we have

$\bar{\varsigma}=\bar{\varsigma}(\xi,f)=\bar{\varsigma}(\xi,f_{\varepsilon}).$

###  7.1. Blenders in the renormalisation scheme

Recall the definitions of $\Psi_{k}$ and of $U_{k}$ in (6.9) and of
$\mathcal{R}_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho}(f_{\varepsilon})$ in (6.10).
Define the maps

$\widehat{\Psi}_{k}\colon\mathbb{R}\times U_{k}\to\mathbb{R}\times
U_{Q},\qquad\mathcal{R}_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho}(f_{\varepsilon})\colon\mathbb{R}\times
M\to\mathbb{R}\times M$

by

$\begin{split}\widehat{\Psi}_{k}(\mu,X)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{(}\,\mu,\Psi_{k}(X)\,\big{)},\\\
\widehat{\mathcal{R}}_{k,\rho}(f_{\varepsilon})(\mu,X)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{(}\,\mu,\mathcal{R}_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho}(f_{\varepsilon})(X)\,\big{)}\end{split}$

and consider (with slight abuse of notation on the domain of definitions) the
maps $\Phi_{k}$ and $\widehat{\Phi}_{k}$ defined by

(7.2)
$\begin{split}X\in\mathbb{R}^{3}\,\,&\mapsto\,\,\Phi_{k}(X)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}{\Psi}_{k}\circ\Theta_{\bar{\varsigma}}(X)\in U_{Q},\\\
(\mu,X)\in\mathbb{R}\times\mathbb{R}^{3}\,\,&\mapsto\,\,\widehat{\Phi}_{k}(\mu,X)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\widehat{\Psi}_{k}\circ\widehat{\Theta}_{\bar{\varsigma}}(\mu,X)\\\
&\hskip
64.58788pt=\big{(}\varsigma_{2}^{-1}\mu,\Phi_{k}(X)\big{)}\in\mathbb{R}\times
U_{Q},\end{split}$

where $\Theta_{\bar{\varsigma}}$ and $\widehat{\Theta}_{\bar{\varsigma}}$ are
the conjugations in (5.5).

The following explicit form of the maps $\Phi_{k}^{-1}$ will be used in
Section 9.1.

######  Remark 7.1.

Note that for $(1+x,y,1+z)\in U_{Q}$ close to $\widetilde{Y}=(1,0,1)$ we have
that

$(\tilde{x},\tilde{y},\tilde{z})\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\Phi_{k}^{-1}(1+x,y,1+z),\quad\left\\{\begin{split}\tilde{x}&=\sigma_{Q}^{n_{k}}\,\sigma^{m_{k}}_{P}\,\varsigma_{2}\,\varsigma_{5}^{-1}\,z,\\\
\tilde{y}&=\varsigma_{2}\,\sigma_{Q}^{2n_{k}}\,\sigma^{2m_{k}}_{P}\,(y-\sigma_{Q}^{-n_{k}}),\\\
\tilde{z}&=\sigma_{Q}^{n_{k}}\,\sigma^{m_{k}}_{P}\,\varsigma_{2}\,\varsigma_{1}^{-1}\,x.\end{split}\right.$

Finally, consider the sequence of maps
$\mathfrak{R}_{k,\rho}(f_{\varepsilon})\colon\mathbb{R}\times\mathbb{R}^{3}\to\mathbb{R}\times\mathbb{R}^{3}$

$(\mu,X)\,\,\mapsto\,\,{\mathfrak{R}}_{k,\rho}(f_{\varepsilon})(\mu,X)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\widehat{\Phi}_{k}^{-1}\circ\widehat{\mathcal{R}}_{k,\rho}(f_{\varepsilon})\circ{\widehat{\Phi}_{k}}(\mu,X).$

Finally, note that for each fixed $\mu$ the projection of
${\mathfrak{R}}_{k,\rho}(f_{\varepsilon})(\mu,\cdot)$ in the “second
coordinate” $\mathbb{R}^{3}$ is exactly the map
$\mathcal{R}_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho}(f_{\varepsilon})$.

