# Germ-typicality of the coexistence of infinitely many sinks

Pierre Berger111Partially supported by the ERC project 818737 _Emergence of
wild differentiable dynamical systems._ , Sylvain Crovisier222Partially
supported by the ERC project 692925 _NUHGD_., Enrique Pujals

###### Abstract

In the spirit of Kolmogorov typicality, we introduce the notion of _germ-
typicality_ : in a space of dynamics, it encompass all these phenomena that
occur for a dense and open subset of parameters of any generic parametrized
family of systems.

For any $2\leq r<\infty$, we prove that the Newhouse phenomenon (the
coexistence of infinitely many sinks) is locally $C^{r}$-germ-typical, nearby
a _dissipative bicycle_ : a dissipative homoclinic tangency linked to a
special heterodimensional cycle.

During the proof we show a result of independent interest: the stabilization
of some heterodimensional cycles for any regularity class
$r\in\\{1,\dots,\infty\\}\cup\\{\omega\\}$ by introducing a new
renormalization scheme. We also continue the study of the paradynamics done in
[Ber16, Ber17, BCP17] and prove that parablenders appear by unfolding some
heterodimensional cycles.

Figure 1: Bicycle

###### Contents

  1. 1 Concepts involved in the proof
    1. 1.1 Preliminaries
    2. 1.2 Homocycle
    3. 1.3 Heterocycles
    4. 1.4 Blenders
    5. 1.5 Bicycles and robust bicycles
    6. 1.6 Paraheterocycles
    7. 1.7 Parablenders
  2. 2 Structure of the proofs of the theorems
    1. 2.1 Proof of B
    2. 2.2 Proof of D
    3. 2.3 Proof of C: chains of heterocycles
    4. 2.4 Proof of A
  3. 3 From heterocycles to basic sets and strong heterocycles
    1. 3.1 Local coordinates for a heterocycle
    2. 3.2 Basic sets induced by a heterocycle
    3. 3.3 Proof of Proposition 2.1: from heterocycles to strong heterocycles
    4. 3.4 Proof of Proposition 2.1 in the analytic case
    5. 3.5 Proof of Proposition 2.3: from paraheterocycles to strong paraheterocycles
  4. 4 From chains of heterocycles to paraheterocycles
    1. 4.1 Notations and local coordinates
    2. 4.2 Compositions nearby a paraheterocycle
    3. 4.3 Families parametrized by $k$-parameters
  5. 5 Nearly affine (para)-blender renormalization
    1. 5.1 Coordinates for generic perturbations of strong heterocycles
    2. 5.2 Unfolding of the strong heterocycle
    3. 5.3 Choice and renormalization of inverse branches
    4. 5.4 Bounds on the renormalized maps
    5. 5.5 Tunning iterates
    6. 5.6  Proof of Proposition 2.2: from strong heterocycles to blenders
    7. 5.7 Proof of the Proposition 2.2 in the analytic case
    8. 5.8 Proof of Proposition 2.4: from strong paraheterocycles to parablenders

One of the most complex and rich phenomenon in differentiable dynamical
systems was discovered by Newhouse [New74, New79]. He showed the existence of
locally Baire-generic sets of dynamics displaying infinitely many sinks which
accumulate onto a Smale’s horseshoe (a stably embedded Bernoulli shift). This
property is the celebrated Newhouse phenomenon. It appears in many classes of
dynamics [Buz97, BD99, DNP06, Buz97, Dua08, Bie20]. Following Yoccoz, this
phenomenon provides more a lower bound on the wildness and complexity of the
dynamics, rather than a complete understanding on the dynamics. Indeed from
the topological or statistical viewpoints, these dynamics are presently
extremely far to be understood; it is not clear that the current dynamical
paradigm would even allow to state a description of such dynamics.

Since then the problem of the typicality of the Newhouse phenomenon has been
fundamental. But the notion of Baire-genericity among dynamical systems is a
priori independent to other notions of typicality involving probability. That
is why many important works and programs [TLY86, PT93, PS96, Pal00, Pal05,
Pal08, GK07] wondered if the complement of the Newhouse phenomenon could be
typical in some probabilistic senses inspired by Kolmogorov.

In his plenary talk ending the ICM 1954, Kolmogorov introduced the notion of
typicality for analytic or finitely differentiable dynamics of a manifold $M$.
He actually gave two definitions: one was designed to decide that a phenomenon
is negligible, the other one to decide that a phenomena is typical. He called
_negligible_ a phenomenon which only holds on a subset dynamics sent into a
Lebesgue null subset of $\mathbb{R}^{n}$ by a finite number of [non trivial]
real valued functionals $(\mathcal{F}_{i})_{0\leq i\leq n}$ on the space of
dynamics. To decide if a phenomenon $\mathcal{B}$ is _typical_ , he proposed
to start with a dynamics $f_{0}$ presenting the behavior, and then to consider
a deformation $f_{a}$ of the form

$f_{a}(z)=f_{0}(z)+a\cdot\phi(x,a)\;,$

where $\phi$ is a function of both $x$ and $a$, of the same regularity as $f$
(e.g. analytic, smooth or finitely differentiable). Then he called the
behavior $\mathcal{B}$ _typical_ , or _stably realizable_ if, for every $a$
small enough, the system $f_{a}$ displays this behavior. This was presented as
a criterion for detecting the importance of a phenomenon:

Any type of behavior of a dynamical system for which there exists at least one
example
of stable realization should be recognized as being important and not
negligible.

Kolmogorov, ICM 1954.

Recently, [Ber16, Ber17] showed the existence of locally Baire-generic sets of
$C^{r}$-families of dynamics $(f_{a})_{a\in\mathbb{R}^{k}}$, $r,k<\infty$,
such that for every $\|a\|\leq 1$, the map $f_{a}$ displays the Newhouse
phenomenon. In particular, this showed that the complement of the Newhouse
phenomenon is not typical for some interpretations of Kolmogorov’s typicality.
In [Ber17], it has been also conjectured that some dynamics with complex
statistical behavior should be typical in many senses.

In this work we show that the Newhouse phenomenon is typical according to the
following notion inspired by Kolmogorov idea and subsequent developements
[IL99, HK10, Mat03, KZ20]:

######  Definition 0.1 (Germ-typicality).

A behavior $\mathcal{B}$ is _$C^{r}$ -germ-typical_ in $\mathcal{U}\subset
C^{r}(M,M)$, if there exist a Baire-generic333i.e. a set which contains a
countable intersection of open and dense sets. set ${\mathcal{R}}$ in the
space of $C^{r}$-families444In Section 1.1, we will precise the topological
space of $C^{r}$-families involved. $\hat{f}=(f_{a})_{a\in\mathbb{R}}$ of maps
in $\cal U$ and a lower semi-continuous function
$\delta\colon{\mathcal{R}}\to(0,+\infty)$ such that for every
$\hat{f}\in\mathcal{R}$ and for all $|a|<\delta(\hat{f})$, the map $f_{a}$
presents the behavior $\mathcal{B}$.

To precise our setting, we consider an open set $U$ with boundary inside a
surface $M$ and the space $\mathrm{Diff}^{r}_{loc}(U,M)$ of local
$C^{r}$-diffeomorphisms from $\overline{U}$ to $M$. Working inside this space,
we reveal that any homoclinic configuration called _bicycle_ (see Fig. 1) is
included in the closure of these open sets $\cal U$:

######  Definition 0.2.

A local diffeomorphism displays a _bicycle_ if one of its saddle point has a
homoclinic tangency and a heterocycle. A saddle point $P$ displays a
_heterocycle_ if $W^{u}(P)$ contains a projectively hyperbolic source $S$ and
if the strong unstable manifold $W^{uu}(S)$ intersects $W^{s}(P)$. The bicycle
is _dissipative_ if the dynamics contracts the area along the orbit of $P$.

Since a bicycle is a simple configuration, in many cases it may be easy to
obtain, as we will see in Example 1.12 for the planar dynamics
$(x,y)\mapsto(x^{2}-2,y)$.

The main theorem of this work is the following:

######  Theorem A.

For every $2\leq r<\infty$ and for every local $C^{r}$-diffeomorphism of
surface $f\in\mathrm{Diff}^{r}_{loc}(U,M)$, which displays a dissipative
bicycle, there exists a (non empty) open set
$\mathcal{U}^{r}\subset\mathrm{Diff}^{r}_{loc}(U,M)$ whose closure contains
$f$ and where the Newhouse phenomenon is $C^{r}$-germ-typical.

Following Kolmogorov viewpoint, this theorem strengthens the evidence of the
importance of the Newhouse phenomenon. As we mentioned previously, [Ber16]
discovered the stable coexistence of infinitely many sinks for a generic
subset of an open set of parametrized families, i.e. showed that the Newhouse
phenomenon can not be neglected when one crosses a region of the space of
systems along some “well-chosen directions”. In comparison A goes one step
further and establishes the typicality of the Newhouse phenomeon in a sense
which only depends on a neighborhood inside the space of systems (and not on
the neighborhood of a specific family).

For the sake of clarity, we restrict the scope of the present work to the case
of surface local diffeomorphisms and to a notion of germ-typicality involving
only one-parameter families. However we are confident that our result could be
generalized to a broader setting: for instance [Ber17] applies also to
diffeomorphisms in higher dimension and to any finitely dimensional parameter
families.

### Locus of robust phenomena: stabilization of heterodimensional cycles

Another main point of the present work is to bring to light a very simple
configuration nearby which germ-typicallity of the Newhouse phenomenon holds
true. The idea to associate a homoclinic configuration to a phenomenon goes
back to Newhouse. In [New70], he first showed that it is possible to get a
(non-empty) open set of diffeomorphisms of surface exhibiting homoclinic
tangencies (these diffeomorphisms exhibit _$C^{2}$ -robust homoclinic
tangencies_), and then in [New74] that this open set can encompasses a Baire-
generic subset formed by dynamics displaying the Newhouse phenomenon
(infinitely many attracting cycles). To obtain such open sets of
diffeomorphisms with robust homoclinic tangencies, Newhouse considered
horseshoes with large fractal dimension (large thickness in his own
nomenclature). Later, in [New79], Newhouse proved that from [the configuration
defined by] a homoclinic tangency, a perturbation of the dynamics displays a
robust homoclinic tangency (see Theorem 1.1).

For local diffeomorphisms, these thick horseshoes can be replaced by more
topological object, called _blenders_. They were introduced by Bonatti and
Diaz [BD96] for diffeomorphisms in dimension larger than or equal to three and
can be recasted in the context of local diffeomorphisms of surface as
hyperbolic compact sets such that the union of their local unstable manifolds
covers $C^{r}$-robustly a (non-empty open set of the surface (see Definition
1.4). In the same spirit as Newhouse’s work, one can wonder, nearby which
homoclinic configurations blenders appear. Bonatti, Diaz and Kiriki [BDK12]
proved that heterodimensional cycles (which, in the case of local
diffeomorphisms of surface correspond to cycles between a saddle and a source)
play that role when one considers the $C^{1}$-topology: a $C^{1}$-perturbation
of the heterodimensional cycle generates open sets of dynamics exhibiting
blenders and $C^{1}$-robust heterodimensional cycles. In the present paper, we
extend this result to the context of more regular dynamics:

######  Theorem B.

For every $1\leq r\leq\infty$ or $r=\omega$, consider
$f\in\mathrm{Diff}_{loc}^{r}(U,M)$ exhibiting a heterocycle associated to a
saddle $P$. Then there exists $\tilde{f}$, that is $C^{r}$-close to $f$, with
a basic set $K$ containing the hyperbolic continuation of $P$, and which has a
$C^{r}$-robust heterocycle.

While communicating our result, Li and Turaev have informed us that they
independently achieved to prove a more general version of Theorem B for higher
dimensional systems, by using different techniques [LT]. Diaz and Perez have
also recently obtained [DP] a similar stabilization of heterodimensional
cycles for $C^{r}$-diffeomorphisms in dimension $3$, assuming in addition that
one of the periodic points exhibits a homoclinic tangency.

### Renormalization nearby heterocycles

In order to prove Theorem B (in Section 2.1), we first show in Proposition 2.1
that nearby heterocycles there are heterocycles satisfying an additional
property. These configurations are called _strong heterocycles_ and are
defined in Definition 1.3. Then Proposition 2.2 introduces a renormalization
nearby strong heterocycles to obtain nearly affine blenders.

This renormalization consists in selecting two inverses branches $g^{+}$ and
$g^{-}$ of larges iterates of the dynamics, which are defined on boxes nearby
the heterocycle and then to rescale $\mathcal{R}g^{-}=\phi^{-1}\circ
g^{-}\circ\phi$, $\mathcal{R}g^{+}=\phi^{-1}\circ g^{+}\circ\phi$ the two
latter branches via a same coordinate change $\phi$. The maps
$\mathcal{R}g^{-},\mathcal{R}g^{+}$ are close to affine maps and define a
blender, which will be called _nearly affine blender_ , see Definition 1.7.

Theorem B is restated more precisely in Section 1.4. Propositions 2.1 and 2.2
are proved in respectively Sections 3 and 5. This renormalization is one of
the main technical novelty of the present work. It is further developed to
obtain D (in Section 1.7), a parametric counterpart of Theorem B. D is
essential to prove Theorem A. It states that nearby paraheterocycles there are
nearly affine parablenders. These are objects of paradynamics.

### Paradynamics

To explain the role of these parametric blenders we have to go back to the
paper [Ber16]: it considered parameter families of local diffeomorphisms on
surfaces and introduced the notion of _paratangencies_ : a homoclinic tangency
that is “sticky” (or unfolded in “slow motion”). That phenomenon implies that
the attracting periodic points created by the unfolding of the tangency have
“a long life in the parameter space”. Moreover, if any perturbation of a
parameter family still exhibits a dissipative homoclinic paratangency for all
parameters (in other words the family exhibits _robust homoclinic
paratangencies_ , the analog in the space parameter families of the robust
homoclinic tangencies in the space of local diffeomorphisms) then, after small
perturbation, the new family displays infinitely many attracting periodic
points for all parameters (see Lemmas 2.16 and 2.15).

To provide robust paratagencies, [Ber16] introduced a parametric version of
the blenders, called _$C^{r}$ -parablenders_, see Definition 1.17. To grasp
the idea behind this notion, first recall that any hyperbolic compact set of a
map has a unique continuation for a nearby system. Any point in the hyperbolic
set has a unique continuation as well (see Section 1.1 for details) and the
same holds true for its local stable and unstable manifolds. When the
parameter family is of class $C^{r}$, the continuation of a point defines a
curve of class $C^{r}$. The key property of a $C^{r}$-parablender, is that for
an open set of parametrized points in the surface, of the local unstable
manifold of the parablender moves in slow motion with respect to the
parametrized point. This property can be push forward to the unfolding of
homoclinic tangencies and allows to create robust homoclinic paratangencies.
For that purpose, it is easier to assume that the collection of local unstable
manifolds covers a source homoclinically linked to the parablender.

In [BCP17], the notion of parablender has been recasted: parameter families of
maps naturally induce an action on $C^{r}$-jets and the parablenders can be
viewed as blenders for this dynamics on the space of jets. This viewpoint
allowed us to systematize the construction of parablenders: in [BCP17], using
Iterated Function Systems, a special type of parablenders called _nearly
affine parablenders_ (see Definition 1.18) is introduced.

In the present paper, we tried to follow Newhouse’s approach and looked for a
simple bifurcation that generates “robust paratangencies”. According to
[Ber17], it suffices to obtain a parablender covering a source and linked to a
dissipative homoclinic tangency. Similarly to [BDK12], one can wonder if the
parametric unfolding of a heterodimensional cycle may generate a parablender.
We answer by proving that the unfolding of a homoclinic tangency related to a
heterocycle (a _bicycle_) is the sough configuration which produces robust
paratangencies.

To precise, first we prove that combining a homoclinic tangency with the
heterocycle, one obtains alternate chain of heterocycles (a chain of
heterocycles involving saddles with negative eigenvalues, see Definition 2.7).
The unfolding of that special chain produces a _paraheterocycle_ (a
heterocycle that is unfolded in “slow motion”, see Definition 1.15 and theorem
C) and which then gives birth to nearly affine parablenders (see theorem D)
using the aforementioned renormalization technique.

### Open problems

Paradynamics has been useful to prove that several complex and interesting
phenomena are robust along a locally Baire-generic set of families of
dynamics, see [Ber, IS17, BRb, BRa]. The tools brought by our work should
enable to show the $C^{r}$-germ-typicality of these phenomena.

Note that if a behavior $\mathcal{B}$ is $C^{r}$-germ-typical in $\mathcal{U}$
then it occurs on an open and dense set of parameters for a Baire-generic set
of $C^{r}$-families $(f_{a})_{a}$ of dynamics $f_{a}\in\cal U$. But it does
not imply that the Lebesgue measure of this open and dense set of parameters
is full. In particular, it remains open whether the Newhouse phenomenon is
locally typical with respect tosome interpretations of Kolmogorov typicality
given by [HK10], [Ber20] or [IL99, Chapter 2, section 1]. The latter is
slightly stronger than:

######  Definition 0.3 (Arnold prevalence (soft version)).

For $r,k\geq 1$, a behavior $\mathcal{B}$ is $C^{r}$-$k$-Arnold prevalent in
$\mathcal{U}\subset C^{r}(M,M)$, if there exists a Baire-generic set
$\mathcal{R}$ of $C^{r}$-families $(f_{a})_{a\in\mathbb{R}^{k}}$ formed by
maps $f_{a}\in\cal U$ such that for every $(f_{a})_{a}\in\mathcal{R}$, for
Lebesgue almost every parameter $a\in\mathbb{R}^{k}$ , $f_{a}$ presents the
behavior $\mathcal{B}$.

Another notion of typicality has been introduced by Hunt, York and Sauer
[HSY92], and then developed by Kaloshin-Hunt in [HK10]; it was used by
Gorodeski-Kaloshin [GK07] to study the typicality of the Newhouse phenomenon,
but leaves open the problem of the typicality of Newhouse phenomenon following
the latter notions555The original notion of typicality defined by [HSY92] is
defined for Banach spaces; its counterpart for Banach manifolds (such as the
space of dynamics on a compact manifold) is so far not unique (there is no
version of this notion which is invariant by coordinate change, contrarily to
germ-typicality or Arnold prevalence)..

Finally let us emphasize that the important Arnold-prevalence or the germ-
typicality of the Newhouse phenomenon are open for the $C^{\infty}$ or
analytic topologies. Hopefully the tools developed in this present work seem
to us useful to progress on these important problems.

##  1 Concepts involved in the proof

In this section we state the main results which are used to obtain A.

In Section 1.1 we recall classical definitions about hyperbolicity in the
particular context of local diffeomorphisms. In Section 1.2 andSection 1.3 we
recall the concepts of homoclinic tangency and heterodymensional cycle between
fixed points with different indices and the classical results of Newhouse and
Bonatti-Diaz associated to these bifurcations. In Section 1.4 we recall the
notions of blenders and nearly affine blenders and we state the main theorem
that relates cycles and blenders (B). In Section 1.5, we state precisely the
definition of bicycle (that combines a homocycle and a heterocycle) and we
show in Corollary B′ that from bicycles one can obtain robust bicycles (it is
worth to mention that this is done in any $C^{r}$-regularity including the
analytic one).

In Section 1.6 and Section 1.7 we give the parametric version of the previous
results. In Section 1.6 we introduce the notion of paraheterocycle and explain
how by unfolding heterocycles associated to saddles with negative eigenvalues
one can obtain a paraheterocycle (C). In Section 1.7 we introduce the notions
of affine and nearly affine parablenders and explain how they emerge from
paraheterocycles (D).

###  1.1 Preliminaries

In the following $M$ is a compact surface, $U$ an open subset whose boundary
is a smooth submanifold and $\mathrm{Diff}_{loc}^{r}(U,M)$ for
$r\in\mathbb{N}\cup\\{\infty\\}$, denotes the restrictions to $U$ of
$C^{r}$-map $f\colon\overline{U}\to M$ whose differential $D_{x}f$ is
invertible at every $x\in\overline{U}$. Endowed with the $C^{r}$-topology,
this is a Baire space.

For some results, one will also assume that $M$ is a real analytic surface and
let $\tilde{M}$ be a complex extension. One then considers the space
$\mathrm{Diff}_{loc}^{\omega}(U,M)$ of real analytic maps endowed with the
analytic topology defined as the inductive limit of the spaces of holomorphic
maps defined on neighborhoods of $M$ in $\tilde{M}$.

