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# Homoclinic tangencies leading to robust heterodimensional cycles

Pablo G. Barrientos, Lorenzo J. Díaz, and Sebastián A. Pérez  Institute of
Mathematics and Statistics, University Federal Fluminense-UFF, Gragoata
Campus, Rua Prof. Marcos Waldemar de Freitas Reis, S/n-Sao Domingos, Niteroi -
RJ, 24210-201, Brazil<EMAIL_ADDRESS>Departamento de Matemática PUC-
Rio, Marquês de São Vicente 225, Gávea, Rio de Janeiro 225453-900, Brazil
<EMAIL_ADDRESS>Instituto de Matemáticas, Pontificia Universidad
Católica de Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso, Chile.
<EMAIL_ADDRESS>
###### Abstract.

We consider $C^{r}$ ($r\geqslant 1$) diffeomorphisms $f$ defined on manifolds
of dimension $\geqslant 3$ with homoclinic tangencies associated to saddles.
Under generic properties, we show that if the saddle is homoclinically related
to a blender then the diffeomorphism $f$ can be $C^{r}$ approximated by
diffeomorphisms with $C^{1}$ robust heterodimensional cycles. As an
application, we show that the classic Simon-Asaoka’s examples of
diffeomorphisms with $C^{1}$ robust homoclinic tangencies also display $C^{1}$
robust heterodimensional cycles. In a second application, we consider
homoclinic tangencies associated to hyperbolic sets. When the entropy of these
sets is large enough we obtain $C^{1}$ robust cycles after $C^{1}$
perturbations.

###### Key words and phrases:

Blender, Cycles, Entropy, Heterodimensional cycle, Homoclinic tangency,
Hyperbolic measure, Lyapunov exponent, Robust properties

###### 2020 Mathematics Subject Classification:

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

PGB and LJD were supported [in part] by CAPES finance code 001 (Brazill), CNPq
Projeto Universal and CNPq grants (Brazil). PGB was partially supported by
grant PID2020-113052GB-I00 funded by MCIN/AEI/10.13039/501100011033 (Spain).
LJD was partially supported by grant E-16/2014 INCT/FAPERJ (Brazil). SP was
supported by Programa Postdoctorado FONDECYT 3190174 (Chile).

##  1\. Introduction

Homoclinic tangencies and heterodimensional cycles associated to saddles are
considered the main bifurcation mechanisms generating nonhyperbolic dynamics.
Palis conjecture [41] claims that nonhyperbolic diffeomorphisms can be
approached by systems displaying these bifurcations. The goal of this paper is
to study the interplay between these two types of bifurcations. A saddle
(i.e., a hyperbolic periodic point with nontrivial stable and unstable
bundles) of a diffeomorphism has a homoclinic tangency if its stable and
unstable manifolds have some nontransverse intersection. Two saddles with
different _$u$ -indices_ (i.e., dimension of the unstable bundle) have a
heterodimensional cycle if their stable and unstable manifolds intersect
cyclically (by dimension deficiency, one of these intersections is necessarily
nontransverse). Homoclinic tangencies may occur in manifolds of dimension two
or higher, while the occurrence of heterodimensional cycles requires dimension
at least three. In what follows, we will use the term cycle to refer either to
a homoclinic tangency or to a heterodimensional cycle.

Note that cycles associated to saddles are breakable by small perturbations:
by Kupka-Smale theorem, periodic points of generic diffeomorphisms are
hyperbolic and their invariant manifolds are in general position (either they
intersect transversely or are disjoint). Even more, cycles may be unbreakable
if they are associated to nontrivial hyperbolic basic sets (i.e., not just
periodic orbits). This leads to the notions of _$C^{r}$ persistent_ and
_$C^{r}$ robust cycles._ In the persistent case, there is a $C^{r}$ open set
where the diffeomorphisms with cycles associated to saddles are $C^{r}$ dense.
In the robust case, there is a hyperbolic set with a cycle that cannot be
destroyed by small $C^{r}$ perturbations, see Definition 1.1111Any robust
cycle provides a persistent cycle. As far as we know, it is unknown the
existence of persistent cycles that are not robust (meaning that the saddle
belongs to a hyperbolic set with a robust cycle).. Bearing in mind the
importance of robust dynamical properties, Bonatti proposed in [15] a stronger
version of Palis conjecture: every nonhyperbolic diffeomorphism can be
approached by diffeomorphisms with a robust cycle.

Palis conjecture holds for $C^{1}$ surface diffeomorphism [44] and there is
some progress in higher dimensions, see [23, 24]. Note that there are no
surface diffeomorphisms with $C^{1}$ robust homoclinic tangencies associated
to basic sets, see [36]. In higher dimensions and in the $C^{1}$ topology,
Bonatti conjecture is true for the so-called _tame diffeomorphisms,_ [18]. For
both conjectures, the $C^{r}$ case, $r\geqslant 2$, is widely open. The first
reason for that is that they rely on the density of periodic points in the
limit set, which is only known in the $C^{1}$ case, see [43, 42]. A second
reason is the lack of connecting-like lemmas allowing intersections between
the invariant manifolds of saddles in the presence of recurrences, see [30,
17]. In view of these obstacles, the following question (that can be posed in
any regularity) provides a bridge between these conjectures and a step towards
the one by Bonatti:

######  Question 1.

When does a cycle associated to saddles yield (by small) perturbations robust
cycles?

The answer to this question depends on the type of cycle, the dimension of the
ambience, and the required regularity. A strategy to find robust cycles is to
see that a cycle can become robust when the saddle in the cycle belongs to (or
when the unfolding of the cycle generates) some massive horseshoe. This vague
term means thick horseshoes or blenders, according to the type of cycle. We
observe that $C^{1}$ diffeomorphisms cannot display robustly thick horseshoes,
[53], while blenders may occur in dimension three or higher in a $C^{1}$
robust way. This illustrates some disparities between the different settings.
Blenders will play a key role in this paper and will be discussed in Section
2. For the discussion in this introduction, keep in mind that a blender is a
special type of hyperbolic set with some $C^{1}$ robust geometrical properties
guaranteeing intersections between the local unstable manifold of the blender
and a family of “strong stable” disks.

There are partial answers to Question 1. When $r\geqslant 2$, a homoclinic
tangency of a $C^{r}$ diffeomorphism yields robust tangencies, see [39] in
dimension two and [27, 40, 47] in higher dimensions. The generation of robust
tangencies in the $C^{1}$ setting in higher dimensions depends crucially on
the geometry of the tangency, see the discussion below below. Finally, the
question _does a heterodimensional cycle yield robust heterodimensional
cycles?_ has a positive answer in the coindex (i.e., the absolute value of the
difference of the $u$-indices of the hyperbolic sets in the cycle) one case,
see [18] for the $C^{1}$ case and [33] for the $C^{r}$ case, $r\geqslant 2$.
The case of heterodimensional cycles of higher coindex is widely open. We
observe that all robust cycles generated in this paragraph are of the same
type as the ones in the initial cycle.

An issue related to Question 1 is to determine the interplay between
heterodimensional cycles and homoclinic tangencies. This leads to the
following refinement of Question 1:

######  Question 2.

(a) When does a homoclinic tangency associated to a saddle lead to robust
heterodimensional cycles? (b) When does a heterodimensional cycle associated
to a pair of saddles lead to robust homoclinic tangencies?