Recalling the definition of $\bar{\eta}(\bar{\varsigma})$ in (5.4) and of
$\bar{\varsigma}(\xi,f)=\bar{\varsigma}(\xi,f_{\varepsilon})$ in (2.8) we
define (with slight abuse of notation) the map

$\bar{\eta}(\xi,f_{\varepsilon})\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\bar{\eta}(\bar{\varsigma}(\xi,f_{\varepsilon}))=\bar{\eta}(\xi,f).$

Recalling Proposition 6.5, Theorem 5.6 (and the definition of the set
$\mathcal{O}_{\mathrm{BH}}$ there), Remark 5.5, and the definition of
$\widehat{G}_{\xi,\bar{\eta}}$ in (5.3) we get the following:

######  Proposition 7.2.

Consider $f\in\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$, $\xi>1$, small
$\varepsilon>0$, and $f_{\varepsilon}$ as in Proposition 4.1. There is
$\rho(\varepsilon)$ such that the sequence of maps
$({\mathfrak{R}}_{k,\rho(\varepsilon)}(f_{\varepsilon}))$ converges in the
$C^{r}$ topology and on compact sets of $\mathbb{R}^{4}$ to
$\widehat{G}_{\xi,\bar{\eta}(\xi,f)}$.

As a consequence, for every $k$ large enough the map
$\mathcal{R}_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho(\varepsilon)}(f_{\varepsilon})$
has a blender-horseshoe
${\Lambda}_{{\bar{\upsilon}^{\xi}_{k}(\mu),\rho(\varepsilon)}}$ in
$\Delta=[-4,4]^{2}\times[-40,22]$.

We now describe more precisely the blenders
${\Lambda}_{{\bar{\upsilon}^{\xi}_{k}(\mu),\rho(\varepsilon)}}$. For
$\mu\in(-10,-9)$, large $k$, and small $\varepsilon$, consider the
diffeomorphism

$f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\Omega_{\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}\circ
f_{\varepsilon},$

where $\Omega_{\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}$ is defined as in
(6.1). Proposition 7.2 implies that
$f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}$ has a blender-
horseshoe defined as follows. Let

(7.3) $\begin{split}\Delta(\mu)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\\{\mu\\}\times\Delta,\\\
\Delta_{k}(\mu)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\Delta_{k}=\Phi_{k}(\Delta),\\\
\widehat{\Delta}_{k}(\mu)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\widehat{\Phi}_{k}\big{(}\Delta(\mu)\big{)}=\big{(}\varsigma_{2}^{-1}\,\mu,\Phi_{k}\big{(}\Delta\big{)}\big{)}=\big{(}\varsigma_{2}^{-1}\,\mu,\Delta_{k}\big{)}.\end{split}$

Recalling that
$\mathcal{R}_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho(\varepsilon)}(f_{\varepsilon})=f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}^{N_{2}+m_{k}+N_{1}+n_{k}}$,
see (6.10), we have that

(7.4)
$\begin{split}\Upsilon_{f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}}&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\bigcap_{\ell\in\mathbb{Z}}\big{(}\mathcal{R}_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho(\varepsilon)}(f_{\varepsilon})\big{)}^{\ell}(\Delta_{k}(\mu))\\\
&=\bigcap_{\ell\in\mathbb{Z}}f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}^{\,\,(N_{2}+m_{k}+N_{1}+n_{k})\,\ell}(\Delta_{k}).\end{split}$

is a blender-horseshoe of
$f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}^{N_{2}+m_{k}+N_{1}+n_{k}}$.
Note that, by construction,

(7.5)
${\Lambda}_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho(\varepsilon)}=\Phi_{k}^{-1}\big{(}\Upsilon_{f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}}\big{)}.$

######  Notation 7.3.

The reference saddles
$P^{\pm}_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho(\varepsilon)}$ of
${\Lambda}_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho(\varepsilon)}$ are the
continuations of the saddles $P^{\pm}_{\xi,\mu,\bar{\eta}}$ of the blender of
$G_{\xi,\mu,\bar{\eta}}$ in (5.7).