Now let us precise the space of $C^{r}$-families parametrized by the interval
$\mathbb{I}=(-1,1)$. For the sake of clarity, we will focus only on the space
$\mathcal{D}^{r}(\mathbb{I}\times U,M)$ of families $(f_{a})_{a\in\mathbb{I}}$
which are the restriction of a map $(a,x)\mapsto f_{a}(x)$ of class $C^{r}$ on
$\overline{\mathbb{I}}\times\overline{U}$, that we endow with the uniform
$C^{r}$-topology. However all our arguments will be also valid for the smaller
space $C^{r}(\overline{\mathbb{I}},\mathrm{Diff}^{r}_{loc}(U,M))$ endowed with
the topology of $C^{r}$-maps from $\overline{\mathbb{I}}$ into
$\mathrm{Diff}^{r}_{loc}(U,M)$.

An _inverse branch_ for $f\in\mathrm{Diff}_{loc}^{r}(U,M)$ is the inverse of a
restriction $f|V$ of $f$ to a domain $V\subset U$ such that $f^{n}|V$ is a
diffeomorphism onto its image.

A compact set $K$ is _(saddle) hyperbolic_ for $f$ if it is $f$-invariant
(i.e. $f(K)=K$) and there exists a continuous, $Df$-invariant subbundle
$E^{s}$ of $TM|K$ which is uniformly contracted and normally uniformly
expanded. More precisely, there exists $N\geq 1$ satisfying:

$\|D_{z}f^{N}|E^{s}_{z}\|<1/2\quad\text{and}\quad\|p_{E^{s\bot}}\circ
D_{z}f^{N}(v)\|\geq 2\|v\|\;,\quad\forall z\in K,\;v\in E^{s\bot}_{z}\;,$

where $E^{s\bot}$ is the subbundle of $TM|K$ equal to the orthogonal
complement of $E^{s}_{z}$ and $p_{E^{s\bot}}$ the orthogonal projection onto
it. The hyperbolic set $K$ is a _basic set_ if it is transitive and locally
maximal. Then $K$ is equal to the closure of its subset of periodic points.

Any point $x\in K$ has a stable manifold $W^{s}(x)$ (also denoted
$W^{s}(x;f)$) which is an injectively immersed curve. The map $f|K$ being in
general not injective, a single point $x\in K$ has in general as many unstable
manifolds as preorbits $\underline{x}$. We denote such a submanifold by
$W^{u}(\underline{x})$, or $W^{u}(\underline{x};f)$. The space of preorbits
$\underline{x}$ is denoted by
$\overleftarrow{K}:=\\{\underline{x}=(x_{i})_{i\leq 0}\in
K^{\mathbb{Z}^{-}}:\;f(x_{i-1})=x_{i}\\}$. The space $\overleftarrow{K}$ is
canonically endowed with the product topology. The zero-coordinate projection
is denoted by $\pi_{f}\colon\overleftarrow{K}\to M$; it semi-conjugates the
shift dynamics $\sigma$ on $\overleftarrow{K}$ with $f$.

It is well known (see for instance [BR13]) that a hyperbolic compact set is
_$C^{1}$ -inverse limit stable_: for every $C^{1}$-perturbation $f^{\prime}$
of $f$, there exists a (unique) map
$\pi_{f^{\prime}}\colon\overleftarrow{K}\to M$ which is $C^{0}$-close to
$\pi_{f^{\prime}}$ and so that:

$\pi_{f^{\prime}}\circ\sigma=f^{\prime}\circ\pi_{f^{\prime}}\;.$

The image $K_{f^{\prime}}:=\pi_{f^{\prime}}(\overleftarrow{K})$ is also a
hyperbolic set. Note that $K_{f}=K$. Also $K_{f^{\prime}}$ is called the
_hyperbolic continuation_ of $K$.

Two basic sets are _(homoclinically) related_ if there exists an unstable
manifold of the first which has a transverse intersection point with a stable
manifold of the second, and vice-versa. Then by the Inclination Lemma, the
local unstable manifolds of one basic set are dense in the unstable manifolds
of the other.

An $f$-invariant compact space is _projectively hyperbolic expanding_ if there
exists a continuous $Df$-invariant subbundle $E^{cu}$ of $TM|K$ which is
uniformly expanded and normally uniformly expanded. More precisely, there
exists $N\geq 1$ satisfying:

$\|D_{z}f^{N}|E^{cu}_{z}\|>2\quad\text{and}\quad\|p_{E^{cu\bot}}\circ
D_{z}f^{N}(v)\|\geq 2\cdot\|v\|\cdot\|D_{z}f^{N}|E^{cu}_{z}\|\;,\quad\forall
z\in K,\;v\in E^{cu\bot}_{z}.$

If it is transitive and locally maximal, it is equal to the closure of its
subset of periodic points. To any $\underline{x}\in\overleftarrow{K}$, one
associates a strong unstable manifold $W^{uu}(\underline{x})$ as the set of
points which converge to the orbit of $\underline{x}$ in the past
transversally to the bundle $E^{cu}$.

A saddle periodic point $P$ of period $p\geq 1$, is _dissipative_ if
$|\mathrm{det}D_{P}f^{p}|<1$.

A source periodic point $S$ is _projectively hyperbolic_ if the tangent space
at $S$ split into two $Df$-invariant directions, $T_{S}M=E^{cu}\oplus E^{uu}$,
the direction $E^{cu}$ –called _center unstable_ – being less expanded than
the direction $E^{uu}$ –called _strong unstable_. Its strong unstable manifold
$W^{uu}(S)$ is the set of points which converge to the orbit of $S$ in the
past in the direction of $E^{uu}$.

###  1.2 Homocycle

Given $f\in\mathrm{Diff}_{loc}^{r}(U,M)$, a saddle periodic point $P\in U$ has
a _homoclinic tangency_ or _homocycle_ for short, if its stable manifold has a
non-transverse intersection point $T\in U$ with its unstable manifold.

(Homocycle) $\exists T\in TW^{s}(P)\cap TW^{u}(P)\;.$

More generally, a basic set $K\subset U$ has a homoclinic tangency if there
exist $P\in K$ and $\underline{Q}\in\overleftarrow{K}$ (not necessarily
periodic) such that $W^{s}(P)$ is tangent to $W^{u}(Q)$. A basic set $K$ has a
_$C^{r}$ -robust homoclinic tangency_ if for every $C^{r}$-perturbation of the
dynamics, the hyperbolic continuation of $K$ still has a homoclinic tangency.
If $r\geq 2$ and if the phase space is a surface, the tangency $T$ is
_quadratic_ , if the curvature of $W^{s}(P)$ and $W^{u}(\underline{Q})$ at $T$
are not equal.

Figure 2: Homocycle

Here is a famous theorem by Newhouse [New79], which stabilizes the homoclinic
tangencies.

######  Theorem 1.1 (Newhouse).

For $2\leq r\leq\infty$ or $r=\omega$, consider
$f\in\mathrm{Diff}_{loc}^{r}(U,M)$ and a saddle periodic point $P$ exhibiting
a homoclinic tangency $T$. Then there exists $\tilde{f}$ $C^{r}$-close to $f$,
with a basic set $K$ containing the hyperbolic continuation of $P$, and which
has a $C^{r}$-robust homoclinic tangency.

The open set $\mathcal{N}^{r}$ of dynamics displaying a $C^{r}$-robust
homoclinic tangency is called the _Newhouse domain_. We denote by
$\mathcal{N}^{r}(P)\subset\mathcal{N}^{r}$ the open set of dynamics for which
the hyperbolic continuation of $P$ belongs to a basic set displaying a
$C^{r}$-robust homoclinic tangency. By the Inclination Lemma, the stable and
unstable manifolds of $P$ are dense in the stable and unstable sets of $K$.
Thus a $C^{r}$-small perturbation of any dynamics in $\mathcal{N}^{r}(P)$
creates a homoclinic tangency for $P$. This proves:

######  Proposition 1.2.

For every $1\leq r\leq\infty$ or $r=\omega$, there exists a $C^{r}$-dense set
in $\mathcal{N}^{r}(P)$, made by maps for which the hyperbolic continuation of
$P$ has a homoclinic tangency.

Let $\mathcal{N}_{diss}^{r}(P)\subset\mathcal{N}(P)$ be the open set formed by
dynamics for which the hyperbolic continuation of $P$ is dissipative. As a
periodic sink of arbitrarily large period can be obtained by a small
perturbation of a dissipative homoclinic tangency, the latter proposition then
implies the Baire-genericity in $\mathcal{N}_{diss}^{r}(P)$ of dynamics
exhibiting a Newhouse phenomenon (see [New74] for more details).

###  1.3 Heterocycles

In the present section we first recast for the case of surface endomorphisms,
the notion of heterodimensional cycle introduced in [D9́5, BD96], and present
two stronger versions of it called _heterocycle_ and _strong heterocycle_.

######  Definition 1.3.

A map $f\in\mathrm{Diff}_{loc}(U,M)$ displays a _heterodimensional cycle_ if
it has a saddle periodic point $P$ and a periodic source $S$ such that
$W^{u}(S)$ intersects $W^{s}(P)$ and $S$ is in $W^{u}(P)$:

(Heterodimensional cycle) $S\in W^{u}(P)\quad\text{and}\quad W^{s}(P)\cap
W^{u}(S)\not=\varnothing\;.$

The heterodimensional cycle forms a _heterocycle_ if the source is
projectively hyperbolic and $W^{uu}(S)$ intersects $W^{s}(P)$:

(Heterocycle) $S\in W^{u}(P)\quad\text{and}\quad W^{s}(P)\cap
W^{uu}(S)\not=\varnothing\;.$

This heterocycle is _strong_ if furthermore $W^{uu}(S)$ contains $P$:

(Strong heterocycle) $S\in W^{u}(P)\quad\mathrm{and}\quad P\in W^{uu}(S)\;.$

We will see in Proposition 2.1 that any map displaying a heterocycle can be
smoothly perturbed to display a strong heterocycle between a saddle point
$P^{\prime}$ homoclinically related to the initial one $P$, and the initial
source $S$.

Figure 3: Heterocycle for a surface map.

A heterocycle is a one-codimensional phenomenon. To show its local density, we
shall generalize it as follow. A basic set $K$ and a projectively hyperbolic
periodic source $S$ of a surface map display a _heterocycle_ if there exists
$P\in K$ (not necessary periodic) such that $W^{s}(P)\cap
W^{uu}(S)\not=\varnothing$ and there exists
$\underline{P}\in\overleftarrow{K}$ such that $P=\pi_{f}(\underline{P})$ and
$S\in W^{u}(\underline{P})$. The _heterocycle is $C^{r}$-robust_ if for every
$C^{r}$-perturbation of the dynamics, the hyperbolic continuations of $K$ and
$S$ still have a heterocycle. The $C^{r}$-open set of surface maps which
display a robust heterocycle is called the _Bonatti-Diaz domain_ and is
denoted by $\mathcal{BD}^{r}$. We denote by
$\mathcal{BD}^{r}(P,S)\subset\mathcal{BD}^{r}$ the open set of dynamics for
which the hyperbolic continuation of $P$ belongs to a basic set displaying a
$C^{r}$-robust heterocycle with the continuation of $S$.

###  1.4 Blenders

Let us again consider a robust heterocycle between a basic set $K$ and a
source $S$. As by a perturbation of the dynamics, $S$ can be moved
independently of $K$ and its unstable manifold, this implies that $K$ must be
a _blender_ :

######  Definition 1.4 ($C^{r}$-Blender).

A _$C^{r}$ -blender_ for $f\in\mathrm{Diff}_{loc}^{r}(U,M)$ is a basic set $K$
such that the union of its local unstable manifolds has $C^{r}$-robustly non-
empty interior: there exists a continuous family of local unstable manifolds
whose union contains a non-empty open set $V\subset U$ and the same holds true
for their continuations for any $C^{r}$-perturbations $\tilde{f}$ of $f$.

The set $V$ is called an _activation domain_ of the blender $K$.

As the periodic points are dense in $K$, the unstable manifolds of periodic
points are also dense in the activation domain. Hence for a small
$C^{r}$-perturbation supported by a small neighborhood of the blender, there
exists a periodic point whose unstable manifold contains the source, defining
a heterocycle. This proves the following counterpart of Proposition 1.2:

######  Proposition 1.5.

For every $1\leq r\leq\infty$ or $r=\omega$, there exists a $C^{r}$-dense set
in $\mathcal{BD}^{r}(P,S)$ made by maps for which the hyperbolic continuation
of $P$ and $S$ have a heterocycle.

Bonatti and Diaz have introduced the notion of blender and obtained the first
semi-local constructions of robust heterocycles [BD96].

######  Question 1.6.

All the known $C^{r}$-blenders are also $C^{1}$-blender. _Is
$\mathcal{BD}^{r}$ equal to $\mathcal{BD}^{1}$?_

The following notion has been introduced in [BCP17] and will play a key role
in a renormalization that we will perform nearby heterocycles.

######  Definition 1.7 (Nearly affine blender).

For $r\in[1,\infty)$, $\Delta>1$, $x_{0}\in(-2,2)$, $\delta>0$, $f$ has a
_$\delta$ -$C^{r}$-nearly affine blender_ with contraction $\Delta^{-1}$ if
there is a $C^{r}$-chart $H\colon\mathbb{R}^{2}\hookrightarrow M$ such that:

  * –

there is an inverse branch $g^{+}$ of an iterate $f^{N^{+}}$ of $f$ such that
${\mathcal{R}}g^{+}:=H^{-1}\circ g^{+}\circ H$ is well defined on $[-2,2]^{2}$
and is $\delta$-$C^{r}$-close to $(x,y)\mapsto(x_{0},\Delta(y-1)+1)$;

  * –

there is an inverse branch $g^{-}$ of an iterate $f^{N^{-}}$ of $f$ such that
${\mathcal{R}}g^{-}:=H^{-1}\circ g^{-}\circ H$ is well defined on $[-2,2]^{2}$
and is $\delta$-$C^{r}$-close to $(x,y)\mapsto(x_{0},\Delta(y+1)-1)$.

Observe that the maximal invariant set of the map:

$({\mathcal{R}}g^{+})^{-1}\sqcup({\mathcal{R}}g^{-})^{-1}:{\mathcal{R}}g^{+}([-2,2]^{2})\sqcup({\mathcal{R}}g^{-}([-2,2]^{2})\to\mathbb{R}^{2}$

is a basic set $K$. The following is easy, see for instance [BCP17, Section 6]
for details.

######  Proposition 1.8.

For every $\Delta>1$ close to $1$, $x_{0}\in(-2,2)$ and $\eta\in(0,1)$, if
$\delta>0$ is sufficiently small, then the set $K$ is a $C^{1}$-blender and
$(-2,2)\times\left[-1+\eta,1-\eta\right]$ is an activation domain.

In Section 2 we will prove the following analogous of Newhouse Theorem 1.1,
which stabilizes the heterocycles. It will be obtained by introducing a
renormalization for a perturbation of $f$ leading to a nearly affine blender.

######  Theorem B.

For every $1\leq r\leq\infty$ or $r=\omega$, consider
$f\in\mathrm{Diff}_{loc}^{r}(U,M)$ exhibiting a heterocycle formed by a saddle
$P$ and a projectively hyperbolic source $S$. Then for every $\delta>0$ and
any number $\rho\leq r$, there exists $\tilde{f}$, $C^{r}$-close to $f$, such
that $P_{\tilde{f}}$ is homoclinically related to a $\delta$-$C^{\rho}$-nearly
affine blender whose activation domain contains $S_{\tilde{f}}$.

######  Question 1.9.

To what extend the previous results generalize to heterodimensional cycles?

In that direction, [BDK12] proved for diffeomorphisms that it is possible to
stabilize by $C^{1}$-perturbation any _classical_ heterodimensional cycle
between saddles whose stable dimension differs by one, provided that at least
one of the saddle involved in the cycle belongs to a nontrivial hyperbolic
set. An analogue in any regularity class is done in [LT].

###  1.5 Bicycles and robust bicycles

Let us precise the definition of bicycle mentioned in the introduction:

######  Definition 1.10.

A saddle $P$ and a projectively hyperbolic source $S$ display a _bicycle_ if
they form a heterocycle and if $P$ has a homocycle. The bicycle is
_dissipative_ if the orbit of $P$ is dissipative.

The notion of bicycle can be extended to basic sets.

######  Definition 1.11.

A basic set for $f\in\mathrm{Diff}_{loc}^{r}(U,M)$ displays a _$C^{r}$ -robust
bicycle_ if it displays a $C^{r}$-robust homocycle and forms a $C^{r}$-robust
heterocycle with a projectively hyperbolic source.

It is easy to build a bicycle by perturbation of some explicit example:

######  Example 1.12.

For every $r\geq 2$, the map $f:=(x,y)\in\mathbb{R}^{2}\mapsto(x^{2}-2,y)$ is
$C^{r}$-accumulated by maps $f_{\varepsilon}$ exhibiting a bicycle. Hence by
A, there is an open set of $C^{r}$-perturbations $\mathcal{U}^{r}$ of
$f_{\varepsilon}$ in which the coexistence of infinitely many sinks is
$C^{r}$-germ-typical.

###### Proof of Example 1.12.

First, we choose the parameter $a$ close to $-2$ such that the map
$g(x)=x^{2}+a$ admits two homoclinically related repelling periodic points
$s$, $p$, the orbit of the critical point contains $p$ (there exists $n\geq 1$
such that $g^{n}(0)=p$) and belongs to the unstable set of $p$ (there exists a
sequence of backward iterates of $0$ which accumulates on $p$): usually such a
parameter $a$ is called a Misiurewicz parameter).

Then we consider a function $\rho$ close to $1$ which is equal to
$1+\varepsilon$ on a small neighborhood of the orbit of $s$ and to
$1-\varepsilon$ in a small neighborhood of the orbit of $p$. We now consider
the following small perturbation of $f$:

$f_{a,\varepsilon}(x,y)\mapsto(x^{2}+a,\rho(x)y).$

Observe that it has a projectively hyperbolic source $S:=(s,0)$ and
dissipative saddle point $P:=(p,0)$, such that the unstable manifold of each
point intersects the other point and the image of the critical point still is
preperiodic. One now can perform a small perturbation in a neighborhood of the
critical point that makes the map a local diffeomorphism and that also
preserves the image of the critical point, which then defines a homoclinic
tangency of $p$. In a such way, one obtains a map with a bicycle involving $P$
and $S$. ∎

Similarly to Proposition 1.2 and Proposition 1.5 we have:

######  Proposition 1.13.

For every $1\leq r\leq\infty$ or $r=\omega$, consider an open set of maps
$f\in\mathrm{Diff}_{loc}^{r}(U,M)$ displaying a $C^{r}$-robust bicycle
involving a saddle $P$ and a projectively hyperbolic source $S$. It contains a
$C^{r}$-dense subset of maps for which the hyperbolic continuation of $P$ and
$S$ form a bicycle.

Combining Theorems 1.1 and B, one can stabilize the bicycles:

######  Corollary B′.

For $2\leq r\leq\infty$ or $r=\omega$, consider
$f\in\mathrm{Diff}_{loc}^{r}(U,M)$ and a saddle $P$ exhibiting a bicycle. Then
there exists $\tilde{f}$, $C^{r}$-close to $f$, with a hyperbolic basic set
$K$ containing the hyperbolic continuation of $P$ which exhibits a
$C^{r}$-robust bicycle.

###  1.6 Paraheterocycles

Let us fix $1\leq r\leq\infty$, and a $C^{r}$-family
$(f_{a})_{a\in\mathbb{R}}$ of local diffeomorphisms
$f_{a}\in\mathrm{Diff}_{loc}^{r}(U,M)$.

##### Hyperbolic sets for families of dynamics

It is well known that if $f_{0}$ has a hyperbolic fixed point $P$, then its
hyperbolic continuation $(P_{a})_{a\in I}$ is a $C^{r}$ function of the
parameter $a$ on a neighborhood $I\subset\mathbb{R}$ of $0$. More generally,
if $K$ is a hyperbolic set for $f_{0}$, with $\overleftarrow{K}$ the inverse
limit of $K$, its hyperbolic continuation $(K_{a})_{a\in I}$ by the range
$K_{a}=\pi_{a}(\overleftarrow{K})$ of a family of maps
$\pi_{a}:=\pi_{f_{a}}:\overleftarrow{K}\to M$ (see Section 1.1) with the
following regularity:

######  Proposition 1.14 (see Prop 3.6 [Ber16]).

There exists a neighborhood $I$ of $0$ where $(\pi_{a})_{a\in I}$ is well
defined. For any $\underline{z}\in\overleftarrow{K}$, the map $a\in
I\mapsto\pi_{a}(\underline{z})$ is of class $C^{r}$ and depends continuously
on $\underline{z}$ in the $C^{r}$-topology.