The answers to these questions involves global dynamical aspects. In this
paper, we focus on item (a), providing settings where homoclinic tangencies
associated to saddles lead to $C^{1}$ robust cycles after small $C^{r}$
perturbations222Note that if $f\in\mathrm{Diff}^{r}(M)$, $r\geqslant 1$, then
every $C^{1}$ neighbourhood of $f$ contains a $C^{r}$ neighbourhood of $f$.
Hence, if $f$ has a $C^{1}$ robust cycle then it also has a $C^{r}$ robust
cycle., $r\geqslant 1$. Our constructions are motivated by the ones in [35],
where heterodimensional cycles are obtained from homoclinic tangencies with
some symmetry conditions. To describe our setting, we first comment some
global aspects and restrictions concerning Question 2.

The presence of a homoclinic tangency implies the existence of an invariant
space of dimension at least two that cannot be decomposed into one-dimensional
invariant directions in a dominated way (see the appendix in Section 9 for the
precise definitions). Thus partially hyperbolic diffeomorphisms having
dominated splittings with one-dimensional bundles may display
heterodimensional cycles but cannot exhibit homoclinic tangencies. This shows
an obvious restriction to item (b). When considering homoclinic tangencies a
key aspect is how they are embedded in the global dynamics. The following
discussion illustrates this aspect and motivates our constructions and
hypotheses.

To simplify the analysis, let us assume that the ambiance has dimension three
and the saddle with the homoclinic tangency has $u$-index one. Under
appropriate assumptions on the multipliers of the saddle, the unfolding of
this tangency yields new hyperbolic periodic points with $u$-indices different
from one. Once saddles of $u$-index two are generated and the resulting
dynamics exhibits saddles of different $u$-indices, it is necessary to get
cyclic heteroclinic relations between them.

The occurrence of such heteroclinic relations is an intricate question that
involves geometrical constraints. For instance, the tangency may occur
“inside” a normally contracting hyperbolic surface and the saddle with the
homoclinic tangency may have a Jacobian whose restriction to that surface is
bigger than one. In this case, due to the persistence of normally hyperbolic
surfaces, the homoclinic tangency yields saddles contained in those surfaces
with a normally contracting direction which are sources for the dynamics
restricted to the surface. In this way, the tangency provides saddles of
$u$-indices one and two. However, in the absence of further recurrences given
by the global dynamics, there is no dynamical interaction between these
saddles and heterodimensional cycles cannot occur. In rough terms, in the
previous configuration, the dynamics can be “reduced” to a surface dynamics
“multiplied” by a contraction, we will refer to it as “essentially two-
dimensional bifurcations”.

We now discuss the setting and the hypotheses of our results. Recall that
hyperbolic basic sets of surface diffeomorphisms cannot display $C^{1}$ robust
homoclinic tangencies. Thus to get $C^{1}$ robust homoclinic tangencies in
higher dimensions essentially two-dimensional dynamics must be avoided. To
bypass this obstruction, in [20] it is considered a type of blender (called
_blender-horseshoe,_ see Remark 2.2) with a homoclinic tangency. Although [20,
Theorem 1.2], see also [16, Theorem 1 and Corollary 2], are $C^{1}$ global
results involving the recurrences provided by a homoclinic class (i.e., the
closure of the set of transverse intersections between the invariant manifolds
of a saddle) they lead to the following heuristic principle: a homoclinic
class with robust _index variability_ (i.e., containing saddles of different
$u$-indices) and a nondominated “central” direction displays $C^{1}$ robust
cycles (simultaneously, homoclinic tangencies and heterodimensional cycles).
In the $C^{1}$ setting in [20, 16] the index variability provides blenders
inside the homoclinic class as well as the following configuration: a saddle
(satisfying some dissipative conditions) with a homoclinic tangency and
homoclinically related to a blender. The tangency provides the nondomination
condition and also (after its unfolding) the index variability. The blender
leads to dynamics that are not essentially two-dimensional.

This sort of conditions motivated the hypotheses in Theorems A and B claiming
the existence of $C^{1}$ robust heterodimensional cycles arising from
homoclinic tangencies after small $C^{r}$ perturbations, $r\geqslant 1$.
Theorems A considers the codimension one case (the saddle with the homoclinic
tangency has $u$-index one) while Theorems B deals with the general setting
(any $u$-index). The precise statements can be found in Section 1.1. The main
difference between these two theorems is that in the codimension one case the
homoclinic tangency provides only saddles of $u$-indices one and two. As a
consequence, only heterodimensional cycles associated to these saddles can
appear and hence these cycles have coindex one. In the general setting, the
homoclinic tangency yields saddles of a wider range of $u$-indices, hence
cycles of higher coindex may appear.

We also provide settings to which our results apply. For that, let us recall
the first examples of $C^{1}$ robust homoclinic tangencies given by Simon [49]
and Asaoka [3] built on the examples of nonhyperbolic diffeomorphisms in [1].
The key ingredient in these constructions is the existence of a hyperbolic set
that is the local product of a nontrivial hyperbolic repeller (a DA-
diffeomorphism in [49] and a Plykin repeller in [3]) and a contraction. We
call this sort of hyperbolic set blenders of DA or Plykin type. A crucial
property of this type of blender is that they contain their one-dimensional
unstable manifolds. This property allows to get $C^{1}$ robust
heterodimensional cycles in [1, 48] (although this terminology is not used
there). The idea in [49, 3] (reviewed in Section 7) is to produce homoclinic
tangencies associated to these blenders. In Corollary C (see also Remark 1.3)
we prove that these examples satisfy the hypotheses of Theorem A.
Consequently, for any $r\geqslant 1$, they lead to the simultaneous occurrence
of $C^{1}$ robust homoclinic tangencies and heterodimensional cycles.
Corollary C answers affirmatively a question posed in [20, Section 1.3] in a
discussion about the $C^{1}$ coexistence of robust cycles and tangencies in
the examples in [3].

Finally, we exhibit settings where entropy-like assumptions imply the
existence of blenders. For that, we revisit the $C^{1}$ constructions in [4],
showing that diffeomorphisms with hyperbolic measures whose entropy is
sufficiently large exhibit blenders. These results are summarised in Theorem
1.4. Corollaries D and E translate our main results to that setting, providing
simultaneous $C^{1}$ robust heterodimensional cycles of a variety of
coindices.

We now state precisely our results.

###  1.1. Robust heterodimensional arising from homoclinic tangencies

In the sequel, $\mathrm{Diff}^{r}(M^{n})$ denotes the space of $C^{r}$
diffeomorphisms of a compact boundaryless manifold $M^{n}$ of dimension $n$
endowed with the uniform $C^{r}$ topology. When we do not want to emphasise
the dimension of the ambiance we will omit the superscript $n$.

The first ingredient in our results is the $cs$-blender. Blenders are
hyperbolic sets with an additional geometric structure guaranteeing
intersections between a family of “stable” disks and the local unstable
manifold of the blender. Blenders are also $C^{1}$ robust sets. See Definition
2.1 for details.

Recall that two saddles of $f\in\mathrm{Diff}^{r}(M)$ are _homoclinically
related_ if the invariant manifolds of their orbits intersect transversely and
cyclically. To be homoclinic related defines an equivalence relation on the
set of saddles of $f$. Two saddles that are homoclinically related have the
same $u$-index. The _homoclinic class_ of a saddle $P$ of $f$ is the closure
of the saddles of $f$ that are homoclinically related to $f$. A homoclinic
class is a _transitive set_ (i.e., it contains a point whose orbit is dense in
the set). As a blender $\Gamma$ of $f\in\mathrm{Diff}^{r}(M)$ is a hyperbolic
basic set (see Definition 2.1), its saddles form a dense subset of it and
every pair of saddles of $\Gamma$ are homoclinically related. A saddle is
homoclinically related to the blender if it is homoclinically related to some
saddle of the blender, see (3) in Remark 2.3.