######  Remark 7.4.

Consider $\big{(}\xi,\mu,\bar{\eta}(\xi,f)\big{)}\in\mathcal{O}_{\mathrm{BH}}$
and write $\bar{\eta}=\bar{\eta}(\xi,f)$. Recall the definition of the disc
$L\subset\mathbb{R}^{3}$ in the superposition region of the blender of
${G}_{\xi,\mu,\bar{\eta}}$, see Remark 5.8. The second part of that remark and
the $C^{r}$ convergence

$\Phi_{k}^{-1}\circ\mathcal{R}_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho(\varepsilon)}(f_{\varepsilon})\circ\Phi_{k}\to{G}_{\xi,\mu,\bar{\eta}}$

on compact subsets of $\mathbb{R}^{3}$ imply that for every large $k$ the set

$\Phi_{k}^{-1}\circ\mathcal{R}_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho(\varepsilon)}(f_{\varepsilon})\circ\Phi_{k}(L)$

contains a disc in the superposition region of the blender
${\Lambda}_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho(\varepsilon)}$.

###  7.2. Unfolding the heteroclinic points $X_{i,\varepsilon}$

By construction, we have that $X_{1,\varepsilon}$ and $X_{2,\varepsilon}$ are
quasi-transverse heteroclinic points of
$f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}$ and (recalling the
definitions in (4.6))

$\widetilde{X}_{i,\varepsilon}\in
W^{\mathrm{s}}_{\mathrm{loc}}(P,f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)})\cap
W^{\mathrm{u}}(Q,f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}).$

######  Remark 7.5.

The choices of $\varepsilon$ and $\rho(\varepsilon)$ imply that the closure of
the orbits of $\widetilde{X}_{i,\varepsilon}$ and the orbit of the blender
$\Upsilon_{f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}}$ are
disjoint. Moreover, the orbit of the blender is also disjoint from the
neighbourhoods ${U}_{i,\varepsilon}$ of $X_{i,\varepsilon}$ (and thus from the
neighbourhoods $\widetilde{U}_{i,\varepsilon}$ of
$\widetilde{X}_{i,\varepsilon}$).

We now consider a “local unfolding of the heteroclinic point
$\widetilde{X}_{1,\varepsilon}$ independent of the renormalisation process”:
this unfolding is given by a perturbation whose support is disjoint from
$B\big{(}\widetilde{X},\rho(\varepsilon)\big{)}$ and
$B\big{(}\widetilde{Y},\rho(\varepsilon)\big{)}$. For that, consider a family
of local perturbations of $f_{\varepsilon}$ given by

(7.6)
$g_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}=\theta_{\varepsilon,k}\circ
f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)},$

where $\theta_{\varepsilon,k}$ is a $C^{r}$ perturbation of identity supported
on
$B\big{(}\widetilde{X}_{1,\varepsilon},2\rho(\varepsilon)\big{)}\subset\widetilde{U}_{1,\varepsilon}$
satisfying $\lim_{k\to\infty}d_{r}(\theta_{\varepsilon,k},\mathrm{id})=0$. To
define $\theta_{\varepsilon,k}$, recall the definitions of the bump function
$\Pi_{\delta}$ in (3.2) and the sequences $m_{k},\,n_{k}$ in Section 6.2.1,
$\tilde{\mathfrak{c}}_{k}$ and $\tilde{\mathfrak{s}}_{k}$ in (6.6), and
$\bar{\nu}_{k}$ in (6.8), and consider the sequence of vectors

(7.7) $\bar{\tau}_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{(}0,\sigma^{-m_{k}}_{P}(\tilde{\mathfrak{c}}_{k}+\tilde{\mathfrak{s}}_{k}),\sigma^{-m_{k}}_{P}(\tilde{\mathfrak{c}}_{k}-\tilde{\mathfrak{s}}_{k})\big{)}\in\mathbb{R}^{3},\qquad\bar{\tau}_{k}\to\mathbf{0}.$

The map $\theta_{\varepsilon,k}\colon M\rightarrow M$ is defined by:

(7.8)
$\begin{split}\theta_{\varepsilon,k}\big{(}Z\big{)}&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}Z+{\Pi}_{\rho(\varepsilon)}(W)\,\bar{\tau}_{k},\quad\mbox{if}\quad
Z=\widetilde{X}_{1,\varepsilon}+W\in
B\big{(}\widetilde{X}_{1,\varepsilon},\,2\rho(\varepsilon)\big{)},\\\
\theta_{\varepsilon,k}(Z)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}Z,\quad\mbox{if}\quad Z\notin
B\big{(}\widetilde{X}_{1,\varepsilon},\,2\rho(\varepsilon)\big{)}.\end{split}$