The local stable and unstable manifolds $W^{s}_{loc}(z;f_{a})$ and
$W^{u}_{loc}(\underline{z};f_{a})$ are canonically chosen so that they depend
continuously on $a$, $z$ and $\underline{z}$ in the $C^{r}$-topology (see Prop
3.6 in [Ber16]). They are called the _hyperbolic continuations_ of
$W^{s}_{loc}(z;f_{0})$ and $W^{u}_{loc}(\underline{z};f_{0})$ for $f_{a}$.

######  Definition 1.15 (Paraheterocycle).

Given $0\leq d\leq r$, the family $(f_{a})_{a\in\mathbb{R}}$ displays a
_$C^{d}$ -paraheterocycle_ at $a_{0}$ if there exist a heterocycle for
$f_{a_{0}}$ involving a saddle $P$ and a projectively hyperbolic source $S$
whose hyperbolic continuations satisfy for some $N\geq 0$

(1.1)
$d(S_{a},f^{N}_{a}(W^{u}_{loc}(P_{a})))=o(|a-a_{0}|^{d^{\prime}}),\quad\text{
for any integer $0\leq d^{\prime}\leq d$.}$

We say it is a _strong $C^{d}$-paraheterocycle_ if furthermore $P$, $S$ form a
strong heterocycle.

Note that if $f_{a_{0}}$ has a heterocycle then $(f_{a})_{a}$ has a
$C^{0}$-paraheterocycle at $a=a_{0}$.

######  Theorem C.

Consider a $C^{\infty}$ family of local diffeomorphisms
$(f_{a})_{a\in\mathbb{R}}$ in $\mathrm{Diff}_{loc}^{\infty}(U,M)$ and a
heterocycle for $f_{0}$ between a saddle point $P$ with period $p$ and a
projectively hyperbolic source $S$. Let us assume furthermore that the stable
eigenvalue of $D_{P}f^{p}_{0}$ is negative.

Then there exists a family $(\tilde{f}_{a})_{a\in\mathbb{R}}$,
$C^{\infty}$-close to $(f_{a})_{a\in\mathbb{R}}$, which displays a
$C^{\infty}$-paraheterocycle at $a=0$ between the continuation of the saddle
$P$ and a projectively hyperbolic source $S^{\prime}$.

We will see in Lemma 2.9 that the assumption on the negative stable eigenvalue
can be obtained when the heterocycle is included in a bicycle.

######  Remark 1.16.

The definition of paraheterocycle, the statement of C and its proof extend
without difficulty to families parametrized by $\mathbb{R}^{k}$, for any
$k\geq 1$, see Remark 2.11 and Section 4.3.

###  1.7 Parablenders

In this section we fix $1\leq r<\infty$.

Parablenders are a parametric counterpart of blenders. The first example of a
parablender was given in [Ber16]; in [BCP17] a new example of parablender was
given and therein the definition of parablender was formulated as:

######  Definition 1.17 ($C^{r}$-Parablender).

The continuation $(K_{a})_{a\in I}$ of a hyperbolic set $K$ for the family
$(f_{a})_{a\in\mathbb{R}}$ is a _$C^{r}$ -parablender_ at
${a_{0}}\in\operatorname{Interior}(I)$ if the following condition is
satisfied.

There exist a continuous family of local unstable manifolds
$(W^{u}_{loc}(\underline{z};f_{a_{0}}))_{\underline{z}\in\overleftarrow{K}}$
and a non-empty open set $O$ of germs at $a_{0}$ of $C^{r}$-families of points
$(\gamma_{a})_{a\in I}$ in $M$ such that for every
$(\tilde{f}_{a})_{a\in\mathbb{R}}$ $C^{r}$-close to
$(f_{a})_{a\in\mathbb{R}}$, there exists $\underline{z}\in\overleftarrow{K}$
satisfying:

$\lim_{a\to a_{0}}|a-a_{0}|^{-r}\cdot
d\bigg{(}\gamma_{a}\;,\;W^{u}_{loc}(\underline{z};\tilde{f}_{a})\bigg{)}=0\;.$

The open set $O$ is called an _activation domain_ for the $C^{r}$-parablender
$(K_{a})_{a\in I}$.

Here is the parametric counterpart of the nearly affine blender introduced in
Def. 1.7.

######  Definition 1.18 (Nearly affine parablender666 The coordinates
considered in [BCP17] were slightly different but the same modulo conjugacy:
the renormalized inverses branches are of the form:
$B_{b}^{\pm}:(X,Y)\mapsto\left(0,(Y\pm 1)/(\Delta^{-1}+b)\right),$ which is
conjugate to the presented form $(A_{a}^{\pm})_{\pm}$ via the coordinates
changes:  $(X,Y)=(x-x_{0},\frac{\Delta-1}{\Delta+a}\cdot
y)\quad\mathrm{and}\quad a=-\frac{b\cdot\Delta^{2}}{1+b\cdot\Delta}\,.$
[BCP17] ).

For $\Delta>1$, $x_{0}\in(-2,2)$ and $\delta>0$, a $C^{r}$-family
$(f_{a})_{a\in\mathbb{R}}$ has a _$\delta$ -nearly affine $C^{r}$-parablender_
with contraction $\Delta^{-1}$ at $a=0$ if there exist a neighborhood $I$ of
$0$ in $\mathbb{R}$, a $C^{r}$-family $(H_{a})_{a\in I}$ of charts
$H_{a}\colon\mathbb{R}^{2}\hookrightarrow M$, a diffeomorphism
$\theta:J\hookrightarrow I$ fixing $0$ and inverse branches $(g^{+}_{a})_{a\in
I}$, $(g^{-}_{a})_{a\in I}$ of iterates $f_{a}^{N^{+}}$, $f_{a}^{N^{-}}$ such
that

${\mathcal{R}}g_{a}^{+}:=H_{a}^{-1}\circ g_{\theta(a)}^{+}\circ
H_{a}\quad\mathrm{and}\quad{\mathcal{R}}g_{a}^{-}:=H_{a}^{-1}\circ
g_{\theta(a)}^{-}\circ H_{a}$

are well defined on $[2,2]^{2}$ and $({\mathcal{R}}g_{a}^{\pm})_{a\in I}$ are
$\delta$-$C^{r}$-close to the two families $(A_{a}^{\pm})_{a\in I}$ defined by

$A_{a}^{+}:(x,y)\mapsto\left(x_{0},(\Delta+a)\cdot
y+\Delta-1\right)\quad\mathrm{and}\quad
A_{a}^{-}:(x,y)\mapsto\left(x_{0},(\Delta+a)\cdot y-\Delta+1\right).$

Note that a nearly affine parablender defines a germ of family of nearly
affine blenders $(K_{a})_{a\in I}$ at $a=0$ and so a germ of family of
blenders by Proposition 1.8. In [BCP17, Section 6], we showed777The activation
domain is not explicited in the statements of the results of [BCP17, Section
6], but appears in the proof as a product $W=B\times A$ (see page 67), where
$B=[-2,2]\times(-\eta,\eta)^{r}$ and where $A$ is a neighborhood of $0$ in
$\mathbb{R}^{r+1}$, obtained as the image of a neighborhood of $0$ by a
surjective linear map (page 63). that it defines also a parablender:

######  Proposition 1.19.

For every $\Delta>1$ close to $1$ and $x_{0}\in(-2,2)$, there is $\eta>0$
arbitrarily small such that if $\delta>0$ is sufficiently small, then
$(K_{a})_{a\in I}$ is a $C^{r}$-parablender at $a=0$. Moreover, its activation
domain contains:

$\left\\{(z_{a})_{a\in I}\in
C^{r}(I,\mathbb{R}^{2}):z_{0}\in[-2,2]\times(-\eta,\eta)\quad\mathrm{and}\quad\|\partial_{a}^{k}z_{a}|_{a=0}\|<\eta,\quad\forall
1\leq k\leq r\right\\}.$

We will show that nearly affine $C^{r}$-parablenders appear as
renormalizations of the dynamics nearby paraheterocycles. This will enable us
to show:

######  Theorem D.

Let us consider a $C^{\infty}$ family of local diffeomorphisms
$(f_{a})_{a\in\mathbb{R}}$ in $\mathrm{Diff}_{loc}^{\infty}(U,M)$ and, for
$r\geq 1$, a family of saddles $(P_{a})_{a\in\mathbb{R}}$ and a family of
projectively hyperbolic sources $(S_{a})_{a\in\mathbb{R}}$ exhibiting a
$C^{r}$-paraheterocycle at $a=0$.

Then there exists $(\tilde{f}_{a})_{a\in\mathbb{R}}$, $C^{\infty}$-close to
$(f_{a})_{a\in\mathbb{R}}$ displaying a $C^{r}$-parablender at $a=0$ which is
homoclinically related to $P_{0}$ and whose activation domain contains the
germ of $(S_{a})_{a\in\mathbb{R}}$ at $a=0$. In particular
$(\tilde{f}_{a})_{a\in\mathbb{R}}$ displays a robust $C^{r}$-robust
paraheterocycle at $a=0$.

##  2 Structure of the proofs of the theorems

###  2.1 Proof of B

The strategy of the proof breaks down into two steps. In a first step, we
obtain, by perturbation of the heterocycle, a strong heterocycle. This is done
in Section 3.

######  Proposition 2.1.

For $\rho\in\\{\infty,\omega\\}$, let $f\in\mathrm{Diff}_{loc}^{\rho}(U,M)$
with a projectively hyperbolic source $S$ and a saddle point $P$ forming a
heterocycle. Then there exists a map $\tilde{f}$ arbitrary $C^{\rho}$-close
with a saddle point $Q$ homoclinically related to $P_{\tilde{f}}$ and which
forms with $S_{\tilde{f}}$ a strong heterocycle.

In a second step we perturb the strong heterocycle in order to exhibit a
nearly affine blender displaying a robust heterocycle. See Section 5.

######  Proposition 2.2.

For $\rho\in\\{\infty,\omega\\}$, let $f\in\mathrm{Diff}_{loc}^{\rho}(U,M)$
with a projectively hyperbolic source $S$ and a saddle point $Q$ forming a
strong heterocycle. Fix $\infty>r\geq 1$ and take $\Lambda>1$ close to $1$.

Then, for every $\delta>0$ there exists a $C^{\rho}$-perturbation $\tilde{f}$
exhibiting a $\delta-C^{r}$-nearly affine blender which is homoclinically
related to $Q_{\tilde{f}}$ and whose activation domain contains
$S_{\tilde{f}}$.

Note that the conjunction of these two propositions implies B for the
topologies $C^{\infty}$ and $C^{\omega}$. When the initial diffeomorphism is
$C^{r}$, $1\leq r<\infty$, we first perturb in the $C^{r}$-topology in order
to get a $C^{\infty}$-diffeomorphism taking care that the source $S$ still
belongs to the unstable manifold of the saddle $P$, and we then apply the
result for $C^{\infty}$-diffeomorphisms. ∎

###  2.2 Proof of D

Similarly to the proof of B, the proof consists in two steps that are the
parametric counterparts of Proposition 2.1 and Proposition 2.2. They are
detailed in Sections 3 and 5.

######  Proposition 2.3.

Consider a $C^{\infty}$ family of local diffeomorphisms
$(f_{a})_{a\in\mathbb{R}}$ in $\mathrm{Diff}_{loc}^{\infty}(U,M)$, and, for
$r\geq 1$, a family of saddles $(P_{a})_{a\in\mathbb{R}}$ and a family of
projectively hyperbolic sources $(S_{a})_{a\in\mathbb{R}}$ exhibiting a
$C^{r}$-paraheterocycle at $a=0$.

Then there exist $(\tilde{f}_{a})_{a\in\mathbb{R}}$, $C^{\infty}$-close to
$(f_{a})_{a\in\mathbb{R}}$ with a family of saddles $(Q_{a})_{a\in\mathbb{R}}$
homoclinically related to $(P_{a})_{a\in\mathbb{R}}$ which forms with
$(S_{a})_{a\in\mathbb{R}}$ a strong $C^{r}$-paraheterocycle at $a=0$.

######  Proposition 2.4.

Consider a $C^{\infty}$ family of local diffeomorphisms
$(f_{a})_{a\in\mathbb{R}}$ in $\mathrm{Diff}_{loc}^{\infty}(U,M)$, and, for
$r\geq 1$, a family of saddles $(Q_{a})_{a\in\mathbb{R}}$ and a family of
projectively hyperbolic sources $(S_{a})_{a\in\mathbb{R}}$ exhibiting a strong
$C^{r}$-paraheterocycle at $a=0$.

Then there exists $(\tilde{f}_{a})_{a\in\mathbb{R}}$, $C^{\infty}$-close to
$(f_{a})_{a\in\mathbb{R}}$, displaying a $C^{r}$-parablender at $a=0$
homoclinically related to $Q_{0}$ and whose activation domain contains the
germ of $(S_{a})_{a\in\mathbb{R}}$ at $a=0$.

This completes the proof of D. ∎

######  Remark 2.5.

One can choose the parablender and the family of local unstable manifolds
defining its activation domain in such a way that each local unstable manifold
does not have $S_{0}$ as an endpoint and is not tangent to the weak unstable
direction of $S_{0}$. See Remark 5.15.

###  2.3 Proof of C: chains of heterocycles

We begin with some preparation lemmas. The first one is proved in section
3.2.1.

######  Lemma 2.6.

Let $S$ and $P$ be a projectively hyperbolic source and a saddle point forming
a heterocycle for a smooth map $f$. Then for a $C^{\infty}$-small perturbation
of the dynamics, the source $S$ belongs to a Cantor set $R$ which is a
projectively hyperbolic expanding set.

We introduce the following notion.

######  Definition 2.7.

A _$N$ -chain of alternate heterocycles_ for a map
$f\in\mathrm{Diff}_{loc}(U,M)$ is the data of $N$ saddle points
$P^{1},\dots,P^{N}$ and $N$ projectively hyperbolic sources
$S^{1},\dots,S^{N}$ such that:

  * •

the orbits of $P^{1},\dots,P^{N},S^{1},\dots,S^{N}$ are pairwise disjoint,

  * •

the stable eigenvalues of the saddles $P^{i}$ are negative,

  * •

$W^{u}(P^{i})$ contains $S^{i}$ and is transverse to $E^{cu}_{S^{i}}$ for each
$1\leq i\leq N$,

  * •

$W^{uu}(S^{i})$ intersects transversally $W^{s}(P^{i+1})$ for $1\leq i<N$ and
$W^{uu}(S^{N})$ intersects transversally $W^{s}(P^{1})$.

Figure 4: 2-Chain of heterocycles.

Chains of alternate heterocycles may be obtained as follows.

######  Lemma 2.8.

Consider $f\in\mathrm{Diff}_{loc}^{\infty}(U,M)$ with a heterocycle between a
saddle $P$ with period $p$ and a source $S$ such that the stable eigenvalue of
$D_{P}f^{p}$ is negative.

Then, for any $N\geq 1$, there exists $\tilde{f}$, $C^{\infty}$-close to $f$,
with an $N$-chain of alternate heterocycles whose saddles
$P^{1}=P,P^{2},\dots,P^{N}$ are homoclinically related to the continuation
$P_{\tilde{f}}$.

###### Proof.

By preliminary perturbations one stabilizes the heterocycle and builds a
blender $K$ homoclinically related to $P$, whose activation domain contains
$S$ (B). One also reduces to the case where the source $S$ belongs to a
projectively hyperbolic expanding invariant Cantor set $R$ (Lemma 2.6). One
can also assume that $W^{uu}(S)$ and $W^{s}(P)$ have a transverse intersection
point. In order to simplify, one will assume that $K$ is topologically mixing
(otherwise one has to decompose $K$ into finitely many pieces permuted by the
dynamics and whose return map is topologically mixing on each piece). Note
that $P^{1}=P$ and $S^{1}=S$ define a $1$-chain of alternate heterocycle. One
proves the statement by induction on $N$. Let us assume that $f$ has a
$N-1$-chain of alternate heterocycles whose saddles $P^{i}$ are homoclinically
related to $P$.

One chooses a saddle $P^{N}$ whose orbit is distinct from the orbits of
$P^{1},\dots,P^{N-1}$ and which is homoclinically related to $P$: since
$W^{uu}(S^{N-1})$ intersects transversally $W^{s}(P^{1})$, it also intersects
transversally $W^{s}(P^{N})$. One also chooses a source $S^{N}\in R$ in the
activation domain of $K$ and whose orbit is distinct from the orbits of
$S^{1},\dots,S^{N-1}$; one can furthermore assume that it is arbitrarily close
to $S$, so that $W^{uu}(S^{N})$ intersects transversally $W^{s}(P)$, hence
$W^{s}(P^{1})$. The blender property implies that $S^{N}$ belongs to the
unstable set of $K$. More precisely there exists $x\in K$ and $y\in
W^{u}(x)\setminus\text{\rm Orbit}(S^{N})$ such that $f(y)\in\text{\rm
Orbit}(S^{N})$. Since $K$ is topologically mixing, $W^{u}(P^{N})$ is dense in
the unstable set of $K$, one can find $y^{\prime}\in W^{u}(P^{N})$ arbitrarily
close to $y$ and whose backward orbit is disjoint from a uniform neighborhood
of $y$. One then perturbs $f$ in a small neighborhood of $y$ and get a map
satisfying $\tilde{f}(y^{\prime})=f(y)$. Consequently $P^{N}$ and $S^{N}$
define a heterocycle for $\tilde{f}$ and the properties built at the previous
steps of the induction are preserved. ∎

The existence of a saddle point with negative stable eigenvalue may be
obtained once a saddle belongs to a homocycle, as we recall in the next lemma.

######  Lemma 2.9.

Let $f\in\mathrm{Diff}_{loc}^{\infty}(U,M)$ and $P$ be a saddle point with a
homoclinic tangency $L$. Then for a $C^{\infty}$-small perturbation
$\tilde{f}$ of the dynamics supported on a small neighborhood of $L$, the
saddle $P$ belongs to a basic set which contains a point $Q$ with some period
$\tau$ and such that the stable eigenvalue of $D_{Q}\tilde{f}^{\tau}$ is
negative.

###### Proof.

This is a well-known result. Up to replace $L$ by an iterate, one assumes
$L\in W^{s}_{loc}(P)$. One perturbs $f$ so that the contact of the homoclinic
tangency is quadratic. By unfolding the homoclinic tangency, a horseshoe
containing $P$ appears. Indeed, one considers a thin rectangle $R$ which is a
tubular neighborhood of $W^{s}_{loc}(P)$. A large iterate $f^{\ell}(R)$
crosses $R$ twice, with different orientations. In each component of the
intersection, a $\ell$-periodic point is obtained, and the signs of
$Df^{\ell}$ along the stable direction differ. See [PT93, chapter 3] for
details. ∎

C follows from the next proposition, proved in Section 4.

######  Proposition 2.10.

For any $d\geq 0$, there exists $N=N(d)\geq 1$ with the following property.

Consider a $C^{\infty}$ family $(f_{a})_{a\in\mathbb{R}}$ in
$\mathrm{Diff}_{loc}^{\infty}(U,M)$ such that $f_{0}$ has a $N$-chain of
alternate heterocyles with saddle points $P^{i}$ and sources $S^{i}$. Then
there exists a family $(\tilde{f}_{a})_{a\in\mathbb{R}}$, $C^{\infty}$-close
to $(f_{a})_{a\in\mathbb{R}}$, such that the continuations of $P^{1}$ and
$S^{N}$ form a $C^{d}$-paraheterocycle at $a=0$.

######  Remark 2.11.

This result is still valid for families parametrized by $\mathbb{R}^{k}$,
$k\geq 1$ (see Section 4.3). The length of the chain required is then equal
to:

$N(r,k)={\dim_{\mathbb{R}}\\{P\in\mathbb{R}[X_{1},\dots,X_{k}]:\;\deg P\leq
r,\;P(0)=0\\}}.$

###### Proof of C.

For any large integer $d\geq 1$, Lemma 2.8 and Proposition 2.10 give after a
$C^{\infty}$-perturbation a $C^{d}$-paraheterocycle between the continuation
of the saddle $P$ and a projectively hyperbolic source $S^{\prime}$. Hence
there exists a $C^{d}$-small perturbation $(f^{\prime}_{a})_{a\in\mathbb{R}}$
in $\mathrm{Diff}_{loc}^{\infty}(U,M)$ and an integer $N$ which satisfy
$S^{\prime}_{a}\in f^{N}(W^{u}_{loc}(P_{f^{\prime}_{a}}))$ for any $a$ close
to $0$. Since $d$ has been chosen arbitrarily large, the perturbation can be
chosen $C^{\infty}$-small. ∎

###  2.4 Proof of A

A consequence of the previous results is the:

######  Corollary 2.12.

Consider a $C^{\infty}$ family $(f_{a})_{a\in\mathbb{R}}$ in
$\mathrm{Diff}_{loc}^{\infty}(U,M)$ such that $f_{0}$ displays a bicycle
between a projectively hyperbolic source $S_{0}$ and a dissipative saddle
point $P_{0}$. Let $r\geq 1$. Then up to a $C^{r}$-perturbation of the family,
and up to replacing $S_{0}$ by another projectively hyperbolic source, we can
assume that:

  1. $(H_{0})$

There exists a blender $K_{0}$ for $f_{0}$ whose activation domain contains
$S_{0}$.