Recall that every hyperbolic set $\Lambda$ of a diffeomorphism $f$ has a well
defined _continuation_ for every $C^{1}$ nearby diffeomorphism $g$. We denote
this continuation by $\Lambda_{g}$. If a hyperbolic set $\Lambda$ is
transitive then the dimension of its unstable bundle does not depend on the
point of $\Lambda$. This dimension is called the $u$-index of $\Lambda$.

######  Definition 1.1 (Robust cycles).

Two transitive hyperbolic sets $\Lambda$ and $\Upsilon$ of
$f\in\mathrm{Diff}^{r}(M)$ with different $u$-indices form a
_heterodimensional cycle_ if their invariant stable and unstable sets
intersect cyclically, that is,

$W^{s}(\Lambda,f)\cap W^{u}(\Upsilon,f)\neq\emptyset\quad\mbox{and}\quad
W^{u}(\Lambda,f)\cap W^{s}(\Upsilon,f)\neq\emptyset.$

The coindex of this cycle is the absolute value of the difference of the
$u$-indices of $\Lambda$ and $\Upsilon$. This cycle is $C^{r}$ robust if there
is a $C^{r}$ neighbourhood $\mathscr{N}$ of $f$ such that the continuations
$\Lambda_{g}$ and $\Upsilon_{g}$ have a heterodimensional cycle for every $g$
in $\mathscr{N}$.

A transitive hyperbolic set $\Lambda$ has a homoclinic tangency it its stable
and unstable sets have some nontransverse intersection. This homoclinic
tangency is _$C^{r}$ robust_ if there is a $C^{r}$ neighbourhood of $f$ such
that the continuation $\Lambda_{g}$ of $\Lambda$ has a tangency for every $g$
in that neighbourhood.

We now describe the class of homoclinic tangencies studied in this paper.
Consider $f\in\mathrm{Diff}^{r}(M^{m+n})$, with $r,m,n\geqslant 1$ and
$m+n\geqslant 3$, and a saddle $P$ of $f$ of period $\pi$ and $u$-index $n$.
Let $\lambda_{1},\dots,\lambda_{m},\gamma_{1},\dots,\gamma_{n}$ be the
eigenvalues of $Df^{\pi}(P)$ counted with multiplicity and ordered so that

(1.1)
$|\lambda_{m}|\leqslant\dots\leqslant|\lambda_{1}|=\lambda<1<\gamma=|\gamma_{1}|\leqslant\dots\leqslant|\gamma_{n}|.$

The multipliers of $P$ with modulus equal to $\lambda$ and $\gamma$ are called
_stable and unstable leading multipliers,_ respectively. We denote by
$m_{s}=m_{s}(P)$ and $n_{u}=n_{u}(P)$ the number of stable and unstable
leading multipliers, respectively, and say that $P$ is of _type
$(m_{s},n_{u})$_. The saddle $P$ is simple if one of the following cases
holds:

  * •

$P$ is of type $(1,1)$;

  * •

$P$ is of type $(2,1)$ and $\lambda_{1}$ and $\lambda_{2}$ are nonreal (and
hence conjugate);

  * •

$P$ is of type $(1,2)$ and $\gamma_{1}$ and $\gamma_{2}$ are nonreal (hence
conjugate);

  * •

$P$ is of type $(2,2)$, $\lambda_{1}$ and $\lambda_{2}$ are nonreal, and
$\gamma_{1}$ and $\gamma_{2}$ are nonreal.

We define the _leading Jacobian_ $J_{P}(f)$ of $P$ as the product of the
moduli of all leading multipliers of $P$ counted with multiplicity,

(1.2) $J_{P}(f)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\lambda^{m_{s}}\,\gamma^{n_{u}}.$

We are now ready to state our main results.

######  Theorem A.

Consider $f\in\mathrm{Diff}^{r}(M^{m+1})$, $r\geqslant 1,\,m\geqslant 2$, with
a homoclinic tangency associated to a simple saddle $P$ of $u$-index one such
that:

  1. (1)

$P$ is of type $(m_{s},1)$ with $m_{s}\in\\{1,2\\}$,

  2. (2)

$J_{P}(f)\geqslant 1$ and

  3. (3)

$P$ is homoclinically related to a $cs$-blender $\Gamma$ of central dimension
$m_{s}$.

Then there are diffeomorphisms $g$ arbitrarily $C^{r}$ close to $f$ having
simultaneously $C^{1}$ robust heterodimensional cycles of coindex $i$ for
every $1\leqslant i\leqslant m_{s}$. These cycles are associated to a
hyperbolic basic set containing $P_{g}$ and a hyperbolic basic set of
$u$-index $1+i$.

Let us briefly comment on the hypotheses of Theorem A. Items (1) and (3) are
compatibility conditions between the dominated splittings of the saddle and
the blender. Condition (2) implies that the unfolding of the homoclinic
tangency generates saddles of $u$-index greater than one, enabling the
creation of heterodimensional cycles between these new saddles and the
continuations of the saddle in the initial cycle.

The next result extends the previous theorem when the saddles exhibiting the
homoclinic tangency have $u$-index $n>1$. A key ingredient in this result is
the notion of a double blender, see Definition 2.1.

######  Theorem B.

Consider $f\in\mathrm{Diff}^{r}(M^{m+n})$, $r\geqslant 1,m,n\geqslant 2$, with
a homoclinic tangency associated to a simple saddle $P$ of $u$-index $n$ that
is

  1. (1)

of type $(m_{s},n_{u})$ with $m_{s},n_{u}\in\\{1,2\\}$ and

  2. (2)

homoclinically related to a double blender of central dimensions
$(m_{s},n_{u})$.

Then there are diffeomorphisms $g$ arbitrarily $C^{r}$ close to $f$ having
simultaneously $C^{1}$ robust heterodimensional cycles of coindex $i$ for
every $1\leqslant i\leqslant N$, where

$N=m_{s},\quad\mbox{if}\quad J_{P}(f)\geqslant 1\qquad\mbox{and}\qquad
N=n_{u},\quad\mbox{if}\quad J_{P}(f)\leqslant 1.$

These cycles are associated to hyperbolic basic sets containing $P_{g}$ and a
hyperbolic basic set of $u$-index $n+i$.

In Theorem B, we obtain robust cycles associated to the continuations of the
blender and new saddles of different $u$-indices. It is an open question if
there are robust cycles associated to hyperbolic sets containing these new
saddles.

###  1.2. Simultaneity of robust homoclinic tangencies and robust
heterodimensional cycles

We now study the concurrence of robust cycles of different types. We start
with an observation.