Recalling that
$\|\Pi_{\rho(\varepsilon)}\|_{r}\leqslant(\|b\|_{r})^{3}\,\rho(\varepsilon)^{-r}$,
see (3.3), and that
$|\tilde{\mathfrak{c}}_{k}\pm\tilde{\mathfrak{s}}_{k}|\leqslant 2$, see (6.6),
we get

$d_{r}({\theta}_{\varepsilon,k},\mathrm{id})\leqslant
2\,(\|b\|_{r})^{3}\,\rho(\varepsilon)^{-1}\,\sigma_{P}^{-m_{k}}\to 0,\qquad
k\to\infty.$

######  Remark 7.6.

By definition of $\theta_{\varepsilon,k}$, for every $W\not\in
f_{\varepsilon}^{-1}\big{(}B(\widetilde{X}_{1,\varepsilon},2\rho(\varepsilon)\big{)}$
it holds that

$g_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}(W)=\theta_{\varepsilon,k}\circ
f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}(W)=f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}(W).$

As a consequence, if $W\not\in
f_{\varepsilon}^{-1}\big{(}B(\widetilde{X}_{1,\varepsilon},2\delta)\big{)}$
then

$\mathcal{R}_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho(\varepsilon)}(f_{\varepsilon})(W)=\mathcal{R}_{\bar{\upsilon}^{\xi}_{k}(\mu),\rho(\varepsilon)}(g_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)})(W).$

Hence, by Remark 7.5, the maps
$g_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}$ and
$f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}$ have the common
blender $\Upsilon_{f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}}$
defined in (7.4).

Moreover, by construction, the curve
$L^{\mathrm{u}}_{2,\varepsilon}(\rho(\varepsilon))$ in (4.12) containing
$X_{2,\varepsilon}$ is contained in
$W^{\mathrm{u}}(Q,g_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)})$.

##  8\. Orbits and itineraries associated to the renormalisation

This is a preparatory section to the proof of Theorem 1.1. We study admissible
points and returns: we select a set of points whose itineraries for the
diffeomorphisms $g_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}$ in
(7.6) are associated to the renormalisation scheme, see (8.1).

Recall the charts
$\Phi_{k}=\Phi_{k,\bar{\varsigma}}=\Psi_{k}\circ\Theta_{\bar{\varsigma}}\colon
U_{k}\subset\mathbb{R}^{3}\to U_{Q}$ in (7.2) and that for every compact set
$K\subset\mathbb{R}^{3}$ it holds $\Phi_{k}(K)\to\widetilde{Y}$ as
$k\to\infty$.

Recall the definitions of the neighbourhoods $U_{X}$, $U_{\widetilde{X}}$,
$U_{Y}$, and $U_{\widetilde{Y}}$ of the heteroclinic points of the cycle in
Section 2.1.2, and of the balls $B(\widetilde{X},2\rho(\varepsilon))\subset
U_{\widetilde{X}}$ and $B(\widetilde{Y},2\rho(\varepsilon))\subset
U_{\widetilde{Y}}$ in (7.1).

Consider the subset
$U_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}$ of
$B(\widetilde{Y},2\rho(\varepsilon))\subset\Phi_{k}(U_{k})=U_{Q}$ of points
having the following itinerary for
$f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}$: a point $w\in
U_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}$ if

$w\in
f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}^{-(N_{2}+m_{k}+N_{1}+n_{k})}\big{(}B(\widetilde{Y},2\rho(\varepsilon))\big{)}\cap
B(\widetilde{Y},2\rho(\varepsilon))$

and it satisfies (see Figure 4):

(8.1)
$\begin{split}f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}^{i}(w)\,&\in\,U_{Q},\quad\mbox{for
every}\quad 0\leqslant i\leqslant n_{k},\\\
f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}^{n_{k}}(w)\,&\in\,U_{X},\\\
f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}^{N_{1}+n_{k}}(w)\,&\in\,B(\widetilde{X},2\rho(\varepsilon)),\\\
f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}^{j+N_{1}+n_{k}}(w)\,&\in\,U_{P},\quad\mbox{for
every}\quad 0\leqslant j\leqslant m_{k},\\\
f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}^{m_{k}+N_{1}+n_{k}}(w)\,&\in\,U_{Y},\,\mbox{and}\\\
f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}^{N_{2}+m_{k}+N_{1}+n_{k}}(w)\,&\in\,B(\widetilde{Y},2\rho(\varepsilon)).\end{split}$