  2. $(H_{1})$

$K_{0}$ intersects the repulsive basin of $S_{0}$.

  3. $(H_{2})$

$P_{0}$ is homoclinically related to $K_{0}$ and $W^{s}(P_{0})$ has a robust
tangency with the strong unstable foliation $\mathcal{F}^{uu}$ of $S_{0}$.

  4. $(H_{3})$

The continuation $(K_{a})_{a\in I}$ of $K_{0}$ is a $C^{r}$-parablender at
$a=0$ and the continuation $(S_{a})_{a\in I}$ of $S_{0}$ belongs to its
activation domain.

  5. $(H_{4})$

In the continuous family of local unstable manifolds defining the activation
domain involved in $(H_{0})$ and $(H_{3})$, each local unstable manifold does
not have $S_{0}$ as an endpoint and is not tangent to the weak unstable
direction of $S_{0}$.

######  Remark 2.13.

The properties $(H_{0})$…$(H_{4})$ are $C^{r}$-open.

###### Proof.

With Corollary B′ Corollary B′, one first stabilizes the bicycle. By Lemma
2.9, up to a small $C^{\infty}$-perturbation, one gets a saddle $Q_{0}$
homoclinically related to $P_{0}$ whose stable eigenvalue at the period is
negative. One thus gets a robust heterocycle between $Q_{0}$ and $S_{0}$ and C
C gives a family $(f^{\prime}_{a})_{a\in\mathbb{R}}$, that is
$C^{\infty}$-close, and displaying a $C^{\infty}$-paraheterocycle between the
continuation of $Q_{0}$ and a projectively hyperbolic source saddles
$S^{\prime}$. D D produces a family $C^{r}$-close having a $C^{r}$-parablender
$(K_{a})_{a\in I}$ at $a=0$ which is homoclinically related to $Q_{0}$ (and
$P_{0}$) and whose activation domain contains the family of source
$(S^{\prime}_{a})_{a\in I}$. Denoting the new source by $S_{0}$, we get all
the robust properties $(H_{0})$, $(H_{1})$ and $(H_{3})$. By Remark 2.5,
$(H_{4})$ is also satisfied.

Since $P_{0}$ and $S_{0}$ form a robust heterocycle, one can assume (after a
new perturbation) that the strong unstable manifold $W^{uu}(S_{0})$ intersects
transversally $W^{s}(P_{0})$. From the robust tangency, we can perturb and
produce a homoclinic tangency point $L$ between $W^{u}_{loc}(P_{0})$ and
$W^{s}(P_{0})$. The inclination lemma implies that $W^{uu}(S_{0})$ accumulates
on $W^{u}_{loc}(P_{0})$. A last perturbation near $L$ gives a quadratic
tangency between $W^{uu}(S_{0})$ and $W^{s}(P_{0})$. For maps close, this
tangency admits a continuation which is a quadratic tangency between
$W^{s}(P_{0})$ and the leaves of the strong unstable foliation in the
repelling basin of $S_{0}$: this is $(H_{2})$. ∎

Figure 5: Assumptions $(H_{0})$…$(H_{4})$.

We now use the following result of [Ber17, Theorem A, page 11]:

######  Theorem 2.14.

Consider a $C^{\infty}$ family $(f_{a})_{a\in\mathbb{R}}$ in
$\mathrm{Diff}_{loc}^{\infty}(U,M)$ with a projectively hyperbolic source
$(S_{a})_{a\in\mathbb{R}}$ and a dissipative saddle point
$(P_{a})_{a\in\mathbb{R}}$ satisfying $(H_{0})$…$(H_{4})$. Then, there are
$\delta>0$, a $C^{r}$-neighborhood $\mathcal{V}$ of the family $(f_{a})_{a}$
in the space of $C^{r}$-families and a Baire-generic subset $\cal G\subset\cal
V$ such that for any $(\tilde{f}_{a})_{a}\in\cal G$ and
$a\in(-\delta,\delta)$, the map $\tilde{f}_{a}$ displays infinitely many
sinks.

For completeness we sketch its proof.

###### Idea of the proof of Theorem 2.14.

Since the hypothesis are open, they hold for an open neighborhood
$\mathcal{V}$ of the initial family. Let us consider an arbitrary family
$(f^{\prime}_{a})_{a\in\mathbb{R}}$ in $\mathcal{V}$. The robust heterocycle
provided by $(H_{0})$ and $(H_{1})$ and Lemma 2.6 allow after a perturbation
to assume that there are $\delta>0$ and two distinct sources
$(S_{a})_{a\in[-\delta,\delta]}$, $(S_{a}^{\prime})_{a\in[-\delta,\delta]}$
which satisfy $(H_{0})$…$(H_{4})$ at every $a_{0}\in[-\delta,\delta]$ for each
of these sources and for the family $(f^{\prime}_{a})_{a\in\mathbb{R}}$.

Then we apply the following key lemma (which uses $(H_{3})$):

######  Lemma 2.15 ([Ber17, Prop. 3.6]).

For every $\varepsilon>0$, there exist $\alpha>0$ and an
$\varepsilon$-$C^{r}$-perturbation
$(f^{\prime\prime}_{a})_{a\in[-\delta,\delta]}$ localized at $(S_{a})_{a}$ and
$(S_{a}^{\prime})_{a}$ such that:

  1. 1.

for every $j\in 2\mathbb{Z}$, there exists a continuation of a periodic point
$(P^{(j)}_{a})_{a}$ in the parablender whose local unstable manifold contains
$S_{a}$ for every $a\in[-\delta,\delta]\cap[\alpha j-\alpha/2,\alpha
j+\alpha/2]$,

  2. 2.

for every $j\in 2\mathbb{Z}+1$, there exists a continuation of a periodic
point $(P^{(j)}_{a})_{a}$ in the parablender whose local unstable manifold
contains $S^{\prime}_{a}$ for every $a\in[-\delta,\delta]\cap[\alpha
j-\alpha/2,\alpha j+\alpha/2]$.

We continue with:

######  Lemma 2.16 ([Ber17, Prop. 3.4]).

After a new $C^{\infty}$-small perturbation of $(f^{\prime\prime}_{a})_{a}$,
for every $j\in\mathbb{Z}\cap[-\delta/\alpha,\delta/\alpha]$ the point $P_{a}$
displays a quadratic homoclinic tangency which persists for every
$a\in[-\delta,\delta]\cap[\alpha j-\alpha/2,\alpha j+\alpha/2]$.

###### Idea of proof of Lemma 2.16.

Assume $j$ odd (resp. even) and let us continue with the setting of Lemma
2.15. As $P_{a}$ and $P^{(j)}_{a}$ belong to the same transitive hyperbolic
set and using Proposition 1.14, after a small perturbation a fixed iterate of
the local unstable manifold of $P_{a}$ contains $S_{a}$ for every
$a\in[-\delta,\delta]\cap[\alpha j-\alpha/2,\alpha j+\alpha/2]$. Then we
proceed as depicted in Fig. 6: we denote by $W^{s}_{a}$ a segment of
$W^{s}(P_{a})$ which is included in a basin of $S_{a}$ (resp.
$S_{a}^{\prime}$) and display a tangency with the strong unstable foliation of
the repelling basin of $S^{\prime}_{a}$ (resp. $S_{a}$) by $(H_{2})$. After
perturbation we can assume this tangency quadratic. Then, in the Grassmanian
bundle $\mathbb{P}(TM)$ of $M$, the tangent space $TW^{s}$ of this curve
intersects transversally the unstable manifold of $(S_{a},E^{uu}(S_{a}))$ for
the action $\hat{f}_{a}$ of $Df_{a}$ on the Grassmannian. By the inclination
lemma, the preimages $TW^{s}_{n,a}$ of $TW^{s}_{a}$, by $\hat{f}_{a}^{n}$,
converge to the stable manifold
$\\{S_{a}\\}\times\mathbb{PR}^{1}\setminus\\{E^{cu}(S_{a})\\}$ of
$(S_{a},E^{uu}(S_{a}))$. By property $(H_{4})$, a piece of
$W^{u}_{loc}(P_{a})$ intersects $S_{a}$ with a direction different from
$E^{uu}(S_{a})$, hence the stable manifold of $(S_{a},E^{uu}(S_{a}))$
intersects untangentially a piece $TW^{u}_{a}$ of $TW^{u}_{loc}(P_{a})$ for
every $a\in[-\delta,\delta]\cap[\alpha j-\alpha/2,\alpha j+\alpha/2]$. This
enables to perturb $(f_{a})_{a}$ such that $TW^{s}_{n,a}\subset TW^{s}(P_{a})$
intersects $TW^{s}_{loc}(P_{a})$ for every $a\in[-\delta,\delta]\cap[\alpha
j-\alpha/2,\alpha j+\alpha/2]$.

Figure 6: Inclination lemma used in the bundle $\mathbb{P}(TM)$ at
$E^{uu}(S_{A})$.

∎

In [Ber17, Prop. 3.5] it is shown that for every $N\geq 1$, we can then
perturb the family in the $C^{\infty}$-topology near the homoclinic tangency
of $(P_{a})_{a}$ obtained in Lemma 2.16 so that for every $a\in[\alpha
j-\alpha/2,\alpha j+\alpha/2]$ and
$j\in\mathbb{Z}\cap[-\delta/\alpha,\delta/\alpha]$, the new map
$\tilde{f}_{a}$ displays a periodic sink of period $\geq N$. Hence we have
obtained an open and dense subset in $\mathcal{V}$ of families displaying a
sink of period $\geq N$ at every parameter $a\in[-\delta,\delta]$. By taking
the intersection $\cal G$ of these open and dense subsets over $N\geq 1$, we
obtain Theorem 2.14. ∎

This allows to complete the proof of our main theorem.

###### Proof of A.

Let us consider a $C^{r}$ map $f$ with a dissipative bicycle associated to a
saddle $P$. By LABEL:{robust-bicycle}, there exists a $C^{r}$-open set
$\mathcal{U}\in\mathrm{Diff}_{loc}^{r}(U,M)$, which contains $f$ in its
closure, such that the continuation of $P$ exhibits a robust bicycle for any
map in $\mathcal{U}$.

Let $F:=(f_{a})_{a\in\mathbb{R}}$ be a $C^{r}$-family consisting of maps
$f_{a}\in\mathcal{U}$. By perturbation, one can assume that the family is
$C^{\infty}$ and by Proposition 1.13 that $f_{0}$ displays a bicycle. Then, by
2.12, there exists a new $C^{r}$-perturbation which satisfies
$(H_{0})\cdots(H_{4})$. Theorem 2.14 associates a neighborhood
$\mathcal{V}_{F}$ of this family and a dense Gδ-set ${\mathcal{G}}_{F}$ of
$\mathcal{V}_{F}$ and $\delta_{F}>0$. Let $\\{F_{n}:n\in\mathbb{N}\\}$ be a
dense countable set in the space of families
$(f_{a})_{a\in\mathbb{R}}\in\mathcal{D}^{r}(\mathbb{I}\times U,M)$ consisting
of maps $f_{a}\in\mathcal{U}$. The union
${\mathcal{G}}=\bigcup{\mathcal{G}}_{F_{n}}$ is a dense $G_{\delta}$ subset of
this space. By construction, for any family $F=(f_{a})_{a\in\mathbb{R}}$ in
${\mathcal{G}}$ and any $|a|$ smaller than a semi-countinuous function
$\delta_{F}$ of $F$, the map $\tilde{f}_{a}$ exhibits infinitely many sinks
for any parameter $a$ close to $0$. ∎

##  3 From heterocycles to basic sets and strong heterocycles

In this section we prove Proposition 2.1, Proposition 2.3 and Lemma 2.6.

We consider a $C^{\infty}$ map $f\in\mathrm{Diff}_{loc}^{\infty}(U,M)$ with a
projectively hyperbolic source $S$ and a saddle point $P$ forming a
heterocycle, and we show that by perturbation it can be improved to a strong
heterocycle.

In Section 3.1, first we establish local coordinates around $P$ and $S$. To
obtain these coordinates, we need to perturb the dynamics, to assume the
eigenvalues non-resonant, but also to ensure two transversality conditions
$(T_{1})$-$(T_{2})$. Then nearby $P$ and $S$, the inverse dynamics
${\mathcal{P}}$ and ${\mathcal{S}}$ are linear in local coordinates.
Furthermore, the heterocycle defines inverse branches of the dynamics that are
transitions from one linearizing chart to the other.

As a direct application of these linearizing charts, we build an IFS and from
there an expanding projective hyperbolic set containing the source: this
allows to prove Lemma 2.6 at the beginning of Section 3.2). Later, using again
these coordinates, we obtain the existence of a non-trivial basic set $K$
which contains $P$ (Lemma 3.1). After a small perturbation, which consists in
perturbing the stable eigenvalues of $P$, the strong unstable manifold of $S$
intersects $K$, whereas $S$ belongs to $W^{u}(K)$. This will imply Proposition
2.1. The proof of Proposition 2.3 follows similar lines.

###  3.1 Local coordinates for a heterocycle

For the sake of simplicity we assume that the periodic points $P$ and $S$ are
fixed and that the eigenvalues of $D_{P}f$ and $D_{S}f$ are positive. Anyway
we can go back to this case by regarding an iterate of the dynamics and
performing the forthcoming perturbations nearby finitely many points belonging
to different orbits.

Up to a smooth perturbation we can assume that the eigenvalues of $D_{P}f$ and
$D_{S}f$ are non-resonant. Then Sternberg Theorem [Ste58] implies the
existence of:

  * •

neighborhoods $V^{\prime}_{S}\subset V_{S}:=f(V^{\prime}_{S})$ of $S$ and
coordinates for which $f|V^{\prime}_{S}$ has the form:

$(x,y)\in V^{\prime}_{S}\mapsto(\sigma_{uu}^{-1}\cdot x,\sigma_{u}^{-1}\cdot
y)\in V_{S}\quad\text{with }0<\sigma_{uu}<\sigma_{u}<1\;.$

  * •

neighborhoods $V^{\prime}_{P}$ and $V_{P}:=f(V_{P}^{\prime})$ of $P$ and
coordinates for which $f|V^{\prime}_{P}$ has the form:

$(x,y)\in V^{\prime}_{P}\mapsto(\sigma^{-1}\cdot x,\lambda^{-1}\cdot y)\in
V_{P}\quad\text{ with }0<\sigma<1<\lambda\;.$

This defines the inverse branches ${\mathcal{P}}:=(f|V_{P}^{\prime})^{-1}$ and
${\mathcal{S}}:=(f|V_{S}^{\prime})^{-1}$:

${\mathcal{S}}:(x,y)\in V_{S}\mapsto(\sigma_{uu}\cdot x,\sigma_{u}\cdot y)\in
V_{S}^{\prime}\quad\mathrm{and}\quad{\mathcal{P}}:(x,y)\in
V_{Q}\mapsto(\sigma\cdot x,\lambda\cdot y)\in V_{Q}^{\prime}\;.$

Up to restricting $V_{P}$ and $V_{P}^{\prime}$ and rescaling the coordinates,
we can assume:

$V_{P}^{\prime}\equiv[-\sigma,\sigma]\times[-1,1]\quad\mathrm{and}\quad
V_{P}\equiv[-1,1]\times[-\lambda^{-1},\lambda^{-1}]\;.$

Let $W^{u}_{loc}(P):=V_{Q}\cap\\{y=0\\}$,
$W^{s}_{loc}(P):=V^{\prime}_{Q}\cap\\{x=0\\}$ and
$W^{uu}_{loc}(S):=\\{y=0\\}\cap V_{S}$.

Let $H$ be a point in $W^{s}(P)\cap W^{uu}(S)$. Up to replacing it by an
iterate, we can assume that $H$ belongs to $V^{\prime}_{P}$ with $H=:(0,h)$ in
the linearizing coordinates of $P$. Up to conjugating the dynamics by
$(x,y)\mapsto(x,-y)$, we can assume moreover that $h>0$. Also, a preimage
$S^{\prime}$ of $S$ has coordinates $S^{\prime}=:(s,0)$ in the linearizing
coordinates of $P$:

$S^{\prime}\equiv(s,0)\quad\mathrm{and}\quad H\equiv(0,h)\;,\quad h>0\;.$

Furthermore up to a smooth perturbation, we can assume that:

  1. $(T_{1})$

The intersection $W^{s}(P)\cap W^{uu}(S)$ is transverse at $H$.

  2. $(T_{2})$

The line $T_{S}W^{u}(P)$ is in direct sum with the weak unstable direction
$E^{cu}$ of $S$.

  3. $(T_{3})$

The line $T_{S}W^{u}(P)$ is in direct sum with the strong unstable direction
$E^{uu}$ of $S$.

Let $V^{\prime\prime}_{S}\Subset V^{\prime}_{S}$ and $V_{H}\Subset
V^{\prime}_{P}$ be small neighborhoods of $S$ and $H$; and let ${\cal
T}^{\scriptscriptstyle{\cal S}}:V_{S}^{\prime\prime}\hookrightarrow V_{P}$ and
${\cal T}^{\scriptscriptstyle{\cal H}}:V_{H}\hookrightarrow V_{S}$ be inverse
branches of iterates of $f$ such that ${\cal T}^{\scriptscriptstyle{\cal
S}}(S)=S^{\prime}$ and ${\cal T}^{\scriptscriptstyle{\cal H}}(H)\in
W^{uu}_{loc}(S)$.

Figure 7: Inverse branches ${\mathcal{S}},{\mathcal{P}},{\cal
T}^{\scriptscriptstyle{\cal S}},{\cal T}^{\scriptscriptstyle{\cal H}}$ induced
by the heterocycle.

###  3.2 Basic sets induced by a heterocycle

We now build two hyperbolic sets: one expanding projective hyperbolic set
containing the source, and a saddle hyperbolic set containing the saddle.

####  3.2.1 Proof of Lemma 2.6: expanding Cantor set linked to the heterocycle

Note that for $n$ large, the point $(s,\lambda^{-n}h)$ belongs to the range of
${\cal T}^{\scriptscriptstyle{\cal S}}$. We perturb $f$ near the point
$S^{\prime}$ and define a map $\tilde{f}$ which satisfies in coordinates

$\tilde{f}(s,\lambda^{-n}h)=f(S^{\prime}).$

This in turn defines a perturbation $\widetilde{\cal
T}^{\scriptscriptstyle{\cal S}}$ of the inverse branch ${\cal
T}^{\scriptscriptstyle{\cal S}}$.

As the point $(s,\lambda^{-n}h)$ is sent by ${\mathcal{P}}^{n}$ to the
$(\sigma^{n}\cdot s,h)\in V_{H}$, the map ${\cal T}^{\scriptscriptstyle{\cal
H}}\circ{\mathcal{P}}^{n}\circ\widetilde{\cal T}^{\scriptscriptstyle{\cal S}}$
is well defined on a neighborhood $W$ of $S$. Hence for $N$ large compared to
$n$, the maps ${\mathcal{S}}_{1}:={\mathcal{S}}^{N}\circ{\cal
T}^{\scriptscriptstyle{\cal H}}\circ{\mathcal{P}}^{n}\circ\widetilde{\cal
T}^{\scriptscriptstyle{\cal S}}$ and ${\mathcal{S}}_{2}:={\mathcal{S}}^{N}$
are contractions from $W$ into $W$ with disjoint images. So they define a
transitive expanding Cantor set $R$ for $\tilde{f}$ which contains $S$.