######  Remark 1.2 (Robust tangencies).

Blender-horseshoes are a particular type of $cs$-blenders, see Remark 2.2.
Under the assumptions of Theorems A or B, if the $cs$-blender $\Gamma$ of $f$
is a blender-horseshoe and the saddle $P$ is of type $(1,1)$ then [20, Theorem
4.9] implies that there are diffeomorphisms $g$ arbitrarily $C^{r}$ close to
$f$ having $C^{1}$ robust tangencies associated to a hyperbolic basic set
containing $P_{g}$. This leads to $C^{r}$ open sets whose closure contains $f$
consisting of diffeomorphisms having simultaneously $C^{1}$ robust
heterodimensional cycles and $C^{1}$ robust homoclinic tangencies. We do not
know if under the assumptions of Theorems A or B and with the general
definition of a $cs$-blender (see Definition 2.1) such a concurrence holds.

We observe that (in dimension three) there are several results assuring the
simultaneous occurrence of the two types of robust cycles above. The results
in [34] state conditions on the multipliers of a saddle-focus with a
homoclinic tangency to generate simultaneously $C^{1}$ robust homoclinic
tangencies and heterodimensional cycles by small $C^{1}$ perturbations.
Similarly, in [5] it is proved that both types of robust cycles can be $C^{1}$
approximated by diffeomorphisms with heterodimensional cycles associated to
saddles with nonreal multipliers. A nondominated $C^{r}$ setting, $r\geqslant
2$, with simultaneous occurrence of robust homoclinic tangencies and robust
heterodimensional cycles was explored in [25].

We aim to get the simultaneity of robust cycles in more general settings. We
apply Theorem A to the class of diffeomorphisms with robust homoclinic
tangencies in [49, 3]. We see that these diffeomorphisms also display $C^{1}$
robust heterodimensional cycles. The key ingredient in this result are the
$cs$-blenders of Plykin type obtained as a “local” product of a Plykin
repeller by strong contractions, for details see Section 7.1. These
$cs$-blenders have fixed points of $u$-index one. Roughly, the robust cycles
in the next corollary are obtained bifurcating the homoclinic tangency of a
fixed point in the blenders, see Figure 1.

######  Corollary C.

Consider $f\in\mathrm{Diff}^{r}(M^{m+1})$, $r\geqslant 1,\,m\geqslant 2$, with
a $cs$-blender $\Gamma$ of Plykin type whose fixed point $P$ has leading
multipliers $\lambda$ and $\sigma$ satisfying $\lambda\,\sigma>1$. Suppose
that $f$ has homoclinic tangency associated to $P$. Then there is a $C^{r}$
open set $\mathscr{C}$ whose closure contains $f$ such that every
$g\in\mathscr{C}$ has simultaneously a $C^{1}$ robust homoclinic tangency and
a $C^{1}$ robust heterodimensional cycle associated to the blender
$\Gamma_{g}$.

######  Remark 1.3.

An ipsis litteris version of Corollary C can be stated for $cs$-blenders of DA
type. In this case, the construction is analogous, starting with a DA
diffeomorphism defined on the two-dimensional torus, see for instance [45,
Chapter 7.8]. This provides some constraints on the ambient manifold for the
occurrence of blenders of DA type. Note that Plykin attractors are defined
locally in a ball, therefore there are no topological constrains for their
occurrence.

\begin{overpic}[scale={0.09}]{fomi.jpg} \put(97.0,127.0){\Large{$P_{g}$}}
\put(20.0,160.0){\Large$W^{u}(\Gamma_{g})$} \put(200.0,185.0){\Large$Q_{g}$}
\end{overpic} Figure 1. Robust heterodimensional cycles in Corollary C

###  1.3. $C^{1}$ robust heterodimensional cycles arising from horseshoes with
large entropy

We now use Theorems A and B to prove that homoclinic tangencies associated to
horseshoes with sufficiently large entropy lead to $C^{1}$ robust
heterodimensional cycles by small $C^{1}$ perturbations. For this, we use the
results in [4] claiming that these horseshoes contain blenders.

We start with some preliminaries from ergodic theory. In what follows, all
measures considered are probability ones. A measure $\mu$ of
$f\in\mathrm{Diff}^{r}(M^{d})$ is called hyperbolic if it is $f$-invariant,
ergodic, and all its Lyapunov exponents are different from zero. We list the
Lyapunov exponents of $\mu$ (counted with multiplicity) as follows

(1.3)
$\chi^{-}_{m}(\mu)\leqslant\dots\leqslant\chi^{-}_{1}(\mu)<0<\chi_{1}^{+}(\mu)\leqslant\dots\leqslant\chi^{+}_{n}(\mu),\quad
m+n=d,$

and say that $\mu$ has $u$-index $n$. To emphasise the nature of these
exponents, we write

(1.4)
$\chi^{cs}_{\mu}=\chi^{-}_{1}(\mu)\quad\mbox{and}\quad\chi^{cu}_{\mu}=\chi^{+}_{1}(\mu).$

We also consider the  stable and unstable Jacobians of $\mu$ defined by

(1.5) $\mathrm{J}^{s}_{\mu}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathrm{exp}\left(\sum_{i=1}^{m}\chi^{-}_{i}(\mu)\right)\quad\text{and}\quad\mathrm{J}^{u}_{\mu}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathrm{exp}\left(\sum_{i=1}^{n}\chi^{+}_{i}(\mu)\right).$

The previous discussion applies to any ergodic measure supported on a
horseshoe, in particular to its measure of maximal entropy. By a _horseshoe_
of a diffeomorphism we mean a hyperbolic Cantor set that is transitive and
locally maximal. In this case, the horseshoe $\Lambda$ has only one measure of
maximal entropy ${\mu_{\Lambda,\max}}$. We denote by
$h_{\mathrm{top}}(\Lambda,f)$ the topological entropy of $\Lambda$.

######  Corollary D.

Consider $f\in\mathrm{Diff}^{r}(M^{m+1})$, $r>1,\,m\geqslant 2$, having a
homoclinic tangency associated to a simple saddle $P$ of $u$-index one such
that

  1. (1)

$P$ is of type $(m_{s},1)$ with $m_{s}\in\\{1,2\\}$,

  2. (2)

$J_{P}(f)\geqslant 1$, and

  3. (3)

$P$ is homoclinically related to a horseshoe $\Lambda$ satisfying

$h_{\mathrm{top}}(\Lambda,f)>-\log
J^{s}_{\mu_{\Lambda,\max}}+\frac{1}{2r}\chi^{cs}_{\mu_{\Lambda,\max}}.$

Then there are diffeomorphisms $g$ arbitrarily $C^{1}$ close to $f$ displaying
simultaneously $C^{1}$ robust heterodimensional cycles of coindex $i$ for
every $1\leqslant i\leqslant m_{s}$. These cycles are associated to
$\Lambda_{g}$ and a transitive hyperbolic set of $u$-index $1+i$.

Similarly, as a consequence of Theorem B, we obtain the following.

######  Corollary E.

Consider $f\in\mathrm{Diff}^{r}(M^{m+n})$, $r>1$ and $m,\,n\geqslant 2$,
having a homoclinic tangency associated to a simple saddle $P$ of $u$-index
one such that

  1. (1)

$P$ is of type $(m_{s},n_{u})$ with $m_{s},n_{u}\in\\{1,2\\}$,

  2. (2)

$P$ is homoclinically related to a horseshoe $\Lambda$ with

$h_{\mathrm{top}}(\Lambda,f)>\max\\{-\log
J^{s}_{\mu_{\Lambda,\max}}+\frac{1}{2r}\chi_{\mu_{\Lambda,\max}}^{cs},\ \log
J^{u}_{\mu_{\Lambda,\max}}-\frac{1}{2r}\chi_{\mu_{\Lambda,\max}}^{cu}\\}.$

Then there are diffeomorphisms $g$ arbitrarily $C^{1}$ close to $f$ having
simultaneously $C^{1}$ robust heterodimensional cycles of coindex $i$ for
every $1\leqslant i\leqslant N$, where

$N=m_{s},\quad\mbox{if $J_{P}(f)\geqslant 1$}\qquad\mbox{and}\qquad
N=n_{u},\quad\mbox{if $J_{P}(f)\leqslant 1$.}$

These cycles are associated to $\Lambda_{g}$ and a transitive hyperbolic set
of $u$-index $n+i$.