We say that the points in
$U_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}$ are
$({\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)})$-admissible and
that $n_{k}+N_{1}+m_{k}+N_{2}$ is the
$({\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)})$-admissible return.
We now define the maps

$\begin{split}F_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}&\colon\Phi_{k}^{-1}(U_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)})\to\mathbb{R}^{3},\\\
F_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\Phi_{k}^{-1}\circ
f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}^{N_{2}+m_{k}+N_{1}+n_{k}}\circ\Phi_{k}.\end{split}$

\begin{overpic}[scale={0.13}]{hj.jpg} \put(95.0,80.0){\Large{$Q$}}
\put(25.0,105.0){\Large$n_{k}$} \put(92.0,209.0){\Large$N_{1}$}
\put(160.0,203.0){\Large$m_{k}$} \put(235.0,87.0){\Large$N_{2}$}
\put(180.0,150.0){\Large{$P$}} \end{overpic} Figure 4. The points in the
blender are admissible points

Recall that
$g_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}=\theta_{\varepsilon,k}\circ
f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}$ and that for the
points in $U_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}$ the map
$\theta_{\varepsilon,k}$ is the identity, so
$g_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}=f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}$
for points in $U_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}$.

######  Remark 8.1.

Every point of the blender is
$(\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon))$-admissible:

$\begin{split}\Upsilon_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}&\subset\bigcap_{j\in\mathbb{Z}}f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}^{j\,(N_{2}+m_{k}+N_{1}+n_{k})}(U_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)})\\\
&=\bigcap_{j\in\mathbb{Z}}g_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}^{j\,(N_{2}+m_{k}+N_{1}+n_{k})}(U_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}).\end{split}$

######  Notation 8.2.

In what follows, if there is no possibility of misunderstanding, we will
simply write:

  * •

$f_{\varepsilon,k,\mu}$ and $g_{\varepsilon,k,\mu}$ in the places of
$f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}$ and
$g_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}$,

  * •

$\Lambda_{\varepsilon,k,\mu}$ and $\Upsilon_{\varepsilon,k,\mu}$ in the place
of ${\Lambda}_{\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}$ and
$\Upsilon_{f_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}}$,

  * •

$\mathcal{R}_{\varepsilon,k,\mu}$ in the place of
$\mathcal{R}_{\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}$.

##  9\. Proof of Theorem 1.1: Intersections between the two-dimensional
manifolds

Throughout this and the next section, we will assume that
$f\in\mathcal{H}^{r}_{{\mathrm{BH}}}(M)$, $r\geqslant 2$, and consider the
perturbations $g_{\varepsilon,k,\mu}$ of $f$. By Proposition 7.2 the set
$\Upsilon_{\varepsilon,k,\mu}$ in Remark 8.1 is a blender of
$g_{\varepsilon,k,\mu}$.

We prove that the unstable manifolds of the blenders
$\Upsilon_{\varepsilon,k,\mu}$ of $g_{\varepsilon,k,\mu}$ transversely
intersects the stable manifold of the saddle $Q$.

######  Proposition 9.1.

For every small $\varepsilon>0$, large $k$, and $\mu\in(-10,-9)$ it holds

$W^{\mathrm{u}}\big{(}{\Upsilon}_{g_{\varepsilon,k,\mu}},g_{\varepsilon,k,\mu}\big{)}\pitchfork
W^{\mathrm{s}}\big{(}Q_{g_{\varepsilon,k,\mu}},g_{\varepsilon,k,\mu}\big{)}\neq\emptyset.$

The proof of this proposition is inspired by [23, Proposition 1, Chapter 6.4].
Here there are additional difficulties due to the heterodimensional nature of
the bifurcation. A comparison between the two settings is done in Section
9.1.1.