Let us fix $\eta>0$ small and introduce the cone
$\mathcal{C}:=\\{(u,v):\;|u|<\eta|v|\\}$. Using (T1), (T2) and assuming that
$n,N$ have been chosen large enough, for any $x\in W$, the maps
$D_{x}{\mathcal{S}}_{1}$ and $D_{x}{\mathcal{S}}_{2}$ send
$\overline{\mathcal{C}}$ inside $\mathcal{C}\cup\\{0\\}$. The cone field
criterion (see for instance [Yoc95]) implies that the Cantor set $R$ is
projectively hyperbolic. The Lemma 2.6 is proved. ∎

####  3.2.2 Basic sets linked to the heterocycle

The heterocycle configuration implies under the transversality assumptions
$(T_{1})$ and $(T_{2})$ that the saddle $P$ has a transverse homoclinic
intersection.

######  Lemma 3.1.

For all $n$ large, the subsegment:

$W^{s}_{loc}(\bar{H}):={\cal T}^{\scriptscriptstyle{\cal
S}}\circ{\mathcal{S}}^{n}\circ{\cal T}^{\scriptscriptstyle{\cal
H}}(W^{s}_{loc}(P)\cap V_{H})$

of $W^{s}(P)$ intersects transversally the local unstable manifold
$W^{u}_{loc}(P)$ at a point $\bar{H}$ which is $\asymp\sigma_{uu}^{n}$-close
to $S^{\prime}$. The endpoints of $W^{s}_{loc}(\bar{H})$ are
$\asymp\sigma_{u}^{n}$-distant from $W^{u}_{loc}(P)$.

###### Proof.

Let $\Gamma:=W^{s}_{loc}(P)\cap V_{H}$. This curve is sent by ${\cal
T}^{\scriptscriptstyle{\cal H}}$ to a curve which intersects transversally
$W^{uu}_{loc}(S)$ by $(T_{1})$. By projective hyperbolicity, the image by
${\mathcal{S}}^{n}$ of ${\cal T}^{\scriptscriptstyle{\cal H}}(\Gamma)$ is a
curve which is tangent to a thin vertical cone field, which is
$\asymp\sigma_{uu}^{n}$-close to $S$ and which has length
$\asymp\sigma_{u}^{n}$. As $({\cal T}^{\scriptscriptstyle{\cal
S}})^{-1}(\\{y=0\\})$ intersects transversally $\\{x=0\\}\cap V_{S}$ at $S$ by
$(T_{2})$, it must intersect transversally ${\mathcal{S}}^{n}\circ{\cal
T}^{\scriptscriptstyle{\cal H}}(\Gamma)$ for $n$ large. Consequently the curve
${\cal T}^{\scriptscriptstyle{\cal S}}\circ{\mathcal{S}}^{n}\circ{\cal
T}^{\scriptscriptstyle{\cal H}}(\Gamma)$ intersects the local unstable
manifold $\\{y=0\\}\cap V_{P}$ of $P$. ∎

By Smale’s horseshoe theorem (see [PT93, chapter 2]), one deduces:

######  Corollary 3.2.

There exists a basic set $K$ containing $P$ and $\bar{H}$.

We will make it more precise. If $N$ is large, $K$ can be spanned by the
inverse branches

$\mathcal{G}_{1}:={\mathcal{P}}^{N}\quad\text{and}\quad\mathcal{G}_{2}:={\cal
T}^{\scriptscriptstyle{\cal S}}\circ{\mathcal{S}}^{n}\circ{\cal
T}^{\scriptscriptstyle{\cal H}}\circ{\mathcal{P}}^{N}.$

Figure 8: The box $B$ and its images.

Let $\varepsilon>0$ be small enough so that
$\\{0\\}\times[h-\varepsilon,h+\varepsilon]$ is included in $V_{H}$ and let
(see Fig. 8):

$B:=[-1,1]\times\left[\frac{h-\varepsilon}{\lambda^{N}},\frac{h+\varepsilon}{\lambda^{N}}\right]\;.$

######  Lemma 3.3.

For every $n,N$ large, the map $\mathcal{G}_{2}$ is well defined on $B$. If
$\varepsilon\sigma^{n}_{u}\lambda^{N}\gg 1$, the map $\mathcal{G}_{2}$
displays a saddle fixed point $\bar{Q}$ in $B\cap K$, which is homoclinically
related to $P$.

###### Proof.

The box $B$ is sent by ${\mathcal{P}}^{N}$ to
$(0,h)+[-\sigma^{N},\sigma^{N}]\times[-\varepsilon,\varepsilon]$ which is
included in $V_{H}$ for $N$ large enough. As ${\mathcal{S}}^{n}\circ{\cal
T}^{\scriptscriptstyle{\cal H}}(V_{H})$ is included in $V_{S}^{\prime\prime}$
for $n$ large enough, the map $g$ is well defined on $B$. Let us decompose the
boundary of $B$:

$\partial^{s}B:=\\{-1,1\\}\times\left[\frac{h-\varepsilon}{\lambda^{N}},\frac{h+\varepsilon}{\lambda^{N}}\right]\quad\mathrm{and}\quad\partial^{u}B=\partial
B\setminus\partial^{s}B.$

Both curves of ${\mathcal{S}}^{n}\circ{\cal T}^{\scriptscriptstyle{\cal
H}}\circ{\mathcal{P}}^{N}(\partial^{s}B)$ are $\sigma_{uu}^{n}$ close to the
vertical arc $W^{c}_{loc}(S):=\\{0\\}\times[-1,1]$ and their endpoints are
$\asymp\varepsilon\cdot\sigma_{u}^{n}$ distant to $W^{uu}_{loc}(S)$ by
transversality $(T_{1})$ at ${\cal T}^{\scriptscriptstyle{\cal H}}(H)$ and by
projective hyperbolicity of $S$. Thus they intersect transversally $({\cal
T}^{\scriptscriptstyle{\cal S}})^{-1}(W^{u}_{loc}(P))$ by property $(T_{2})$.

Consequently $\mathcal{G}_{2}(B)$ intersects $W^{u}_{loc}(P)$, and
$\mathcal{G}_{2}(\partial^{u}B)$ is $\asymp\varepsilon\cdot\sigma_{u}^{n}$
distant to $W^{u}_{loc}(P)$. By assumption, $\lambda^{-N}$ is small compared
to $\varepsilon\cdot\sigma^{n}_{u}$, then $\mathcal{G}_{2}(B)$ crosses $B$: it
does not meet the vertical boundary $\partial^{s}B$, whereas $B$ does not meet
the horizontal boundary $\mathcal{G}_{2}(\partial^{u}B)$. Thus
$\mathcal{G}_{2}$ displays a fixed point $\bar{Q}$ in
$B\cap\mathcal{G}_{2}(B)$.

Note that $D\mathcal{G}_{2}$ expands vectors in a vertical cone by a factor
$\asymp\lambda^{N}\cdot\sigma_{u}^{n}$, which is large, and the image of these
vectors are uniformly transverse to the horizontal. On the other hand by
projective hyperbolicity $D\mathcal{G}_{2}^{-1}$ sends the vectors in an
horizontal cone to uniformly horizontal vectors and expands them by a factor
$\sigma^{-n}_{uu}\cdot\sigma^{-N}$. The point $\bar{Q}$ is a saddle, its local
unstable manifold is an horizontal graph in $B$ over $[-1,1]$ whereas its
local stable manifold connects the two curves in
$\mathcal{G}_{2}(\partial^{u}B)$ and so crosses the horizontal
$W^{u}_{loc}(P)$. This shows that $\bar{Q}$ and $P$ are homoclinically related
as required. ∎

####  3.2.3 Replacement of the saddle point

Let us consider a saddle periodic point $Q$ homoclinically related to $P$. The
following allows to replace the saddle $P$ by $Q$ in the heterocycle.

######  Lemma 3.4.

Let $Q$ be a periodic saddle point that is homoclinically related to $P$. Then
there exists a map $\tilde{f}$ that is $C^{\infty}$ close to $f$ such that $S$
and $Q$ form a heterocycle.

One can choose $\tilde{f}$ to coincide with $f$ outside an arbitrarily small
neighborhood of $f^{-1}(S)\setminus\\{S\\}$. In particular if $W^{uu}(S;f)$
contains $Q$, then $S$ and $Q$ form a strong heterocycle for $\tilde{f}$.

###### Proof.

By assumption, there exists a point $z\in W^{u}(P)\cap
f^{-1}(S)\setminus\\{S\\}$. Let $Q_{-1}$ be the forward iterate of $Q$ which
satisfies $f(Q_{-1})=Q$. Since $Q_{-1}$ is homoclinically related to $P$,
there exists $z^{\prime}\in W^{u}(Q_{-1})$ arbitrarily close to $z$ having a
backward orbit which converges to the orbit of $Q$ and which avoids a uniform
neighborhood of $z$.

Hence, there exists a $C^{\infty}$-small perturbation of $f$ supported on a
small neighborhood of $z$ satisfying $\tilde{f}(z^{\prime})=f(z)$. In
particular $W^{u}(Q)$ contains $S$. ∎

We state a parametric version of the previous lemma.

######  Lemma 3.5.

Consider a $C^{\infty}$ family $(f_{a})_{a\in\mathbb{R}}$ in
$\mathrm{Diff}_{loc}^{\infty}(U,M)$, and, for $r\geq 1$, families of saddles
$(P_{a})_{a\in\mathbb{R}}$ and of projectively hyperbolic sources
$(S_{a})_{a\in\mathbb{R}}$ exhibiting a $C^{r}$-paraheterocycle at $a=0$. If
$(Q_{a})_{a\in\mathbb{R}}$ is a family of saddles homoclinically related to
$(P_{a})_{a\in\mathbb{R}}$, then there exists
$(\tilde{f}_{a})_{a\in\mathbb{R}}$, $C^{\infty}$-close to
$(f_{a})_{a\in\mathbb{R}}$ such that $Q_{0}$ and $S_{0}$ form a
$C^{r}$-paraheterocycle at $a=0$.

One can choose $(\tilde{f}_{a})_{a\in\mathbb{R}}$ to coincide with
$(f_{a})_{a\in\mathbb{R}}$ outside an arbitrarily small neighborhood of
$f^{-1}_{0}(S_{0})\setminus\\{S_{0}\\}$. Hence if $Q_{0}\in
W^{uu}(S_{0};f_{0})$, then $S_{0}$, $Q_{0}$ form a strong heterocycle for
$\tilde{f}_{0}$.

###### Proof.

Let $(K_{a})_{a\in I}$ be a basic set that contains $P_{a}$ and $Q_{a}$ for
$a$ in a neighborhood $I$ of $0$. Let $\underline{P}$ and $\underline{Q}$ be
the periodic lifts of $P$ and $Q$ in $\overleftarrow{K}$. By assumption, there
exists a choice of local unstable manifolds $W^{u}_{loc}(\underline{z},f_{a})$
for $\underline{z}\in\overleftarrow{K}$ and $N\geq 1$ such that
$d(S_{a},f^{N}(W^{u}_{loc}(\underline{P}_{a})))=o(|a|^{r})$. Since $P$ and $Q$
are homoclinically related, there exists a sequence of points
$\underline{z}_{n}\in\overleftarrow{K}$ which converges to $\underline{P}$ and
which belong to $W^{u}_{loc}(\underline{Q})$. Since
$W^{u}_{loc}(\underline{z};f_{a})$ varies continuously with $\underline{z}$
for the $C^{\infty}$-topology, when $n$ is large there exists a family
$(\tilde{f}_{a})_{a\in\mathbb{R}}$, which is $C^{\infty}$-close to
$(f_{a})_{a\in\mathbb{R}}$, such that
$d(S_{a},\tilde{f}^{N}_{a}(W^{u}_{loc}(\underline{z}_{n},\tilde{f}_{a})))=o(|a|^{r})$.
There exists a large integer $\ell\geq 1$ such that
$\tilde{f}^{N}_{a}(W^{u}_{loc}(\underline{z}_{n},\tilde{f}_{a}))\subset\tilde{f}^{\ell}_{a}(W^{u}_{loc}(\underline{Q}_{a},\tilde{f}_{a}))$,
hence
$d(S_{a},\tilde{f}^{\ell}_{a}(W^{u}_{loc}(\underline{Q}_{a},\tilde{f}_{a})))=o(|a|^{r})$
as in the definition of $C^{r}$-paracycle. Note that the perturbation can be
supported on a neighborhood of a point in
$f_{0}^{-1}(S_{0})\setminus\\{S_{0}\\}$. ∎

###  3.3 Proof of Proposition 2.1: from heterocycles to strong heterocycles

The main step in the proof of Proposition 2.1 is contained in the following
lemma.

######  Lemma 3.6.

Let us assume that both stable branches of $P$ intersect $W^{u}(P)$
transversally. Then there exists a map $\tilde{f}$, $C^{\infty}$-close to $f$,
with a saddle $Q$ homoclinically related to $P_{\tilde{f}}$ such that:

  * •

$f$ and $\tilde{f}$ coincide on $W^{u}_{loc}(P)$ and outside a small
neighborhood of $P$,

  * •

$W^{uu}(S,\tilde{f})$ contains $Q$.

###### Proof.

From $(T_{1})$, the curves $W^{uu}_{loc}(S)$ and ${\mathcal{S}}^{n}\circ{\cal
T}^{\scriptscriptstyle{\cal H}}(W^{s}_{loc}(P)\cap V_{H})$ intersect
transversally at a point whose image under ${\cal T}^{\scriptscriptstyle{\cal
S}}$ is denoted as $[S^{\prime},\bar{H}]$, see Fig. 9.

Figure 9: The two cases for the position of $[S^{\prime},\bar{H}]$.

We can reduce to the case depicted on the left part of Fig. 9, where
$[S^{\prime},\bar{H}]$ belongs to the half upper plane $\\{y>0\\}$ (for the
chart of $V_{P}$). Indeed if we are in the other case (depicted on the right
part of Fig. 9), we use the fact that the stable branch $\\{0\\}\times[-1,0]$
of $P$ has backward iterates which accumulate on $W^{s}_{loc}(P)$ in order to
replace $H$ by a point $H^{\prime}=(0,h^{\prime})$, $h^{\prime}<0$, which is a
transverse intersection between $W^{s}(P)$ and $W^{uu}(S)$. The new point
$[S^{\prime},\bar{H}^{\prime}]$ is close to $[S^{\prime},\bar{H}]$, hence
belongs to the lower half plane. It remains to conjugate the chart by
$(x,y)\mapsto(x,-y)$ in order to find the desired configuration.

Let us consider some large integers $n,N$, the map $\mathcal{G}_{2}$ and the
box $B$ defined at Section 3.2.2. The transversality conditions $(T_{2})$ and
$(T_{3})$ imply that ${\cal T}^{\scriptscriptstyle{\cal H}}(W^{uu}_{loc}(S))$
crosses the box $\mathcal{G}_{2}(B)$ along a small curve whose vertical
coordinate belongs to an interval
$[c_{1}.\sigma_{uu}^{n},c_{2}.\sigma_{uu}^{n}]$, where $c_{1},c_{2}$ are
independent from the choice of $n,N$.

We choose $n,N$ such that

(3.1)
$(h-\varepsilon)\lambda^{-N-1}<c_{2}\sigma_{uu}^{n}<(h-\varepsilon)\lambda^{-N}.$

Note that the condition $\varepsilon\sigma^{n}_{u}\lambda^{N}\gg 1$ is
satisfied and Lemma 3.3 associates a saddle point $\bar{Q}\in B$ whose
vertical coordinates is in
$[(h-\varepsilon)\lambda^{-N},(h+\varepsilon)\lambda^{-N}]$. By the previous
estimates, $\bar{Q}$ is “above” the graph ${\cal T}^{\scriptscriptstyle{\cal
H}}(W^{uu}_{loc}(S))$.

Now we consider a family $(f_{t})_{t\in[0,1]}$ such that $f_{0}=f$, and for
every $t$, the restrictions of $f_{t}$ to $W^{u}_{loc}(P)$ and to the
complement of a neighborhood of $V_{P}^{\prime}$ coincide with $f$, while the
restriction of the map $f_{t}|V_{P}^{\prime}$ is still linear with eigenvalues
$(\lambda_{t},\sigma)$ such that:

$\lambda_{t}=\frac{\lambda}{\sqrt[N]{1+t\cdot(C-1)}}\quad\text{ with
}C=\frac{c_{1}}{c_{2}}\frac{h-\varepsilon}{h+\varepsilon}\cdot\lambda^{-1}\;.$

Note that $(f_{t})_{t\in[0,1]}$ is a smooth family which is $C^{\infty}$-close
to be constantly equal to $f$ since $n$ is large. The map ${\mathcal{S}}$,
${\cal T}^{\scriptscriptstyle{\cal S}}$, ${\cal T}^{\scriptscriptstyle{\cal
H}}$ are unchanged, while ${\mathcal{P}}_{t}^{p}:=(x,y)\in
V_{P}\mapsto(\sigma\cdot x,\lambda_{t}\cdot y)$ depends on $a$. Any map of
this family satisfies the assumptions of Section 3.2.2. Let
$(\bar{Q}_{t})_{t\in[0,1]}$ be the hyperbolic continuation of $\bar{Q}$. The
vertical coordinate of $\bar{Q}_{1}$ is bounded by

$(h+\varepsilon)\lambda_{1}^{-N}=C\cdot(h+\varepsilon)\cdot\lambda^{-N}=(h-\varepsilon)\cdot\lambda^{-N-1}\frac{c_{1}}{c_{2}},$

From (3.1), it is smaller than $c_{1}.\sigma_{uu}^{n}$, hence $\bar{Q}_{1}$ is
“below” the graph ${\cal T}^{\scriptscriptstyle{\cal H}}(W^{uu}_{loc}(S))$.
One deduces that there exists a parameter such that $\bar{Q}_{t}$ belongs to
${\cal T}^{\scriptscriptstyle{\cal H}}(W^{uu}_{loc}(S))$. This implies that
$\bar{Q}_{t}$ has an iterate $Q$ which belongs to $W^{uu}_{loc}(S)$. ∎

###### Proof of Proposition 2.1 in the $C^{\infty}$ case.

One considers a basic set $K$ provided by Corollary 3.2. It contains a
periodic saddle $P^{\prime}$ homoclinically related to $P$ such that both of
its stable branches intersects $W^{u}(P^{\prime})$ transversally. The Lemma
3.4 allows by a first perturbation $\tilde{f}_{1}$ to replace $P$ by the
saddle $P^{\prime}$ so that the assumptions of the Lemma 3.6 are satisfied.
One can then build a new perturbation $\tilde{f}_{2}$ such that
$W^{uu}(S,\tilde{f}_{2})$ contains a saddle $Q$ which is homoclinically
related to $P$ and $P^{\prime}$, whereas the heterocycle between $S$ and
$P^{\prime}$ is not destroyed (since the perturbation does not modify $S$ nor
$W^{u}_{loc}(P^{\prime})$). After a third perturbation $\tilde{f}_{3}$
provided by Lemma 3.4, a strong heterocycle between $Q$ and $S$ is obtained. ∎

###  3.4 Proof of Proposition 2.1 in the analytic case

Now we assume $f\in\mathrm{Diff}_{loc}^{\omega}(U,M)$ and as before $f$
displays a heterocycle between a saddle $P$ and a source $S$. To prove
Proposition 2.1 in the analytic case, it suffices to show the following
counterparts of Lemmas 3.4 and 3.6.

######  Lemma 3.7.

Let $Q$ be a periodic saddle point that is homoclinically related to $P$. Then
there exists a map $\tilde{f}$ that is $C^{\omega}$ close to $f$ such that $S$
and $Q$ form a heterocycle.

If $W^{uu}(S;f)$ contains $Q$, then, one can choose $\tilde{f}$ so that $S$
and $Q$ form a strong heterocycle.

######  Lemma 3.8.

Let us assume that both stable branches of $P$ intersect $W^{u}(P)$
transversally. Then there exists a map $\tilde{f}$, $C^{\omega}$-close to $f$,
with a saddle $Q$ homoclinically related to $P_{\tilde{f}}$ such that:

  * •

$W^{uu}(S,\tilde{f})$ contains $Q$.

  * •

$W^{u}(P,\tilde{f})$ contains $S$.

###### Proof of Lemma 3.7.

First recall that $M$ is analytically embedded into an Euclidean space
$\mathbb{R}^{N}$, see [Gra58]. Hence there exists an analytic retraction
$\pi:U\to M$ of a neighborhood $U$ of $M$ in $\mathbb{R}^{N}$. Let
$W^{u}_{loc}(P)$ be a local unstable manifold of $P$ which contains $S$ in its
interior and let $S^{\prime}\neq S$ in $W^{u}_{loc}(P)$ such that
$f(S^{\prime})=S$. Let $V_{S^{\prime}}$ be a small neighborhood of
$S^{\prime}$ such that the backward orbit of $S^{\prime}$ inside
$W^{u}_{loc}(P)$ does not meet $S^{\prime}$. One takes an analytic chart
$\phi:V_{S^{\prime}}\to[-1,1]^{2}$ sending $S^{\prime}$ to $0$ and
$V_{S^{\prime}}\cap W^{u}_{loc}(P)$ to $[-1,1]\times\\{0\\}$.