We now state the variation of the results in [4] mentioned above (see Theorem
1.4). A first step of the proof of this variations involves linearisation and
diagonalisation results (see [4, Theorems B and 3.2 ]). These results claim
that given any diffeomorphism $f$ with a horseshoe $\Lambda$ there is a
$C^{1}$ local perturbation $g$ of $f$ having an affine horseshoe
$\widetilde{\Lambda}$ close to $\Lambda$ in the Hausdorff distance and whose
topological entropy is also close to the one of $\Lambda$. Moreover, the set
$\widetilde{\Lambda}$ has a dominated splitting consisting of one-dimensional
bundles. A second step, also borrowed from [4], is that hyperbolic measures
with “large entropy” (in the sense of Equation (1.6), see also Remark 1.6)
provide blenders, see [4, Theorems B’ and C].

Finally, recall that given any hyperbolic measure $\mu$ of a diffeomorphism
$f$, Pesin theory provides stable and unstable manifolds, denoted by
$W_{\mu}^{\ast}(x,f)$, $\ast=s,u$, for $\mu$-almost every point $x$. Note that
this requires $f$ to be $C^{r}$ with $r>1$. We say that a transitive
hyperbolic set $\Lambda$ and a hyperbolic measure $\mu$ are homoclinically
related if there is a periodic orbit $\mathcal{O}\subset\Lambda$ such that
$W^{s}(\mathcal{O},f)\pitchfork W^{u}_{\mu}(x,f)\not=\emptyset$ and
$W^{u}(\mathcal{O},f)\pitchfork W_{\mu}^{s}(x,f)\not=\emptyset$ for
$\mu$-almost every point $x\in\Lambda$.

######  Theorem 1.4.

Consider $f\in\mathrm{Diff}^{r}(M^{m+n})$, $r>1$, $m\geqslant 2,n\geqslant 1$,
and a hyperbolic measure $\mu$ of $f$ with $u$-index $n$ such that

(1.6) $h_{\mu}(f)>-\log\mathrm{J}^{s}_{\mu}+\frac{1}{2r}\chi^{cs}_{\mu}.$

Then there is a sequence $(g_{k})$, $g_{k}\in\mathrm{Diff}^{r}(M^{m+n})$, of
local perturbations of $f$ supported on arbitrarily small neighbourhoods of
the support of $\mu$ and converging to $f$ in the $C^{1}$ topology such that
every $g_{k}$ has a $cs$-blender $\Gamma_{k}$ such that

  1. (1)

$\Gamma_{k}$ has central dimension $d_{cs}=m-1$ and a dominated splitting
consisting of one-dimensional subbunddles,

  2. (2)

$\Gamma_{k}$ converges to the support of $\mu$ in Hausdorff distance, and

  3. (3)

$h_{\mathrm{top}}(\Gamma_{k},g_{k})$ converges to $h_{\mu}(f)$.

Moreover, if $f$ has a saddle $P$ that is homoclinically related to the
measure $\mu$ then the sequence $(g_{k})_{k}$ can be chosen such that the
continuation $P_{k}$ of $P$ for $g_{k}$ is homoclinically related to the
blender $\Gamma_{k}$. In addition, if $P$ has a homoclinic tangency, then
$g_{k}$ can be chosen with a homoclinic tangency associated to $P_{k}$.

######  Remark 1.5.

According to item (2) in Remark 2.3, the blenders $\Gamma_{k}$ in Theorem 1.4
are $cs$-blenders of central dimension $k$ for every $k=1,\dots,m-1$ (here $m$
is the number of negative Lyapunov exponents of $\mu$ counted with
multiplicity).

######  Remark 1.6.

A result analogous to Theorem 1.4 holds under the assumption

(1.7) $h_{\mu}(f)>\log\mathrm{J}^{u}_{\mu}-\frac{1}{2r}\chi^{cu}_{\mu}$

called in [4] almost Pesin formula. This condition leads to $cu$-blenders with
similar properties as in Theorem 1.4. In particular, if $\mu$ is a hyperbolic
measure satisfying the Pesin entropy formula (that is,
$h_{\mu}(f)=\log\mathrm{J}^{u}_{\mu}$) then there are arbitrarily small
$C^{1}$ perturbations $g$ of $f$ having $cu$-blenders $\Gamma_{g}$ satisfying
conditions (1)–(3) in Theorem 1.4. Clearly, in item (1) the central dimension
$d_{cu}$ of the blender is $k=1,\dots,n-1$, where $n$ is the $u$-index of
$\mu$.

######  Remark 1.7.

Double blenders (see Definition 2.1) with central dimensions $(m-1,n-1)$,
where $m$ and $n$ are as in the previous remarks, and dominating splittings
into one-dimensional subbundles are obtained when

$h_{\mu}(f)>\max\\{-\log\mathrm{J}^{s}_{\mu}+\frac{1}{2r}\chi^{cs}_{\mu},\
\log\mathrm{J}^{u}_{\mu}-\frac{1}{2r}\chi^{cu}_{\mu}\\}.$

######  Terminology 1.8.

Throughout this paper, we will use the term _perturbation_ only to refer to
arbitrarily small ones.

### Organisation

This paper is organised as follows. In Section 2, we introduce and discuss
$cs$ and $cu$-blenders. In Section 3, we study the generation of simple
saddles with some prescribed domination properties. In Section 4, we analise
the return maps in a neighbourhood of the homoclinic tangency and explore the
expanding-like properties derived from the expanding leading Jacobian
hypothesis. Section 5 introduces unfolding families and uses them to get new
saddles of different $u$-indices. Theorems A and B are proved in Section 6,
for that we relate the saddles with different $u$-indices previously obtained
with the blender. Section 7 is dedicated to Simon-Asaoka’s examples. Section 8
is an appendix dedicated to the proof of Theorem 1.4. Finally, Section 9 is
another appendix about dominated splittings.

##  2\. Blenders

There are several definitions of blenders, each one adapted to an appropriate
context and emphasising some properties. We do not aim to discuss and compare
these definitions. As mentioned above, blenders are hyperbolic sets with a
geometric structure guaranteeing intersections between a family of “stable”
disks and the local unstable manifold of the blender. A fundamenta property of
blenders is their $C^{1}$ robustness. A relevant point in our Definition 2.1
is the existence of a distinctive saddle. As far as we know, distinctive
saddles exist in any known example of blenders, their existence is an explicit
requirement in [10, Definition A1]333 Let us summarise some settings where
blenders play key roles. Blenders were initially defined as having central
dimension one. Currently, there are versions (as the ones here) where the
central dimensions are greater than one. Applications of blenders of central
dimension one include: robustly transitive dynamics [19], robust
heterodimensional cycles [18, 33], robust homoclinic tangencies [20], stable
ergodicity [46], and construction of nonhyperbolic ergodic measures [14],
among others. Blenders with larger central dimensions were introduced in [37,
8] to study instability problems in symplectic dynamics, in [6] to obtain
robust heterodimensional cycles of large coindex, and in [7, 2] to get robust
tangencies of large codimension. Blenders of large central dimension also
appear in the study of ergodicity of conservative partial hyperbolic systems
[4], of holomorphic dynamics [51, 26, 13], and of parametric families of maps
(endomorphisms in [11],[12] and diffeomorphisms [9])..