######  Notation 9.2.

Recall definitions of $f_{\ast}$, $F_{\ast}$, $\Upsilon_{\ast}$, and
$U_{\ast}$, with
$\ast=(\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon))$, in Section 8.
If there is no possible misunderstanding, in what follows $\varepsilon$ and
$\rho(\varepsilon)$ will be omitted, simply writing
$f_{\bar{\upsilon}_{k}(\mu)}$, $F_{\bar{\upsilon}_{k}(\mu)}$,
$\Upsilon_{\bar{\upsilon}_{k}(\mu)}$, and $U_{\bar{\upsilon}_{k}(\mu)}$.

By Proposition 7.2, the sequence of maps
$F_{\bar{\upsilon}_{k}(\mu)}=F_{\varepsilon,\bar{\upsilon}_{k}(\mu),\rho(\varepsilon)}$
converges (on compacta) to the family of endomorphisms
$G_{\xi,\mu,\bar{\eta}}$ for some fixed $\xi$ and $\bar{\eta}$. Hence we write
$G_{\mu}$ in the place of $G_{\xi,\mu,\bar{\eta}}$.

This section is organised as follows. In Section 9.1, we introduce an
auxiliary one-dimensional foliation $\mathcal{D}$ in $U_{P}$ which subfoliates
the reference domain of the blenders. The main property of this foliation is
that the strong unstable foliation of the blender approaches to $\mathcal{D}$.
We translate the foliation $\mathcal{D}$ to $U_{\widetilde{Y}}$ by
$f^{N_{2}}_{\bar{\upsilon}_{k}(\mu)}$ and thereafter by $\Phi_{k}^{-1}$ to
$\mathbb{R}^{3}$, obtaining in this way a foliation
$\widetilde{\mathcal{D}}_{\bar{\upsilon}_{k}(\mu)}$. In Lemma 9.1, we see that
the leaves of $\widetilde{\mathcal{D}}_{\bar{\upsilon}_{k}(\mu)}$ converge to
parabolas when $k\to\infty$. In Section 9.2, using the foliation
$\mathcal{D}$, for the admissible points and returns in Section 8, we study
the expansion of vectors and how the angles change, see Lemma 9.2. In Section
9.4, we translate these estimates for the map $F_{\bar{\upsilon}_{k}(\mu)}$.
In Section 9.3, we study the separatrices of the saddles of the blenders
nearby heterodimensional tangencies. Finally, in Section 9.5 we conclude the
proof of Proposition 9.1.

###  9.1. The auxiliary one dimensional foliation

We start with some preliminary constructions. We consider first auxiliary
foliations $\mathcal{F}^{{\mathrm{u}}}_{R,\bar{\upsilon}_{k}(\mu)}$, $R=P,Q$,
defined on the neighbourhoods $U_{P}$ and $U_{Q}$ as the natural extensions of
the invariant local manifolds of the saddles in the cycle. The leaf
$\mathcal{F}^{{\mathrm{u}}}_{P,\bar{\upsilon}_{k}(\mu)}(A)$ of the point $A\in
U_{P}$ of the foliation
$\mathcal{F}^{{\mathrm{u}}}_{P,\bar{\upsilon}_{k}(\mu)}$ is the intersection
of the set $U_{P}$ and the plane parallel to the coordinate plane $yz$
containing $A$. Similarly, the leaf
$\mathcal{F}^{{\mathrm{s}}}_{P,\bar{\upsilon}_{k}(\mu)}(A)$ of
$\mathcal{F}^{{\mathrm{s}}}_{P,\bar{\upsilon}_{k}(\mu)}$ is the intersection
of $U_{P}$ and the straight line parallel to the axis $x$ containing $A$. Thus
these leaves are “parallel” to $W^{\mathrm{u}}_{\mathrm{loc}}(P)$ and
$W^{\mathrm{s}}_{\mathrm{loc}}(P)$, respectively. The leaves of the foliations
$\mathcal{F}^{{\mathrm{u}}}_{Q,\bar{\upsilon}_{k}(\mu)}$ and
$\mathcal{F}^{{\mathrm{s}}}_{Q,\bar{\upsilon}_{k}(\mu)}$ are defined similarly
and are “parallel” to $W^{\mathrm{u}}_{\mathrm{loc}}(Q)$ and
$W^{\mathrm{s}}_{\mathrm{loc}}(Q)$. Note that the foliations
$\mathcal{F}^{\ast}_{P,Q,\bar{\upsilon}_{k}(\mu)}$,
$\ast={\mathrm{s}},{\mathrm{u}}$, do not depend on $\bar{\upsilon}_{k}(\mu)$.