Now consider a $C^{\infty}$-family $(f_{p})_{p\in[-\varepsilon,\varepsilon]}$
such that $f_{0}=f$ and each $f_{p}$ is equal to $f$ outside $V_{S^{\prime}}$
while on a smaller neighborhood of $S^{\prime}$, the map $f_{p}$ coincides
with the composition of $f$ with a translation of vector $(0,p)$. In
particular the continuation of $W^{u}_{loc}(P)$ for $f_{p}$ inside
$V_{S^{\prime}}$ is equal to $W^{u}_{loc}(P)$, while the continuations $S_{p}$
of $S$ and of its preimage $S^{\prime}_{p}=f^{-1}(S)\cap V_{S^{\prime}}$
satisfy that $\partial_{p}S^{\prime}_{p}|_{p=0}$ has non-zero second
coordinate. Remark that $\chi:=Df^{-1}\circ(\partial_{p}f_{p}|_{p=0})$ is a
smooth vector field defined on the compact subset
$\bar{U}\subset\mathbb{R}^{N}$. Then by Stone-Weierstrass Theorem, there
exists a polynomial vector fields
$\tilde{\chi}\in\mathbb{R}[X_{1},\dots,X_{N}]$ whose restriction to $\bar{U}$
is $C^{1}$-close $\chi$. Also by reducing $\varepsilon>0$, the following is
well defined for any $|p|<\varepsilon$:

$\tilde{f}_{p}:=x\in U\mapsto\pi\big{(}f(x)+p\cdot
Df\circ\tilde{\chi}(x)\big{)}\;.$

Note that $\partial_{p}\tilde{f}_{p}|_{p=0}=Df\circ\tilde{\chi}$ is
$C^{1}$-close to $\partial_{p}f_{p}|_{p=0}$. In particular the hyperbolic
continuation $(\tilde{S}^{\prime}_{p})_{p\in[-\varepsilon,\varepsilon]}$ of
$S^{\prime}$ for $(\tilde{f}_{p})_{p}$ is family $C^{1}$\- close to
$(\tilde{S}^{\prime}_{p})_{p\in[-\varepsilon,\varepsilon]}$. Also the
hyperbolic continuation
$(W^{u}_{loc}(P,\tilde{f}_{p}))_{p\in[-\varepsilon,\varepsilon]}$ is a family
of curves $C^{1}$\- close to the family constantly equal to
$[-1,1]\times\\{0\\}$. Hence assuming that the $C^{1}$-size of the
perturbation is small, the curve
$\Gamma:=\bigcup_{p\in[-\varepsilon,\varepsilon]}\\{\tilde{S}^{\prime}_{p}\\}\times\\{p\\}$
intersects transversaly the surface
$\Sigma:=\bigcup_{p\in[-\varepsilon,\varepsilon]}W^{u}_{loc}(P,\tilde{f}_{p})\times\\{p\\}$
at $\\{S^{\prime}\\}\times\\{0\\}$.

By the inclination lemma with parameter, see [Ber17, Lemma 3.2], there exists
a sequence $(W_{n,p})_{n}$ of $p$-families of segments $W_{n,p}\subset
W^{u}(Q,f_{p})$ such that the sequence of surfaces
$\Sigma_{n}:=\bigcup_{p\in[-\varepsilon,\varepsilon]}W_{n,p}\times\\{p\\}$
converges to $\Sigma$ in the $C^{1}$-topology as $n\to\infty$. Thus when $n$
is large, the curve $\Gamma$ intersects $\Sigma_{n}$ at a point close to
$\\{S^{\prime}\\}\times\\{0\\}$. Hence there is $p$ arbitrarily small such
that the continuations of $S^{\prime}$ and $Q$ form a heterocycle for
$\tilde{f}_{p}$. This proves the first part of the lemma since $\tilde{f}_{p}$
is $C^{\omega}$-close to $f$ when $p$ is small.

In the second part of the lemma, the saddle $Q$ belongs to a local strong
unstable manifold $W^{uu}_{loc}(S)$ of $S$ and one performs a similar
construction. Let $Q^{\prime}\neq Q$ in $W^{uu}_{loc}(S)$ which satisfies
$f(Q^{\prime})=Q$, let $V_{Q^{\prime}}$ be a small neighborhood of
$Q^{\prime}$, and consider a chart $\psi\colon V_{S^{\prime}}\to[-1,1]^{2}$
sending $Q^{\prime}$ to $0$ and $V_{Q^{\prime}}\cap W^{uu}_{loc}(Q)$ to
$[-1,1]\times\\{0\\}$. One considers a $C^{\infty}$ family of maps which are
equal to $f$ outside $V_{Q^{\prime}}$ and which coincide with the composition
of $f$ with a translation of vector $(0,q)$ on a small neighborhood of
$Q^{\prime}$: it induces a vector field $\xi$, that can be approximated by a
polynomial vector field $\tilde{\xi}$. Up to shrinking $\varepsilon>0$, for
every $(p,q)\in[-\varepsilon,\varepsilon]^{2}$, the following is well defined:

$\tilde{f}_{p,q}:=x\in M\mapsto\pi\big{(}f(x)+p\cdot
Df\circ\tilde{\chi}(x)+q\cdot Df\circ\tilde{\xi}(x)\big{)}\;.$

Similarly we can consider the continuation $\tilde{S}_{p,q}$ of $S$,
$\tilde{Q}_{p,q}$ of $Q$, $W^{u}_{loc}(P,\tilde{f}_{p,q})$ of
$W^{u}_{loc}(P,f)$, $W_{n,p,q}$ of $W_{n,p}$ and
$W^{u}_{loc}(Q,\tilde{f}_{p,q})$ of $W^{u}_{loc}(Q)$.

From the first part of the proof, $W^{u}_{loc}(P,\tilde{f}_{p,q})$ contains
$\tilde{S}_{p,q}$ when $(p,q)$ belongs to graphs $\gamma_{n}$ that are
arbitrarily $C^{1}$-close to the curve $p=0$ when $n\to\infty$. By a similar
argument, $W^{uu}_{loc}(S,\tilde{f}_{p,q})$ contains $\tilde{Q}_{p,q}$ when
$(p,q)$ belongs to a one-dimensional submanifold $\sigma$ that contains $0$,
is $C^{1}$-close to the curve $q=0$. In particular $\sigma$ is transverse to
the graphs $\gamma_{n}$. Thus the conclusion of the lemma holds for some map
$\tilde{f}_{p,q}$ with $(p,q)\in\gamma_{n}\cap\sigma$ which is
$C^{\omega}$-close to $f$ when $n$ is large and $p,q$ are small. This implies
the second part of the lemma. ∎

###### Proof of Lemma 3.8.

The proof of Lemma 3.6 was obtained using a smooth family which changes the
stable eigenvalue of $P$, without changing the relative position of $S$ w.r.t.
$W^{u}_{loc}(P;f)$. To obtain the analytic setting, as above, we approximate
this family by an analytic one and we add an extra parameter which varies the
relative position of $S$ w.r.t. $W^{u}_{loc}(P;f)$. While the first parameter
enables to find a saddle $Q$ homoclinically related to $P$ such that $Q\in
W^{uu}_{loc}(S)$, in the analytic setting this unfolding might unfold also the
heterocycle. However the new second parameter enables to restore it. ∎

###  3.5 Proof of Proposition 2.3: from paraheterocycles to strong
paraheterocycles

We follow the proof of the Proposition 2.1 in the $C^{\infty}$ case. After a
first $C^{\infty}$-small perturbation of $f_{0}$ (and hence of the family
$(f_{a})_{a\in\mathbb{R}}$), there exists a saddle $Q$ homoclinically related
to $P$ which belongs to $W^{uu}_{loc}(S)$. The paracycle property (1.1)
between $S$ and $P$ may not hold anymore, but by a new perturbation, with a
similar size, it can be restored. Note that it is supported near
$f^{-1}(S)\setminus\\{S\\}$, hence the property $Q\in W^{uu}_{loc}(S)$ is not
destroyed. Finally one applies lemma 3.5, and gets a $C^{\infty}$-small
perturbation of the family $(f_{a})_{a\in\mathbb{R}}$ in order to get a strong
$C^{r}$-paraheterocycle at $a=0$ between $S$ and $Q$. ∎

##  4 From chains of heterocycles to paraheterocycles

We prove Proposition 2.10 in this section: an $N$-chain of alternate
heterocycles whose saddles are homoclinically related, can be perturbed as a
$C^{d}$-paraheterocycle, provided that $N$ is large enough with respect to
$d$. This is shown by induction on $d$. The case $d=0$ follows from the
continuity of the family (without any perturbation). The induction step is
given by:

######  Proposition 4.1.

Consider a $C^{\infty}$ family $(f_{a})_{a\in\mathbb{R}}$ in
$\mathrm{Diff}_{loc}^{\infty}(U,M)$ and $d\geq 0$ such that $f_{0}$ has a
$2$-chain of alternate heterocyles with saddle points $P^{1},P^{2}$ and
sources $S^{1},S^{2}$ such that $(P^{1},S^{1})$ and $(P^{2},S^{2})$ form two
$C^{d}$-paraheterocycles at $a=0$. Then there is a $C^{\infty}$-perturbation
of $(f_{a})_{a\in\mathbb{R}}$ such that the continuation of $(P^{1},S^{2})$
forms a $C^{d+1}$-paraheterocycle at $a=0$.

Moreover the perturbation is supported on a small neighborhood of $\text{\rm
orbit}(S^{1})\cup\text{\rm orbit}(S^{2})$.

###### Proof of Proposition 2.10.

One considers a $2^{d}$-chain of alternate heterocycles with periodic points
$P^{1},S^{1},\dots,P^{2^{d}},S^{2^{d}}$. Proposition 4.1 allows to perform a
perturbation at $\text{\rm orbit}(S^{1})\cup\text{\rm orbit}(S^{2})$, such
that $P^{1}$ and the continuation of $S^{2}$ form a $C^{1}$-paraheterocycle.

Note that $P^{1},S^{2},P^{3},S^{3},\dots,P^{2^{d}},S^{2^{d}}$ is still a
$2^{d}-2$-chain of alternate heterocycles. By induction, one gets a
$2^{d-1}$-chain of alternate heterocycles
$P^{1},S^{2},\dots,P^{2^{d}-1},S^{2^{d}}$ such that $P^{2i+1},S^{2i+2}$ form a
$C^{1}$-paraheterocycle at $a=0$, for each $0\leq i<2^{d-1}$.

By a new perturbation supported near the sources, one gets a $2^{d-2}$-chain
of alternate heterocycles $P^{1},S^{4},\dots,P^{2^{d}-3},S^{2^{d}}$ such that
each pair $P^{4i+1},S^{4i+4}$ forms a $C^{2}$-paraheterocycle at $a=0$.
Repeating this construction inductively, one gets a $C^{d}$-paraheterocycle at
$a=0$ between $P^{1}$ and the continuation of $S^{2^{d}}$. ∎

Proposition 4.1 is proved in the next two subsections. In Section 4.3 we
discuss the case where there are several parameters.

###  4.1 Notations and local coordinates

The setting is similar to Section 3.1 and depicted Fig. 10. We chooses a large
integer $r$ and a small number $\varepsilon>0$, we look for a smooth
perturbation of $(f_{a})_{a\in\mathbb{R}}$ which is
$\varepsilon$-$C^{r}$-small and such that the continuation of $(P^{1},S^{2})$
forms a $C^{d+1}$-paraheterocycle at $a=0$.

As in Section 3 we shall assume that the points $P^{2}$ and $S^{1}$ are fixed.
We denote by $|\sigma_{a}|<1$ and $\lambda_{a}<-1$ (resp. by
$|\sigma^{uu}_{a}|<|\sigma_{a}^{u}|<1$) the inverse of the eigenvalues of the
tangent map of $f_{a}$ at $P^{2}_{a}$ (resp. at $S^{1}_{a}$).

After a small perturbation we can assume that the eigenvalues are non-resonant
and:

$\frac{\log|\sigma^{u}_{0}|}{\log|\lambda_{0}|}\in\mathbb{R}\setminus\mathbb{Q}\;.$

Then by [Tak71], there exist:

  * •

neighborhoods $V^{\prime}_{S}(a)\subset V_{S}(a):=f_{a}(V^{\prime}_{S}(a))$ of
$S^{1}_{a}$ endowed with coordinates depending $C^{r}$ on the parameter and
for which the inverse branche ${\mathcal{S}}_{a}:=(f_{a}|V_{S}^{\prime})^{-1}$
has the form:

${\mathcal{S}}_{a}:(x,y)\in V_{S}\mapsto(\sigma^{uu}_{a}\cdot
x,\sigma^{u}_{a}\cdot y)\in V_{S}^{\prime}$

  * •

neighborhoods $V^{\prime}_{P}(a)$ and $V_{P}(a):=f_{a}(V_{P}^{\prime}(a))$ of
$P_{a}^{2}$ endowed with coordinates depending $C^{r}$ on the parameter and
for which the inverse branch has the form:

${\mathcal{P}}_{a}:(x,y)\in V_{P}(a)\mapsto(\sigma_{a}\cdot x,\lambda_{a}\cdot
y)\in V_{P}^{\prime}(a)\;.$

Up to restricting $V_{P}$, $V_{P}^{\prime}$ and $V_{S}$ we can assume them
equal to filled rectangles containing $0$ in their interior. We define:

$W^{u}_{loc}(P^{2}_{a})\equiv V_{P}(a)\cap\\{y=0\\}\;,\quad
W^{s}_{loc}(P^{2}_{a})\equiv
V^{\prime}_{P}(a)\cap\\{x=0\\}\quad\mathrm{and}\quad
W^{uu}_{loc}(S^{1}_{a})\equiv\\{y=0\\}\cap V_{S}(a)\;.$

Let $H_{0}$ be a point in $W^{s}(P_{0}^{2})\cap W^{uu}(S_{0}^{1})$. Up to
replacing it by an iterate, we can assume that $H_{0}$ belongs to
$V^{\prime}_{P}(0)$ with $H_{0}\equiv(0,h_{0})$ in the linearizing coordinates
of $P_{0}^{2}$.

Also, a preimage $S^{\prime 2}_{a}$ of $S^{2}_{a}$ by an iterate of $f_{a}$
has coordinates $S^{\prime 2}_{a}=:(x_{a},y_{a})$ in the linearizing
coordinates of $P^{2}_{a}$. Let ${\mathcal{T}}_{a}:V_{H}\hookrightarrow V_{S}$
be an inverse branches of an iterate of $f_{a}$ defined on a neighborhood
$V_{H}\Subset V^{\prime}_{P}$ of $H_{0}$ and such that ${\mathcal{T}}_{0}$
sends $H_{0}$ into $W^{uu}_{loc}(S^{1}_{0})$.

Up to a smooth perturbation, one can require that:

  1. $(T_{1})$

$W^{u}(P^{1})$ is transverse to $E^{cu}_{S^{1}}$ at $S^{1}$,

  2. $(T_{2})$

$W^{uu}_{loc}(S^{1}_{0})$ and $W^{s}_{loc}(P^{2}_{0})$ are transverse at
$H_{0}$.

By $(T_{1})$, $W^{u}(P^{1}_{a})$ contains a graph in the chart at $S^{1}_{a}$,
over a neighborhood $I\subset\mathbb{R}$ of $0$:

$\Gamma_{a}\equiv\\{(x,\gamma_{a}(x));\;x\in I\\}.$

By $(T_{2})$, the transverse intersection $H_{0}$ admits a continuation
$H_{a}$ for $a$ close to $0$. One sets

$H_{a}\equiv(0,h_{a})\quad\mathrm{and}\quad{\mathcal{T}}_{a}(H_{a})=(z_{a},0)\,.$

Since $(P^{1},S^{1})$ and $(P^{2},S^{2})$ form two $C^{d}$-paraheterocycles at
$a=0$, one has for any $0\leq k\leq d$,

$\partial^{k}_{a}\gamma_{a}(0)|_{a=0}=0\quad\text{and}\quad\partial^{k}_{a}y_{a}|_{a=0}=0.$

Up to a small perturbation, one can also assume that

$\partial^{d+1}_{a}\gamma_{a}(0)|_{a=0}\neq
0\quad\text{and}\quad\partial^{d+1}_{a}y_{a}|_{a=0}\neq 0.$

Figure 10 summaries the notations.

Figure 10: Notations.

###  4.2 Compositions nearby a paraheterocycle

Let $\Delta$ be the second coordinate of
$\partial_{y}{\mathcal{T}}_{0}(H_{0})$; it is nonzero by $(T_{2})$.

######  Lemma 4.2.

Given integers $n,m\geq 1$ large such that
$(\sigma^{u}_{0})^{n}\lambda_{0}^{m}=O(1)$, there is a $C^{r}$-perturbation of
$(f_{a})_{a}$ locallized at $S^{2}_{a}$ such that the germ at $a=0$ of
$a\mapsto{\mathcal{S}}_{a}^{n}\circ{\mathcal{T}}_{a}\circ{\mathcal{P}}_{a}^{m}(S^{\prime
2}_{a})$ is $C^{d+1}$-close to

$\left(0,(\sigma^{u}_{0})^{n}\cdot\Delta\cdot\lambda^{m}_{0}\cdot\frac{\partial_{a}^{d+1}y_{a}|_{a=0}}{(d+1)!}a^{d+1}\right).$

###### Proof.

For $m$ large, after a $C^{r}$-small perturbation localized at $S_{a}^{2}$
(which is conjugated to a translation in a small neighborhood of $S_{a}^{2}$),
we can assume $S^{\prime 2}_{a}=(x_{a},y_{a}+\varepsilon_{a})$ where
$a\mapsto\varepsilon_{a}$ is the $C^{\infty}$-small function defined by
$\varepsilon_{a}:=\lambda_{a}^{-m}\cdot h_{a}$ and where as before
$(x_{a},y_{a})$ are the coordinates of $S^{\prime 2}_{a}$ before the
perturbation.

Then observe that ${\mathcal{P}}_{a}^{m}(S^{\prime
2}_{a})=H_{a}+(\sigma_{a}^{m}\cdot x_{a},\lambda_{a}^{m}\cdot y_{a})$ forms a
family whose germ at $a=0$ is $C^{d}$-close to $(H_{a})_{a}$. When $m$ is
large, the germ at $a=0$ of
$a\mapsto{\mathcal{T}}_{a}\circ{\mathcal{P}}_{a}^{m}(S^{\prime 2}_{a})$ is
$C^{d+1}$-close to

${\mathcal{T}}_{a}(H_{a})+D_{H_{a}}{\mathcal{T}}_{a}\left(\sigma_{a}^{m}\cdot
x_{a},\lambda^{m}_{a}\cdot\frac{\partial_{a}^{d+1}y_{a}|_{a=0}}{(d+1)!}a^{d+1}\right)$

and so $C^{d+1}$-close to

$(z_{a},0)+\partial_{y}{\mathcal{T}}_{0}(H_{0})\cdot\lambda^{m}_{a}\cdot\frac{\partial_{a}^{d+1}y_{a}|_{a=0}}{(d+1)!}a^{d+1}\;.$

Consequently, for any $n\geq 0$, the germ at $a=0$ of
$a\mapsto{\mathcal{S}}_{a}^{n}\circ{\mathcal{T}}_{a}\circ{\mathcal{P}}_{a}^{m}(S^{\prime
2}_{a})$ is $C^{d+1}$-close to

$((\sigma_{a}^{uu})^{n}\cdot
z_{a},0)+\mathrm{diag}((\sigma^{uu}_{a})^{n},(\sigma^{u}_{a})^{n})\cdot\partial_{y}{\mathcal{T}}_{0}(H_{0})\cdot\lambda^{m}_{a}\cdot\frac{\partial_{a}^{d+1}y_{a}|_{a=0}}{(d+1)!}a^{d+1}\;.$

If $(\sigma^{u}_{0})^{n}\lambda_{0}^{m}=O(1)$, then both
$(\sigma_{a}^{uu})^{n}$ and $(\sigma^{uu}_{0})^{n}\lambda_{0}^{m}$ are small,
and so we obtain the announced bound. ∎

Since the ratio $\log|\sigma^{u}_{0}|/\log|\lambda_{0}|$ is irrational, and
since $\partial^{d+1}_{a}\gamma_{a}(0)|_{a=0}\neq 0$ and
$\partial^{d+1}_{a}y_{a}|_{a=0}\neq 0$, one can choose some large positive
integers $n,m$ such that

$n\log|\sigma^{u}_{0}|+m\log|\lambda_{0}|-\log|\Delta|+\log|\partial^{d+1}_{a}y_{a}|_{a=0}$

is arbitrarily close to $\log|\partial^{d+1}_{a}\gamma_{a}(0)|_{a=0}$. Since
$\lambda$ is negative, one can choose $m$ to be odd or even so that
$\Delta\cdot(\sigma_{0}^{u})^{n}(\lambda_{0})^{m}\partial^{d+1}y_{a}|_{a=0}$
and $\partial^{d+1}\gamma_{a}(0)|_{a=0}$ have the same sign.