Our definition of a blender involves the concepts of hyperbolicity and partial
hyperbolicity, stable and unstable bundles, strong invariant manifolds, and
domination. For these concepts, see the appendix in Section 9. In what
follows, we write dominated splittings $E_{1}\oplus\cdots\oplus E_{k}$ with
several bundles in such a way the bundle $E_{i}$ is more “contracting” than
the bundle $E_{i+1}$.

We will use the following notation for the local invariant manifolds of a
hyperbolic set $\Lambda$ of a diffeomorphism $f$ relative to a neighbourhood
$U$ of $\Lambda$. We let

(2.1)
$W^{u}_{U,\mathrm{loc}}(\Lambda,f)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\\{x\in U\colon f^{i}(x)\in U\,\mbox{for every $i\leqslant 0$}\\}.$

The set $W^{s}_{U,\mathrm{loc}}(\Lambda,f)$ is defined in the obvious way. A
special case occurs when $\Lambda$ is a periodic orbit.

######  Definition 2.1 ($cs$-, $cu$-blender, and double blender).

Let $f\in\mathrm{Diff}^{1}(M^{d})$, $d\geqslant 3$. A hyperbolic and
transitive set $\Gamma\subset{M}^{d}$ is a _$cs$ -blender _ if the following
holds:

(a) (local maximality) There is an open neighbourhood $U$ of $\Gamma$ such
that

$\Gamma=\bigcap_{i\in\mathbb{Z}}f^{i}(\overline{U}).$

(b) (partial hyperbolicity) There is an $Df$-invariant dominated splitting
with three nontrivial bundles defined over $\Gamma$

$T_{\Gamma}M^{d}=E^{ss}\oplus E^{c}\oplus E^{u},$

where $E^{s}=E^{ss}\oplus E^{c}$ and $E^{u}$ are the stable and unstable
bundles of $\Gamma$. We write

$d_{cs}\stackrel{{\scriptstyle\scriptscriptstyle\rm def}}{{=}}\dim
E^{c}\geqslant 1\quad\mbox{and}\quad
d_{ss}\stackrel{{\scriptstyle\scriptscriptstyle\rm def}}{{=}}\dim
E^{ss}\geqslant 1.$

(c) (open set of embedded disks) There is an open set $\mathscr{D}^{ss}$ of
$C^{1}$-embeddings of $d_{ss}$-dimensional disks into $M^{d}$ such that:

  1. i)

(distinctive saddle) There is a saddle $Q^{\ast}\in\Gamma$ whose strong stable
manifold $W^{ss}(Q^{\ast},f)$ contains a disk of $\mathscr{D}^{{ss}}$
containing $Q^{\ast}$ in its interior.

  2. ii)

(robust intersections) There is a $C^{1}$-neighbourhood $\mathscr{U}$ of $f$
such that for every $g\in\mathscr{U}$ the continuation $\Gamma_{g}$ of
$\Gamma$ satisfies

$W^{u}_{U,\mathrm{loc}}(\Gamma_{g},g)\cap D\not=\emptyset,\quad\mbox{for
every}\quad D\in\mathscr{D}^{{ss}}.$

The set $U$ is called a _reference domain_ of the blender. The sets
$\mathscr{D}^{{ss}}$ and

$B^{ss}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathrm{int}\Big{(}\bigcup_{D\in\mathscr{D}^{ss}}D\cap U\Big{)}$

are called _superposition region_ and _superposition domain_ of the blender,
respectively. The point $Q^{\ast}$ is _a distinctive saddle_ of the blender.
We say that $d_{cs}$ is the central dimension of the blender.

A _$cu$ -blender_ of $f$ is $cs$-blender for $f^{-1}$. For a $cu$-blender,
$d_{cu}$, $\mathscr{D}^{{uu}}$, and $B^{uu}$ are defined as above.

A _double blender of central dimensions $(d_{cs},d_{cu})$_ is a hyperbolic set
that is simultaneously a $cs$-blender of central dimension $d_{cs}$ and
${cu}$-blender of central dimension $d_{cu}$. In particular, a double blender
has a pair of distinctive saddles $Q^{\ast}_{cs}$ and $Q^{\ast}_{cu}$ as well
as $ss$\- and $uu$-superposition regions and domains.

Note that $cs$-blenders with central dimension $d_{cs}\geqslant 1$ can occur
only in dimension $1+d_{cs}+1\geqslant 3$. Similarly, double blenders with
central dimensions $(d_{cs},d_{cu})$ can appear only in dimension
$1+d_{cs}+d_{cu}+1\geqslant 4$.

######  Remark 2.2 (Blender-horseshoes).

Blender-horseshoes were introduced in [20]. They have a one-dimensional
central dimension. Their key property is that iterations of the disks in the
superposition region provide new disks of that region. Every blender-horseshoe
is a $cs$\- or a $cu$-blender with central one-dimensional direction.

######  Remark 2.3 (Properties of blenders).

Consider $f\in\mathrm{Diff}^{r}(M)$, $r\geqslant 1$, with a $cs$-blender
$\Gamma$ of central dimension $d_{cs}$, reference domain $U$, partially
hyperbolic splitting

(2.2) $T_{\Gamma}M^{d}=E^{ss}\oplus E^{c}\oplus E^{u},\qquad d_{cs}=\dim
E^{c},\quad d_{ss}=\dim E^{ss},$

superposition region $\mathscr{D}^{ss}$, and distinctive saddle $Q^{\ast}$.

(1) Continuations of blenders: Having a $cs$-blender is a $C^{1}$ open
condition (and hence $C^{r}$ open). For every $g$ sufficiently $C^{1}$ close
to $f$ the continuation $\Gamma_{g}$ of $\Gamma$ is also a $cs$-blender with
the same reference domain $U$, the same superposition region
$\mathscr{D}^{ss}$, and the same central dimension. Moreover, if $Q^{\ast}$ is
a distinctive saddle of $\Gamma$ then its continuation $Q^{\ast}_{g}$ is also
a distinctive saddle of $\Gamma_{g}$.

(2) Central dimensions of a $cs$-blender: The hyperbolicity of $\Gamma$
implies that if $D$ is a disk in $\mathscr{D}^{ss}$ and $X\in
W^{u}_{U,\mathrm{loc}}(\Gamma,f)\cap D$ then there is $Z\in\Gamma$ such that
$X\in W^{u}_{U,\mathrm{loc}}(Z,f)\cap D$. Moreover, the codimension of

$T_{X}W^{u}_{U,\mathrm{loc}}(Z,f)+T_{X}D$

is at least $d_{cs}$ and is equal to $d_{cs}$ if and only if the intersection
between $W^{u}_{U,\mathrm{loc}}(Z,f)$ and $D$ at $X$ is quasi-transverse
(i.e., $T_{X}W^{u}_{U,\mathrm{loc}}(Z,f)\cap T_{X}D=\\{\bar{0}\\}$).