For $\ast={\mathrm{s}},{\mathrm{u}}$, we “transport” the foliations
$\mathcal{F}^{\ast}_{Q,\bar{\upsilon}_{k}(\mu)}$ from $U_{X}\subset U_{Q}$ to
$U_{\widetilde{X}}\subset U_{P}$ by the transition
$f^{N_{1}}_{\bar{\upsilon}_{k}(\mu)}$ and continue denoting the resulting
foliations by $\mathcal{F}^{\ast}_{Q,\bar{\upsilon}_{k}(\mu)}$. Similarly, we
“transport” the foliations $\mathcal{F}^{\ast}_{P,\bar{\upsilon}_{k}(\mu)}$
from $U_{Y}\subset U_{P}$ to $U_{\widetilde{Y}}\subset U_{Q}$ by
$f^{N_{2}}_{\bar{\upsilon}_{k}(\mu)}$ and continue denoting these foliations
by $\mathcal{F}^{\ast}_{P,\bar{\upsilon}_{k}(\mu)}$. Note that these
extensions do depend on $\bar{\upsilon}_{k}(\mu)$.

We now consider an auxiliary one-dimensional foliation $\mathcal{D}$ in
$U_{P}$. For that consider the family of curves

$\ell_{(s,a)}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{\\{}\,(s,a,-a)+(0,t,t)\,:\,t\in\mathbb{R}\,\big{\\}}\cap
U_{P},\quad s,a\in\mathbb{R}.$

and define the diagonal foliation of $U_{P}=[-a_{P},a_{P}]^{3}$ by

(9.1) $\mathcal{D}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{\\{}\,\ell_{(s,a)}\,\colon\,a,s\in[-a_{P},a_{P}]\,\big{\\}}.$

Note that $\mathcal{D}$ “subfoliates” the leaves of
$\mathcal{F}^{{\mathrm{u}}}_{P,\bar{\upsilon}_{k}(\mu)}$ in $U_{P}$, see
Figure 5.

Consider the domain $\Delta_{k}(\mu)$ of the blender
$\Upsilon_{\bar{\upsilon}_{k}(\mu)}$ in (7.3). By [16, Lemma 3], for every
large $k$, the coordinates $(x,1+y,1+z)\in U_{P}$ of the points in
$f^{m_{k}+N_{1}+n_{k}}_{\bar{\upsilon}_{k}(\mu)}\big{(}\Delta_{k}(\mu)\big{)}$
are close to $Y=(0,1,1)$. Moreover, they have Landau symbols

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

These conditions and equation (6.3) imply that

(9.2)
$f^{m_{k}+N_{1}+n_{k}}_{\bar{\upsilon}_{k}(\mu)}(\Delta_{k}(\mu))\subset\bigcup_{s\in
J_{k}}\big{\\{}\,\ell_{(s,a)}\,:\,a\in[-a_{P},a_{P}]\,\big{\\}}$

where

(9.3) $J_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}[\,-\sigma_{Q}^{-2n_{k}}\,\sigma_{P}^{-2m_{k}}a_{P}\,,\,\sigma_{Q}^{-2n_{k}}\,\sigma_{P}^{-2m_{k}}a_{P}\,].$

As above, we consider the intersection of the leaves of $\mathcal{D}$ with
$U_{Y}$ and “transport” them by $f^{N_{2}}_{\bar{\upsilon}_{k}(\mu)}$,
obtaining the following foliation of $U_{\widetilde{Y}}$ (see Figure 5):

(9.4)
$\mathcal{D}_{\bar{\upsilon}_{k}(\mu)}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{\\{}\,\ell_{(s,a,\bar{\upsilon}_{k}(\mu))}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}f^{N_{2}}_{\bar{\upsilon}_{k}(\mu)}(\ell_{(s,a)}\cap
U_{Y})\,\,\colon\,a,s\in[-a_{P},a_{P}]\,\big{\\}}.$

Similarly, we let

(9.5)
$\widetilde{\mathcal{D}}_{\bar{\upsilon}_{k}(\mu)}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{\\{}\,\widetilde{\ell}_{(s,a,\bar{\upsilon}_{k}(\mu))}\stackrel{{\scriptstyle\scriptscriptstyle\rm