By our assumptions, the $C^{d}$-jets of $a\mapsto\gamma_{a}(0)$ and $a\mapsto
y_{a}$ at $a=0$ vanish. With our choices, this guaranties that the
$C^{d+1}$-jet at $a=0$ of
$a\mapsto\Delta\cdot(\sigma_{0}^{u})^{n}(\lambda_{0})^{m}y_{a}-\gamma_{a}(0)$
is arbitrarily small. By Lemma 4.2, after a $C^{r}$-perturbation of
$(f_{a})_{a}$ localized at $(S^{2}_{a})_{a}$, the germ at $a=0$ of the
following function is $C^{d+1}$-small:

$a\mapsto\eta_{a}:=\gamma_{a}\circ
p_{x}\circ{\mathcal{S}}_{a}^{n}\circ{\mathcal{T}}_{a}\circ{\mathcal{P}}_{a}^{m}(S^{\prime
2}_{a})-p_{y}\circ{\mathcal{S}}_{a}^{n}\circ{\mathcal{T}}_{a}\circ{\mathcal{P}}_{a}^{m}(S^{\prime
2}_{a}).$

A $C^{r}$-small perturbation localized at $S_{a}^{1}$ (which is locally
conjugated to a translation) translates the functions $(\gamma_{a})_{a}$ by
$-\eta_{a}$ for each parameter $a$ close to $0$. Then we have at $a=0$:

$d(\Gamma_{a},{\mathcal{S}}_{a}^{n}\circ{\mathcal{T}}_{a}\circ{\mathcal{P}}_{a}^{m}(S^{\prime
2}_{a}))=o(a^{d+1}).$

As a consequence, the continuation of $P^{1}_{0}$ and $S^{2}_{0}$ form a
$C^{d+1}$-paraheterocycle at $a=0$ for the chosen perturbation. Since the
charts are a priori only $C^{r}$, the resulting perturbation is only $C^{r}$.
In a last step, we thus smooth the family near the sources, keeping the
paraheterocycle we have obtained (the latter being a finite codimensional
condition on the family). Proposition 4.1 is now proved. ∎

###  4.3 Families parametrized by $k$-parameters

When the family $(f_{a})$ is parametrized by $a=(a_{1},\dots,a_{k})$ in
$\mathbb{R}^{k}$, $k>1$, the proof follows the same scheme, by canceling one
by one the partial derivatives
$\partial_{a_{1}}^{i_{1}}\partial_{a_{2}}^{i_{2}}\cdots\partial_{a_{k}}^{i_{k}}$
of the unfolding of the heterocycle. For this end, we proceed by induction on
$\\{\underline{i}=(i_{1},\dots,i_{k})\in\mathbb{N}^{k}:\;\sum_{j}i_{j}\leq
d\\}$ following an order $\prec$ such that:

$\sum_{j}i_{j}<\sum_{j}i_{j}^{\prime}\Rightarrow\underline{i}\prec\underline{i}^{\prime}.$

##  5 Nearly affine (para)-blender renormalization

In this section, we prove Propositions 2.2 and 2.4.

We consider a $C^{\infty}$ map $f\in\mathrm{Diff}_{loc}^{\infty}(U,M)$ with a
projectively hyperbolic source $S$ and a saddle point $Q$ forming a strong
heterocycle, and build by perturbation a nearly affine blender homoclinically
related to $Q$. It is defined by two inverse branches from a neighborhood of
$Q$ to “vertical rectangles” stretching across the local unstable manifold of
the saddle.

In §5.1 and §5.2 we choose nice coordinate systems for the inverse dynamics
nearby the source, the saddle and the heteroclinic orbits. It requires
preliminary perturbation in order to satisfy some non-resonance and
transversality conditions. We also explain how to unfold the strong
heterocycle. The heterocycle induces well-defined inverse branches of the
dynamics (§5.3) that are transitions from one linearizing chart to the other.
§5.4 provides $C^{r}$-estimates on rescalings $g^{-},g^{+}$ of the inverse
branches. In §5.5 and §5.6, we tune the length of the branches and the size of
the unfolding so that the inverse branches defines a nearly affine blender
with a neat dilation $\Delta$; it is homoclinally related to the saddle point
$P$ and that its activation domain contains $S$. In other words, Proposition
2.2 will be proved.

In §5.8, we add a parameter, consider a family $(f_{a})_{a\in\mathbb{R}}$ and
apply the previous discussion to $f_{0}$. The inverse branches admit
continuations $(g^{-}_{a})_{a\in\mathbb{R}}$ and
$(g^{+}_{a})_{a\in\mathbb{R}}$. After having chosen an adapted
reparametrization, we extend the $C^{r}$-bounds to the parametrized families
and check that they define a nearly affine $C^{r}$-parablender, concluding the
proof of Proposition 2.4.

##### Notations.

The proofs will depend on a small number $\varepsilon>0$ and on integers
$n^{+},n^{-},m^{+},m^{-}$. The notation $A=O(\varepsilon)$ (or more generally
$A=O(g(\varepsilon,n^{+},n^{-},m^{+},m^{-}))$) will mean that the quantity $A$
has a norm bounded by $C.\varepsilon$ (or by
$C.|g(\varepsilon,n^{+},n^{-},m^{+},m^{-})|$), where the number $C>0$ depends
on the initial map $f$ but not on the choices made during the construction.

Similarly, one will say that a function $h$ (that may depend on coordinates
$x,y$, and/or parameters $a$ or $\alpha$) is _$C^{r}$ -dominated_ by
$\varepsilon$ if $\partial^{k}h=O(\varepsilon)$ for all its $k^{\text{th}}$
derivatives with respect to $x,y,a,\alpha$ with $0\leq k\leq r$. Note that if
in the $C^{r}$-topology, $h_{i}=h^{\prime}_{i}+O(\varepsilon)$,
$i\in\\{1,2\\}$, then $h_{1}\circ h_{2}=h^{\prime}_{1}\circ
h^{\prime}_{2}+O(\varepsilon)$.

###  5.1 Coordinates for generic perturbations of strong heterocycles

We first fix a system of coordinate as depicted in Fig. 11. As in Section 3.1,
we shall assume that the points $Q$ and $S$ are fixed and the eigenvalues
$1<\sigma_{u}^{-1}<\sigma_{uu}^{-1}$ and $0<\lambda^{-1}<1<\sigma^{-1}$ of
$D_{Q}f$ and $D_{S}f$ respectively are positive and non-resonant. Furthermore
we can assume that:

(5.1) $\frac{\log\sigma_{u}}{\log\lambda}\notin\mathbb{Q}\;.$

The hypothesis of the proposition consists of two finite codimensional
conditions:

(5.2) $S\in W^{u}(Q;f)\quad\mathrm{and}\quad Q\in W^{uu}(S;f)\;.$

So after a small smooth perturbation, we can assume moreover:

(5.3) $T_{Q}W^{uu}(S;f)\oplus T_{Q}W^{s}(Q;f)=T_{Q}M\quad\mathrm{and}\quad
E^{cu}(S)\oplus T_{S}W^{u}(Q;f)=T_{S}M\;.$

As in Section 3.1, the non-resonance of the eigenvalues and the smoothness of
the dynamics imply, by the Sternberg Theorem [Ste58], the existence of:

  * •

Neighborhoods $V^{\prime}_{S}\subset V_{S}:=f(V^{\prime}_{S})$ of $S$ and
coordinates for which $f|V^{\prime}_{S}$ has the form:

$f\colon(x,y)\in V^{\prime}_{S}\mapsto(\sigma_{uu}^{-1}\cdot
x,\sigma_{u}^{-1}\cdot y)\in V_{S}.$

  * •

Neighborhoods $V^{\prime}_{Q}$ and $V_{Q}=f(V_{Q}^{\prime})$ of $Q$ and
coordinates in which $f|V^{\prime}_{Q}$ has the form:

$f\colon(x,y)\in V^{\prime}_{Q}\mapsto(\sigma^{-1}\cdot x,\lambda^{-1}\cdot
y)\in V_{Q}.$

This defines the inverse branches ${\mathcal{Q}}:=(f|V_{Q}^{\prime})^{-1}$ and
${\mathcal{S}}:=(f|V_{S}^{\prime})^{-1}$:

${\mathcal{S}}:(x,y)\in V_{S}\mapsto(\sigma_{uu}\cdot x,\sigma_{u}\cdot y)\in
V_{S}^{\prime}\quad\mathrm{and}\quad{\mathcal{Q}}:(x,y)\in
V_{Q}\mapsto(\sigma\cdot x,\lambda\cdot y)\in V_{Q}^{\prime}\;.$

Up to restrict $V_{Q}$ and $V_{Q}^{\prime}$ and rescale their coordinate, we
can assume:

$V_{Q}\equiv[-2,2]\times[-2\lambda^{-1},2\lambda^{-1}]\quad\mathrm{and}\quad
V_{Q}^{\prime}\equiv[-2\sigma,2\sigma]\times[-2,2]\;.$

Let $W^{u}_{loc}(Q):=V_{Q}\cap\\{y=0\\}$,
$W^{s}_{loc}(Q):=V^{\prime}_{Q}\cap\\{x=0\\}$ and
$W^{uu}_{loc}(S):=\\{y=0\\}\cap V_{S}$.

By Eq. 5.2 there is a neighborhood $V^{\prime\prime}_{S}\Subset
V^{\prime}_{S}$ of $0\equiv S$ and an inverse branch ${\cal
T}^{\scriptscriptstyle{\cal S}}:V_{S}^{\prime\prime}\hookrightarrow V_{Q}$ of
an iterate of $f$ sending $0$ into $[-2,2]\times\\{0\\}$. Similarly, there
exists a neighborhood $V^{\prime\prime}_{Q}\Subset V^{\prime}_{Q}\cap V_{Q}$
of $0\equiv Q$ and an inverse branch ${\cal T}^{\scriptscriptstyle{\cal
Q}}:V_{Q}^{\prime\prime}\hookrightarrow V_{S}$ of an iterate of $f$ sending
$0$ into $V_{S}\cap\\{y=0\\}$. The inverse branches ${\cal
T}^{\scriptscriptstyle{\cal S}}$ and ${\cal T}^{\scriptscriptstyle{\cal Q}}$
are called the _transitions maps_.

Figure 11: Inverse branches given by the strong heterocycle.

Assuming the neighborhoods $V_{S}$ and $V_{Q}$ small enough, it is possible
(up to compose by an iterate of $f$) to choose ${\cal
T}^{\scriptscriptstyle{\cal S}},{\cal T}^{\scriptscriptstyle{\cal Q}}$ such
that

${\cal T}^{\scriptscriptstyle{\cal S}}(0)\in V^{\prime}_{Q}\setminus
V_{Q}\quad\mathrm{and}\quad{\cal T}^{\scriptscriptstyle{\cal Q}}(0)\in
V_{S}\setminus V^{\prime}_{S}.$

Let the coordinates of ${\cal T}^{\scriptscriptstyle{\cal S}}$ and ${\cal
T}^{\scriptscriptstyle{\cal Q}}$ be

${\cal T}^{\scriptscriptstyle{\cal S}}:=({\cal X}^{\scriptscriptstyle{\cal
S}},{\cal Y}^{\scriptscriptstyle{\cal S}})\quad\mathrm{and}\quad{\cal
T}^{\scriptscriptstyle{\cal Q}}:=({\cal X}^{\scriptscriptstyle{\cal Q}},{\cal
Y}^{\scriptscriptstyle{\cal Q}})\;.$

By Eq. 5.3, $\partial_{y}{\cal Y}^{\scriptscriptstyle{\cal Q}}(0)\not=0$. Thus
by rescaling one of the linearizing chart, we can assume:

(5.4) $\partial_{y}{\cal Y}^{\scriptscriptstyle{\cal Q}}(0)=1.$

###  5.2 Unfolding of the strong heterocycle

We will perturb ${\cal T}^{\scriptscriptstyle{\cal S}},{\cal
T}^{\scriptscriptstyle{\cal Q}}$ so that the following points are close to but
_not necessarily_ in $\\{y=0\\}$:

$S^{\prime}=(s^{\prime}_{x},s^{\prime}_{y}):={\cal T}^{\scriptscriptstyle{\cal
S}}(0)\quad\mathrm{and}\quad Q^{\prime}=(q^{\prime}_{x},q^{\prime}_{y}):={\cal
T}^{\scriptscriptstyle{\cal Q}}(0)\;.$

This is enabled by the next claim without changing any derivative of the
inverse branches.

######  Claim 5.1.

For every small numbers $s^{\prime}_{y}$ and $q^{\prime}_{y}$, there exists a
$C^{\infty}$ perturbation of the dynamics such that the inverse branches
${\mathcal{S}}$ and ${\mathcal{Q}}$ remain unchanged, while the continuations
of the inverse branches ${\cal T}^{\scriptscriptstyle{\cal S}}$ and ${\cal
T}^{\scriptscriptstyle{\cal Q}}$ have the same derivatives but satisfy:

${\cal Y}^{\scriptscriptstyle{\cal
S}}(0)=s^{\prime}_{y}\quad\mathrm{and}\quad{\cal Y}^{\scriptscriptstyle{\cal
Q}}(0)=q^{\prime}_{y}.$

###### Proof.

First recall that ${\cal T}^{\scriptscriptstyle{\cal S}}(0)\in V_{Q}\setminus
V_{Q}^{\prime}$. One perturbs $f$ by composing with a translation supported on
a small neighborhood of ${\cal T}^{\scriptscriptstyle{\cal S}}(0)$. This
enables to move the vertical position of ${\cal T}^{\scriptscriptstyle{\cal
S}}(0)$, without affecting the other branches. The modification of ${\cal
T}^{\scriptscriptstyle{\cal Q}}(0)$ is done similarly. ∎

In the following we will prescribe some values of
$s^{\prime}_{y},q^{\prime}_{y}$ and consider the perturbed dynamics. The
inverse branches of the new system will be still denoted by ${\mathcal{Q}}$,
${\mathcal{S}}$, ${\cal T}^{\scriptscriptstyle{\cal S}}=({\cal
X}^{\scriptscriptstyle{\cal S}},{\cal Y}^{\scriptscriptstyle{\cal S}})$ and
${\cal T}^{\scriptscriptstyle{\cal Q}}=({\cal X}^{\scriptscriptstyle{\cal
Q}},{\cal Y}^{\scriptscriptstyle{\cal Q}})$. The next lemma enables to assume
that $\partial_{y}{\cal Y}^{\scriptscriptstyle{\cal S}}(0)$ is positive.

######  Lemma 5.2.

Up to perturbation $f$ and to change ${\cal T}^{\scriptscriptstyle{\cal S}}$,
we can assume moreover that

$\partial_{y}{\cal Y}^{\scriptscriptstyle{\cal S}}(0)>0.$

###### Proof.

If $\partial_{y}{\cal Y}^{\scriptscriptstyle{\cal S}}(0)<0$, we are going to
perturb $f$ and replace ${\cal T}^{\scriptscriptstyle{\cal S}}$ by the inverse
branch $\widetilde{\cal T}^{\scriptscriptstyle{\cal S}}:={\cal
T}^{\scriptscriptstyle{\cal S}}\circ{\mathcal{S}}^{n}\circ{\cal
T}^{\scriptscriptstyle{\cal Q}}\circ{\mathcal{Q}}^{m}\circ{\cal
T}^{\scriptscriptstyle{\cal S}}$ for some large $n$ and $m$. First note that
for any large $n$ and any $m$, the map $\widetilde{\cal
T}^{\scriptscriptstyle{\cal S}}$ is well defined on a small neighborhood of
$0\equiv S$. Also $\partial_{y}{\cal T}^{\scriptscriptstyle{\cal S}}(0)$ is a
vector with negative vertical component. By hyperbolicity, it is sent by
$D{\mathcal{Q}}^{m}$ to a vertical vector. Its vertical component is still
negative since $\lambda>0$. It is pointed at a point
${\mathcal{Q}}^{m}\circ{\cal T}^{\scriptscriptstyle{\cal S}}(0)$ close to
$0\equiv S$. Thus for $m$ sufficiently large, by Eq. 5.4, its image by $D{\cal
T}^{\scriptscriptstyle{\cal S}}$ is a vector with negative vertical component.
By projective hyperbolicity of the source $S$, its image by $D{\mathcal{S}}$
is a vector vertical, pointed at a point nearby $S$ when $n$ is large, and
with negative vertical component. Consequently it is sent by ${\cal
T}^{\scriptscriptstyle{\cal S}}$ to a vector with positive vertical component
at a point nearby ${\cal T}^{\scriptscriptstyle{\cal S}}(0)$. In other words,
the second coordinate of $\partial_{y}\widetilde{\cal
T}^{\scriptscriptstyle{\cal S}}(0)$ is positive.

It remains to perform a perturbation of $f$ so that the second coordinate of
$\widetilde{\cal T}^{\scriptscriptstyle{\cal S}}(0)$ is zero. Let $({\cal
T}^{\scriptscriptstyle{\cal S}}_{t})_{t}$ be the family of perturbations of
${\cal T}^{\scriptscriptstyle{\cal S}}$ given by 5.1 and enabling to move the
$y$-coordinate of ${\cal T}^{\scriptscriptstyle{\cal S}}(0)$. Note that when
$n\gg m$,

$\partial_{t}({\cal T}^{\scriptscriptstyle{\cal
S}}_{t}\circ{\mathcal{S}}^{n}\circ{\cal T}^{\scriptscriptstyle{\cal
Q}}\circ{\mathcal{Q}}^{m}\circ{\cal T}^{\scriptscriptstyle{\cal
S}}_{t})(0)\approx\partial_{t}{\cal T}^{\scriptscriptstyle{\cal
S}}_{t}(0)+D({\cal T}^{\scriptscriptstyle{\cal
S}}\circ{\mathcal{S}}^{n}\circ{\cal T}^{\scriptscriptstyle{\cal
Q}}\circ{\mathcal{Q}}^{m})(\partial_{t}{\cal T}^{\scriptscriptstyle{\cal
S}}_{t}(0))\approx\partial_{t}{\cal T}^{\scriptscriptstyle{\cal S}}_{t}(0)\;.$

Hence there is a small parameter $t$ such that $\widetilde{{\cal
T}^{\scriptscriptstyle{\cal S}}_{t}}:=({\cal T}^{\scriptscriptstyle{\cal
S}}_{t}\circ{\mathcal{S}}^{n}\circ{\cal T}^{\scriptscriptstyle{\cal
Q}}\circ{\mathcal{Q}}^{m}\circ{\cal T}^{\scriptscriptstyle{\cal S}}_{t})$
satisfies moreover that the $y$-coordinate of $\widetilde{{\cal
T}^{\scriptscriptstyle{\cal S}}_{t}}(0)$ is $0$. ∎

###  5.3 Choice and renormalization of inverse branches

Let us fix $\Delta>1$ sufficiently close to $1$ so that a nearly affine
blender of contraction $\Delta^{-1}$ is a blender by Proposition 1.8. The
construction also depends on a small number $\varepsilon>0$ (it will measure
the distance of the rescaled blender to an affine one) and on large integers
$n^{-},m^{-},n^{+},m^{+}$ that will be chosen later.

The nearly affine blender will be displayed in the neighborhood $V_{Q}$ of
$Q$, using two inverse branches $g^{+}$ and $g^{-}$ of different iterates of
$f$. We take them of the form:

$g^{\pm}:={\cal T}^{\scriptscriptstyle{\cal
S}}\circ{\mathcal{S}}^{n^{\pm}}\circ{\cal T}^{\scriptscriptstyle{\cal
Q}}\circ{\mathcal{Q}}^{m^{\pm}}.$

Figure 12: Construction of a nearly affine blender.