Assume now that the bundle $E^{c}$ of $\Gamma$ splits in a dominated way as
follows

(2.3) $E^{c}=E^{c}_{1}\oplus\cdots\oplus E^{c}_{k}.$

For each $1\leqslant r\leqslant\ell\leqslant k$, consider the bundles and the
dimensions

$E^{c}_{r,\ell}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}E^{c}_{r}\oplus\cdots\oplus E^{c}_{\ell},\qquad
d_{r,\ell}\stackrel{{\scriptstyle\scriptscriptstyle\rm def}}{{=}}\dim
E^{c}_{r,\ell}.$

We claim that $\Gamma$ is also a $cs$-blender of central dimension $d_{r,k}$
for each $r=2,\dots,k$. Using (2.3), for each $r\in\\{2,\dots,k\\}$, we get a
dominated splittings of $\Gamma$ of the form

(2.4) $T_{\Gamma}M^{d}=(E^{ss}\oplus E^{c}_{1,r-1})\oplus E^{c}_{r,k}\oplus
E^{u},$

where $E^{ss}\oplus E^{c}_{1,r-1}$ is a strong stable bundle and $E^{c}_{r,k}$
is a center stable bundle of dimension $d_{r,k}$.

To see that $\Gamma$ is a $cs$-blender of central dimension $d_{r,k}$ it is
sufficient to consider the splitting in (2.4) and any family of disks
$\widehat{\mathscr{D}}^{ss}$ of dimension $d_{ss}+d_{1,r-1}$ containing the
initial family $\mathscr{D}^{ss}$ and such that some disk of this family
contains $Q^{\ast}$ in its interior and is contained in the strong stable
manifold $W^{ss}(Q^{\ast})$ (now considered with respect the strong stable
bundle $E^{ss}\oplus E^{c}_{1,r-1}$). By the observations above, the
intersections properties in (c) hold.

In the special case when $E^{c}$ splits into one-dimensional bundles in a
dominated way, we have that $\Gamma$ is a $cs$-blender of central dimensions
$1,\dots,d_{cs}$. This sort of blender was introduced in [37, 6] and called
_superblender_ in [4].

(3) Homoclinic relations: Note that every pair of saddles of $\Gamma$ are
homoclinically related. In particular, every saddle of $\Gamma$ is
homoclinically related to any distinctive saddle of $\Gamma$. Thus any saddle
homoclinically related to the blender is homoclinically related to any
distinctive saddle of the blender.

The following remark is a standard consequence of the inclination lemma and
the comments above.

######  Remark 2.4 (Tangencies associated to distinctive saddles).

Let $f\in\mathrm{Diff}^{r}(M)$ and consider two saddles $R$ and $S$ of $f$
which are homoclinically related. Suppose that $R$ has a homoclinic tangency.
Then there is a $C^{r}$ perturbation $g$ of $f$ such that $S_{g}$ has a
homoclinic tangency. We get the following consequence. Fix a $cs$-blender
$\Gamma$ of $f$, a distinctive saddle $Q^{\ast}$ of $\Gamma$, and a saddle $P$
homoclinically related to $\Gamma$. Then if $P$ has a homoclinic tangency
there is a $C^{r}$ perturbation $g$ of $f$ having a homoclinic tangency
associated to the distinctive saddle $Q^{\ast}_{g}$ of the blender
$\Gamma_{g}$.

A variation of the above assertion is the following. Consider a hyperbolic
basic set $\Lambda$ of $f$ (i.e., locally maximal in an open neighbourhood of
it) with a homoclinic tangency (not necessarily associated to a saddle). Given
any periodic saddle $R\in\Lambda$ there is a $C^{r}$ perturbation $g$ of $f$
with a homoclinic tangency associated to $R_{g}\in\Lambda_{g}$.

##  3\. Homoclinic relations of simple saddles with prescribed type

This section aims to prove Proposition 3.1 about the generation of simple
saddles with some prescribed domination. The relevant cases lead to simple
saddles of type $(2,1)$ and $(2,2)$.

For the next definition, note that if a saddle $P$ of a diffeomorphism $f$ has
$u$-index one then $W^{u}_{\mathrm{loc}}(P,f)\setminus\\{P\\}$ has two
connected components444A local unstable manifold of $P$ is any disk contained
in $W^{u}(P,f)$ of the same dimension as $W^{u}(P,f)$ that contains $P$ in its
interior. A local stable manifold is similarly defined.. In this case, we say
that the saddle $P$ is _biaccumulated by its transverse homoclinic points_ if
$W^{s}(P,f)$ transversely intersects both connected components of
$W^{u}_{\mathrm{loc}}(P,f)\setminus\\{P\\}$. Note that this definition does
not depend on the choice of the local manifold. We will use this property in
Section 4.4 to define return maps associated to a homoclinic tangency of a
saddle with $u$-index one.

For the next proposition, recall that a saddle $P$ of $f$ has _simple
spectrum_ if the eigenvalues of $Df^{\pi}(P)$ are all real, positive, and have
multiplicity one, here $\pi$ is the period of $P$.

######  Proposition 3.1.

Consider $f\in\mathrm{Diff}^{r}(M^{m+n})$, $r\geqslant 1$, $m\geqslant 2$,
$n\geqslant 1$, with a pair of saddles $P$ and $P^{\prime}$ of $u$-index $n$
which are homoclinically related. Assume that $P$ is simple of type
$(m_{s},n_{u})$ and that $P^{\prime}$ has simple spectrum. Then given any
neighbourhood $V$ of $P^{\prime}$ there is a $C^{r}$ perturbation $g$ of $f$
having a saddle $Q\in V$ such that

  1. (1)

$Q$ is simple of type $(m_{s},n_{u})$ and $\log J_{P}(f)\cdot\log J_{Q}(g)>0$,

  2. (2)

$Q$ is homoclinically related to the continuations $P_{g}$ and
$P^{\prime}_{g}$.

  3. (3)

If $n=1$ then $Q$ is biaccumulated by its transverse homoclinic points.

Finally, if $P$ has a homoclinic tangency then $Q$ can be taken with a
homoclinic tangency.

For simplicity, in what follows, let us assume that the saddles have period
one.

We will use this proposition to get saddles with nonreal central eigenvalues.
Thus the relevant case occurs when $m_{s}$ or $n_{s}$ (or both) are different
from one. In the literature many results provide conditions for the abundance
of saddles with simple spectrum after perturbations. In particular, the case
$m_{s}=n_{u}=1$ is well-known (see, for instance, the linearisation result in
[4, Theorem 3.2]). Thus we consider the cases when $m_{s}\,n_{u}>1$ and skip
the case $(1,1)$.

Let us note that the condition on the existence of a saddle homoclinically
related to $P$ with simple spectrum is not a restriction in our context.
Indeed, if a saddle $P$ is homoclinically related to a saddle $\widetilde{P}$,
given any neighbourhood $V$ of $\widetilde{P}$ after a $C^{r}$ perturbation of
$f$ we can assume that there is a saddle $P^{\prime}\in V$ with simple
spectrum that is homoclinically related to $P$ and $\widetilde{P}$.