The inverse branches defining the blender will be rescaled by the map:

${\mathcal{H}}:(x,y)\in\mathbb{R}^{2}\to(x,\varepsilon\cdot\lambda^{-m^{+}}\cdot
y)\in V_{Q}\;.$

Their renormalization are given for $\pm\in\\{-,+\\}$ by:

(5.5) ${\mathcal{R}}g^{\pm}:={\mathcal{H}}^{-1}\circ
g^{\pm}\circ{\mathcal{H}}={\mathcal{H}}^{-1}\circ{\cal
T}^{\scriptscriptstyle{\cal S}}\circ{\mathcal{S}}^{n^{\pm}}\circ{\cal
T}^{\scriptscriptstyle{\cal Q}}\circ{\mathcal{Q}}^{m^{\pm}}\circ{\mathcal{H}}$

######  Lemma 5.3.

For every $n^{-},m^{-},n^{+},m^{+}$ large, with $m^{+}>m^{-}$, the
renormalizations ${\mathcal{R}}g^{-},{\mathcal{R}}g^{+}$ are well defined on
$B:=[-2,2]^{2}$.

###### Proof.

Since $m^{-}<m^{+}$, both maps
${\mathcal{Q}}^{m^{+}}\circ{\mathcal{H}},{\mathcal{Q}}^{m^{-}}\circ{\mathcal{H}}$
are well defined on $B$ and equal to:

${\mathcal{Q}}^{m^{+}}\circ{\mathcal{H}}(x,y)=(\sigma^{m^{+}}\cdot
x,\varepsilon\cdot
y)\quad\mathrm{and}\quad{\mathcal{Q}}^{m^{-}}\circ{\mathcal{H}}(x,y)=(\sigma^{m^{-}}\cdot
x,\varepsilon\cdot\lambda^{m^{-}-m^{+}}\cdot y)\;.$

As $\varepsilon$ is small, their ranges are contained in a small neighborhood
of $0$ and so in the domain of ${\cal T}^{\scriptscriptstyle{\cal Q}}$. Thus
both maps ${\cal T}^{\scriptscriptstyle{\cal
Q}}\circ{\mathcal{Q}}^{m^{\pm}}\circ{\mathcal{H}}$ are well defined on $B$ and
their ranges lie in a small neighborhood of ${\cal T}^{\scriptscriptstyle{\cal
Q}}(0)\in V_{S}$. Then as ${\mathcal{S}}$ contracts $V_{S}$ into itself with a
fixed point at $0$ and since $n^{\pm}$ are large,
${\mathcal{S}}^{n^{\pm}}\circ{\cal T}^{\scriptscriptstyle{\cal
Q}}\circ{\mathcal{Q}}^{m^{\pm}}\circ{\mathcal{H}}$ is well defined on $B$ and
its image is included in the small neighborhood $V_{S}^{\prime\prime}$ of $0$.
Thus ${\cal T}^{\scriptscriptstyle{\cal
S}}\circ{\mathcal{S}}^{n^{\pm}}\circ{\cal T}^{\scriptscriptstyle{\cal
Q}}\circ{\mathcal{Q}}^{m^{\pm}}\circ{\mathcal{H}}$ is well defined on $B$. ∎

###  5.4 Bounds on the renormalized maps

Given $\varepsilon>0$ small, we require the following properties on the large
integers $n^{\pm},m^{\pm}$:

(5.6) $\displaystyle n^{+}>n^{-}\geq\varepsilon^{-1}\quad\mathrm{and}\quad
m^{+}>m^{-}\geq\varepsilon^{-1}\;,$ (5.7)
$\displaystyle\\{\;\sigma_{u}^{n^{-}}\lambda^{m^{-}}\cdot\partial_{y}{\cal
Y}^{\scriptscriptstyle{\cal
S}}(0),\;\sigma_{u}^{n^{+}}\lambda^{m^{+}}\cdot\partial_{y}{\cal
Y}^{\scriptscriptstyle{\cal
S}}(0)\;\\}\;\subset\;[\Delta-\varepsilon,\Delta+\varepsilon]\;,$ (5.8)
$\displaystyle\varepsilon^{-1}\leq\sigma_{u}^{n^{+}-n^{-}}\quad\mathrm{and}\quad\lambda^{m^{+}-m^{-}}\leq\varepsilon^{2}\cdot\min(n^{-},m^{-})\;.$

Let us recall that the inverse eigenvalues satisfy
$\kappa:=\max(\sigma_{u},\sigma_{uu},\frac{\sigma_{uu}}{\sigma_{u}},\lambda^{-1},\sigma)<1.$

######  Fact 5.4.

For every $\varepsilon>0$ small and $n>\varepsilon^{-1}$, it holds
$\kappa^{n}<n^{-(r+4)}$.

In particular, one has $\sigma^{n^{-}}<\varepsilon$ and
$\sigma_{u}^{n^{-}}<\varepsilon$.

We decompose the renormalized maps as

${\mathcal{R}}g^{\pm}=\Psi^{\pm}\circ\Phi^{\pm}=[{\mathcal{H}}^{-1}\circ{\cal
T}^{\scriptscriptstyle{\cal
S}}\circ{\mathcal{S}}^{n^{\pm}}\circ{\mathcal{H}}^{\pm}]\circ[({\mathcal{H}}^{\pm})^{-1}\circ{\cal
T}^{\scriptscriptstyle{\cal
Q}}\circ{\mathcal{Q}}^{m^{\pm}}\circ{\mathcal{H}}]\;,$
$\text{where}\quad{\mathcal{H}}^{\pm}:=(x,y)\mapsto(x,\varepsilon\cdot\lambda^{m^{\pm}-m^{+}}\cdot
y)-Q^{\prime}.$

######  Lemma 5.5.

The maps $(x,y)\mapsto\Phi^{\pm}(x,y)-(0,y)$ are $C^{r}$-dominated by
$\varepsilon$.

###### Proof.

We have $\Phi^{\pm}(x,y)=({\mathcal{H}}^{\pm})^{-1}\circ{\cal
T}^{\scriptscriptstyle{\cal
Q}}\circ{\mathcal{Q}}^{m^{\pm}}\circ{\mathcal{H}}(x,y)$. Since
$Q^{\prime}={\cal T}^{\scriptscriptstyle{\cal Q}}(0)={\cal
T}^{\scriptscriptstyle{\cal
Q}}\circ{\mathcal{Q}}^{m^{\pm}}\circ{\mathcal{H}}(0)$ we get
$\Phi^{\pm}(0)=0$. Recalling that ${\cal T}^{\scriptscriptstyle{\cal
Q}}=({\cal X}^{\scriptscriptstyle{\cal Q}},{\cal Y}^{\scriptscriptstyle{\cal
Q}})$ and that $Q^{\prime}=(q^{\prime}_{x},q^{\prime}_{y})$, we obtain:

$\displaystyle\Phi^{\pm}(x,y)$ $\displaystyle=$ $\displaystyle({\cal
X}^{\scriptscriptstyle{\cal
Q}},\varepsilon^{-1}\cdot\lambda^{m^{+}-m^{\pm}}\cdot{\cal
Y}^{\scriptscriptstyle{\cal Q}})(\sigma^{m^{\pm}}\cdot
x,\varepsilon\cdot\lambda^{m^{\pm}-m^{+}}\cdot
y)+(q^{\prime}_{x},\varepsilon^{-1}\cdot\lambda^{m^{+}-m^{\pm}}\cdot
q^{\prime}_{y})\;.$ $\displaystyle\partial_{x}\Phi^{\pm}(x,y)$
$\displaystyle=$ $\displaystyle\sigma^{m^{\pm}}\cdot(\partial_{x}{\cal
X}^{\scriptscriptstyle{\cal
Q}},\varepsilon^{-1}\cdot\lambda^{m^{+}-m^{\pm}}\cdot\partial_{x}{\cal
Y}^{\scriptscriptstyle{\cal Q}})(\sigma^{m^{\pm}}\cdot
x,\varepsilon\cdot\lambda^{m^{\pm}-m^{+}}\cdot y)\;.$
$\displaystyle\partial_{y}\Phi^{\pm}(x,y)$ $\displaystyle=$
$\displaystyle(\varepsilon\cdot\lambda^{m^{\pm}-m^{+}}\cdot\partial_{y}{\cal
X}^{\scriptscriptstyle{\cal Q}},\partial_{y}{\cal Y}^{\scriptscriptstyle{\cal
Q}})(\sigma^{m^{\pm}}\cdot x,\varepsilon\cdot\lambda^{m^{\pm}-m^{+}}\cdot
y)\;.$

From this, (5.6) and 5.4, the first coordinate of $D\Phi^{\pm}$ is
$C^{r-1}$-dominated by $\varepsilon$. Using also (5.8), we have
$\sigma^{m^{\pm}}\cdot\varepsilon^{-1}\cdot\lambda^{m^{+}-m^{\pm}}<\sigma^{m^{\pm}}\cdot\varepsilon\cdot
m^{-}<\varepsilon$ and the second coordinate of $\partial_{x}\Phi^{\pm}$ is
$C^{r-1}$-dominated by $\varepsilon$. As $\partial_{y}{\cal
Y}^{\scriptscriptstyle{\cal Q}}(0)=1$ by (5.4), the second coordinate of
$\partial_{y}\Phi^{\pm}$ coincides with the constant function $1$, up to an
error term that is $C^{r-1}$-dominated by $\varepsilon$. ∎

######  Lemma 5.6.

The maps $\Psi^{\pm}$ coincide with

$(x,y)\mapsto(0,\Delta\cdot
y)+(s^{\prime}_{x}\;,\;\varepsilon^{-1}\lambda^{m^{+}}\cdot
s^{\prime}_{y}-\varepsilon^{-1}\cdot\lambda^{m^{+}}\sigma_{u}^{n^{\pm}}\cdot\partial_{y}{\cal
Y}^{\scriptscriptstyle{\cal S}}(0)\cdot q^{\prime}_{y}),$

up to an error term that is $C^{r}$-dominated by $\varepsilon$.

###### Proof.

We have $\Psi^{\pm}={\mathcal{H}}^{-1}\circ{\cal T}^{\scriptscriptstyle{\cal
S}}\circ{\mathcal{S}}^{n^{\pm}}\circ{\mathcal{H}}^{\pm}$. With ${\cal
T}^{\scriptscriptstyle{\cal S}}=({\cal X}^{\scriptscriptstyle{\cal S}},{\cal
Y}^{\scriptscriptstyle{\cal S}})$, it holds:

$\Psi^{\pm}(x,y)=({\cal X}^{\scriptscriptstyle{\cal
S}},\varepsilon^{-1}\lambda^{m^{+}}{\cal Y}^{\scriptscriptstyle{\cal
S}})(\sigma_{uu}^{n^{\pm}}\cdot(x-q^{\prime}_{x}),\;\sigma_{u}^{n^{\pm}}\cdot(\varepsilon\cdot\lambda^{m^{\pm}-m^{+}}\cdot
y-q^{\prime}_{y}))\;.$

Thus $\partial_{x}\Psi^{\pm}$ is $C^{r-1}$-dominated by
$\sigma_{uu}^{n^{-}}\cdot\lambda^{m^{+}}\cdot\varepsilon^{-1}$, which by
(5.6), (5.7), (5.8) is dominated by

$(\tfrac{\sigma_{uu}}{\sigma_{u}})^{n^{-}}\cdot\lambda^{m^{+}-m^{-}}\cdot\varepsilon^{-1}<(\tfrac{\sigma_{uu}}{\sigma_{u}})^{n^{-}}\cdot
n^{-}\cdot\varepsilon<\varepsilon.$

The first coordinate of $\partial_{y}\Psi^{\pm}$ is $C^{r-1}$-dominated by
$\varepsilon\cdot\sigma_{u}^{n^{\pm}}\cdot\lambda^{m^{\pm}-m^{+}}<\varepsilon$.
Similarly, using (5.7), the second coordinate of $\partial_{y}\Psi^{\pm}$
coincides with
$\sigma_{u}^{n^{\pm}}\cdot\lambda^{m^{\pm}}\cdot\partial_{y}{\cal
Y}^{\scriptscriptstyle{\cal S}}(0)$, hence with $\Delta$, up to an error term
that is $C^{r-1}$-dominated by $\varepsilon$. We have thus shown that the
derivative of $(x,y)\mapsto\Psi^{\pm}(x,y)-(0,\Delta\cdot y)$ is
$C^{r-1}$-dominated by $\varepsilon$. Moreover:

$\Psi^{\pm}(0)=({\cal X}^{\scriptscriptstyle{\cal
S}},\varepsilon^{-1}\cdot\lambda^{m^{+}}\cdot{\cal Y}^{\scriptscriptstyle{\cal
S}})(-\sigma_{uu}^{n^{\pm}}\cdot q^{\prime}_{x},-\sigma_{u}^{n^{\pm}}\cdot
q^{\prime}_{y})\;.$

The first coordinate is $\varepsilon$-close to ${\cal
X}^{\scriptscriptstyle{\cal S}}(0)=s^{\prime}_{x}$ and the second coordinate
is equal to:

$\varepsilon^{-1}\lambda^{m^{+}}{\cal Y}^{\scriptscriptstyle{\cal
S}}(-\sigma_{uu}^{n^{\pm}}\cdot q^{\prime}_{x},-\sigma_{u}^{n^{\pm}}\cdot
q^{\prime}_{y}))=\varepsilon^{-1}\lambda^{m^{+}}\bigg{(}{\cal
Y}^{\scriptscriptstyle{\cal S}}(0)-\partial_{x}{\cal
Y}^{\scriptscriptstyle{\cal S}}(0)\cdot\sigma_{uu}^{n^{\pm}}\cdot
q^{\prime}_{x}-\partial_{y}{\cal Y}^{\scriptscriptstyle{\cal
S}}(0)\cdot\sigma_{u}^{n^{\pm}}\cdot
q^{\prime}_{y}+O(\sigma_{u}^{2n^{\pm}})\bigg{)}.$

As before
$\varepsilon^{-1}\cdot\lambda^{m^{+}}\cdot\sigma_{uu}^{n^{-}}=O(\varepsilon)$.
By (5.7) and (5.8),
$\varepsilon^{-1}\cdot\lambda^{m^{+}}\cdot\sigma_{u}^{2n^{-}}$ is dominated by
$\varepsilon^{-1}\cdot\lambda^{m^{+}-m^{-}}\cdot\sigma_{u}^{n^{-}}=O(\varepsilon).$
As ${\cal Y}^{\scriptscriptstyle{\cal S}}(0)=s^{\prime}_{y}$, we obtain $(1)$.
∎

###  5.5 Tunning iterates

######  Lemma 5.7.

Given $\varepsilon>0$ small, there exist $n^{-},m^{-},n^{+},m^{+}$ which
satisfy (5.6), (5.7), (5.8).

###### Proof.

By (5.1), there exist $m,n\geq 1$ arbitrarily large such that:

$\lambda^{m}\sigma_{u}^{n}\in[1-\tfrac{\varepsilon}{10},1+\tfrac{\varepsilon}{10}]\;.$

As $\Delta$ and $\partial_{y}{\cal Y}^{\scriptscriptstyle{\cal S}}(0)$ have
the same sign (by Lemma 5.2), there are $n^{-},m^{-}>\varepsilon^{-1}$ such
that:

$n^{-}\geq\varepsilon^{-2}\cdot\lambda^{m}\quad\mathrm{and}\quad\partial_{y}{\cal
Y}^{\scriptscriptstyle{\cal
S}}(0)\cdot\lambda^{m^{-}}\sigma_{u}^{n^{-}}\in\Delta+[-\tfrac{\varepsilon}{10},\tfrac{\varepsilon}{10}]\;.$

Then let $m^{+}:=m+m^{-}$ and $n^{+}:=n+n^{-}$. This gives
$\lambda^{m^{\pm}}\cdot\partial_{y}{\cal Y}^{\scriptscriptstyle{\cal
S}}(0)\cdot\sigma_{u}^{n^{\pm}}=\Delta+[-\varepsilon,\varepsilon]$. ∎

A consequence of the Lemmas 5.5, 5.6 and 5.7 is:

######  Corollary 5.8.

For every $\varepsilon>0$ there exist $n^{-},m^{-},n^{+},m^{+}$ such that the
renormalized maps ${\mathcal{R}}g^{\pm}$ coincide, up to a term that is
$C^{r}$-dominated by $\varepsilon$, with:

$(x,y)\mapsto(0,\Delta\cdot
y)+(s^{\prime}_{x}\;,\;\varepsilon^{-1}\cdot\lambda^{m^{+}}\cdot
s^{\prime}_{y}-\varepsilon^{-1}\cdot\lambda^{m^{+}}\cdot\sigma_{u}^{n^{\pm}}\cdot\partial_{y}{\cal
Y}^{\scriptscriptstyle{\cal S}}(0)\cdot q^{\prime}_{y}).$

###  5.6  Proof of Proposition 2.2: from strong heterocycles to blenders

Let $n^{-},m^{-},n^{+},m^{+}$ be given by 5.8. It remains to choose the values
of $s^{\prime}_{y}$ and $q_{y}^{\prime}$, such that the renormalized maps
${\mathcal{R}}g^{\pm}$ are $C^{r}$-close to:

$(x,y)\mapsto(s^{\prime}_{x},\Delta\cdot y)\pm(\Delta-1).$

In view of 5.8, it is enough to ask:

$\varepsilon^{-1}\cdot\lambda^{m^{+}}\cdot
s^{\prime}_{y}-\varepsilon^{-1}\cdot\lambda^{m^{+}}\cdot\sigma_{u}^{m^{\pm}}\cdot\partial_{y}{\cal
Y}^{\scriptscriptstyle{\cal S}}(0)\cdot
q^{\prime}_{y}=\pm(\Delta-1)+O(\varepsilon).$

This is implied by choosing $s^{\prime}_{y}$ and $q^{\prime}_{y}$ as follows:

(5.9) $\varepsilon^{-1}\cdot\lambda^{m^{+}}\cdot
s^{\prime}_{y}=\Delta-1\quad\mathrm{and}\quad\varepsilon^{-1}\cdot\lambda^{m^{+}}\sigma_{u}^{n^{-}}\cdot\partial_{y}{\cal
Y}^{\scriptscriptstyle{\cal S}}(0)\cdot q^{\prime}_{y}=2(\Delta-1)\;.$

Indeed, one has $\sigma_{u}^{n^{+}-n^{-}}\geq\varepsilon^{-1}$ by (5.8), and
with (5.6), (5.7), 5.4, the choices (5.9) give
$s^{\prime}_{y}=O(\varepsilon^{2})$, $q^{\prime}_{y}=O(\varepsilon)$ and
$\varepsilon^{-1}\cdot\lambda^{m^{+}}\sigma_{u}^{n^{+}}\cdot\partial_{y}{\cal
Y}^{\scriptscriptstyle{\cal S}}(0)\cdot q^{\prime}_{y}=O(\varepsilon)$.

By Proposition 1.8, $\\{{\mathcal{R}}g^{+},{\mathcal{R}}g^{-}\\}$ defines a
nearly affine blender with activation domain containing
$[-2,2]\times[-1/2,1/2]$. Thus, $\\{g^{+},g^{-}\\}$ defines a blender with
activation domain containing
${\mathcal{H}}([-2,2]\times[-1/2,1/2])=[-2,2]\times[-\varepsilon\cdot\lambda^{-m^{+}}/2,\varepsilon\cdot\lambda^{-m^{+}}/2]$.
Choosing $|\Delta-1|<1/4$, one gets
$|s^{\prime}_{y}|<\varepsilon\cdot\lambda^{-m^{+}}/2$ and $S^{\prime}$ belongs
to this activation domain. Also the point $Q\equiv 0$ belongs to this
activation domain. Note that the unstable manifold of $Q$ stretches across
$\\{s_{x}^{\prime}\\}\times[-\varepsilon\cdot\lambda^{-m^{+}}/2,\varepsilon\cdot\lambda^{-m^{+}}/2]$
and so the stable manifolds of the blender. Hence $Q$ is homoclinically
related to the blender. Proposition 2.2 is proved in the $C^{\infty}$ case. ∎

###  5.7 Proof of the Proposition 2.2 in the analytic case

The whole previous proof is still valid in the analytic setting but 5.1. Note
that the proof of Proposition 2.2 does not use that the $r$ first derivatives
of ${\cal T}^{\scriptscriptstyle{\cal S}}$ and ${\cal
T}^{\scriptscriptstyle{\cal Q}}$ remain unchanged but only that they are
bounded. Thus to prove the analytic case of Proposition 2.2, it suffices to
show:

######  Claim 5.9.

For every small numbers $s^{\prime}_{y}$ and $q^{\prime}_{y}$, there exists a
$C^{\omega}$ perturbation of the dynamics such that the inverse branches