The key step in the proof of Proposition 3.1 is the next lemma:

######  Lemma 3.2.

Consider $f\in\mathrm{Diff}^{r}(M^{m+n})$ and saddles $P$ and $P^{\prime}$ as
in Proposition 3.1. Assume that $P$ is of type $(2,n_{u})$ and $C^{r}$
linearisable in a neighbourhood $U$. Given any neighbourhood $V$ of
$P^{\prime}$ there is a $C^{r}$ perturbation $g$ of $f$ with a hyperbolic
transitive set $\Lambda_{g}$ containing $P_{g},P_{g}^{\prime},$ and a simple
saddle $Q$ of type $(2,n_{u})$ such that the orbit of $Q$ intersects $U$ and
$V$ and satisfies

$\log J_{P_{g}}(g)\cdot\log J_{Q}(g)>0.$

Before proving the lemma, let us give a simple two-dimensional geometrical
argument leading to matrices with a pair of conjugate nonreal eigenvalues.

Consider a $2\times 2$ real matrix $A$ with real eigenvalues $\tau>\rho>0$ and
the family of linear maps

(3.1) $A_{\varphi}=A\circ R_{\varphi},\quad\mbox{where}\quad
R_{\varphi}=\left(\begin{matrix}\cos\varphi&-\sin\varphi\\\
\sin\varphi&\cos\varphi\end{matrix}\right),\quad\varphi\geqslant 0.$

######  Claim 3.3 (Saddle-node bifurcations).

There is $\varphi_{0}>0$ such that

  * •

$A_{\varphi}$ has different and positive real eigenvalues for every
$\varphi\in[0,\varphi_{0})$,

  * •

$A_{\varphi_{0}}$ has a real eigenvalue of multiplicity two, and

  * •

$A_{\varphi}$ has a pair of nonreal eigenvalues for $\varphi>\varphi_{0}$
close enough to $\varphi_{0}$.

###### Proof.

Without loss of generality, we can assume that the eigenvectors of $A$ are
$v=(1,0)$ and $w=(\cos\theta,\sin\theta)$, $\theta\in(0,\pi)$, associated
$\tau>\rho>0$. Consider the circle map $\widetilde{A}_{\varphi}$ associated to
$A_{\varphi}$ given by

$\widetilde{A}_{\varphi}(u)=\frac{A_{\varphi}(u)}{||A_{\varphi}(u)||}.$

Note that $\widetilde{A}_{0}$ has the following hyperbolic fixed points: the
“most expanding” eigenvector $v$ of $A$ is attracting for $\widetilde{A}_{0}$
while the “most contracting” eigenvalue $w$ of $A$ is repelling for
$\widetilde{A}_{0}$. See Figure 2. These hyperbolic points of
$\widetilde{A}_{0}$ in $\mathbb{S}^{1}$ have continuations $v_{\varphi}$ and
$w_{\varphi}$ for every small $\varphi>0$. Note that for $\varphi>0$ it holds
$\widetilde{A}_{\varphi}(u)>u$, $u=v,w$. That is, the graph of
$\widetilde{A}_{\varphi}$ moves up as $\varphi$ increases. This implies that
the fixed points $v_{\varphi}$ and $w_{\varphi}$ of $\widetilde{A}_{\varphi}$
“move up and down,” respectively, and hence $v<v_{\varphi}<w_{\varphi}<w$.
Thus there is some $\varphi_{0}$ (associated to a saddle-node bifurcation of
the family $(\widetilde{A}_{\varphi})_{\varphi}$) such that these two points
coincide. The parameter $\varphi_{0}$ corresponds to a real eigenvalue of
multiplicity two of $A_{\varphi_{0}}$. For $\varphi>\varphi_{0}$ close enough
to $\varphi_{0}$, this leads to a pair of conjugate nonreal eigenvalues. ∎

\begin{overpic}[scale={0.1}]{SN.jpg} \put(130.0,82.0){\Large{$v$}}
\put(117.0,72.0){\large{$\bullet$}} \put(-10.0,82.0){\Large{$-v$}}
\put(14.0,72.0){\large{$\bullet$}} \put(84.0,135.0){\Large{$w$}}
\put(81.0,120.0){\large{$\bullet$}} \put(32.0,8.0){\Large{$-w$}}
\put(50.0,23.0){\large{$\bullet$}} \put(65.0,72.0){\large{$\bullet$}}
\put(223.0,86.0){$\widetilde{A}_{\varphi_{0}}$ }
\put(253.0,49.0){$\widetilde{A}_{0}$ } \put(210.0,8.0){\Large{$v$}}
\put(275.0,8.0){\Large{$w$}} \put(211.5,42.0){{$\bullet$}}
\put(277.5,108.0){{$\bullet$}} \end{overpic} Figure 2. Saddle-node bifurcation

We now recall a standard construction involving transverse homoclinic points
and hyperbolic sets.

######  Remark 3.4.

Consider $f\in\mathrm{Diff}^{r}(M^{m+n})$ with a pair of saddles $P$ and
$P^{\prime}$ of $u$-index $n$ as in Proposition 3.1. Consider open sets $U\ni
P$ and $V\ni P^{\prime}$, transverse homoclinic points $X\in
W^{s}(P,f)\pitchfork W^{u}(P^{\prime},f)$ and $Y\in
W^{s}(P^{\prime},f)\pitchfork W^{u}(P,f)$ with (recall the definitions in
(2.1))

$\begin{split}&f^{k_{X}}(X)\in W^{s}_{U,\mathrm{loc}}(P,f),\quad
f^{-k_{X}}(X)\in W^{u}_{V,\mathrm{loc}}(P^{\prime},f)\\\ &f^{k_{Y}}(Y)\in
W^{s}_{V,\mathrm{loc}}(P^{\prime},f),\quad f^{-k_{Y}}(Y)\in
W^{u}_{U,\mathrm{loc}}(P,f)\end{split}$

for some constants $k_{X},k_{Y}>0$, see Figure 3. Then for every pair of small
neighbourhoods $U_{X}$ of $X$ and $U_{Y}$ of $Y$, the maximal invariant set
$\Lambda=\Lambda_{\widehat{V}}$ of $f$ in

$\widehat{U}=U\cup
V\cup\widetilde{U}_{X}\cup\widetilde{U}_{Y},\quad\mbox{where}\quad\widetilde{U}_{Z}=\bigcup_{i=-k_{Z}}^{k_{Z}}f^{i}(U_{Z}),\quad
Z=X,Y,$

is a transitive hyperbolic set with a dominated splitting of the form

$E^{ss}\oplus E^{cs}\oplus E^{cu}\oplus E^{uu},\quad E^{s}=E^{ss}\oplus
E^{cs},\quad E^{u}=E^{cu}\oplus E^{uu},$

where the dimensions of $E^{cs}$ and $E^{cu}$ are $2$ and $n_{u}$,
respectively. This decomposition follows from the type of $P$ and the simple
spectrum of $P^{\prime}$.

We observe that there is a homoclinic version (i.e., with $P$ having
transverse homoclinic points and considering only one transverse homoclinic
point) of the construction above.

###### Proof of Lemma 3.2.

Let $\Lambda=\Lambda_{\widehat{U}}$ be the hyperbolic set provided by Remark
3.4 (applied to $P$, $P^{\prime}$, $U$, $V$ as in the lemma, and the
corresponding homoclinic points). We take a neighbourhood $U_{0}$ of $P$
contained in $U$, a transverse homoclinic point $A\in\Lambda$ of $P$, and
$\ell>0$ such that the orbit $O_{f}(A)$ of $A$ satisfies

(3.2)
$\begin{split}&\mathcal{O}_{f}(A)\setminus\\{f(A),\dots,f^{\ell-1}(A)\\}\subset
U_{0},\\\ &\\{f(A),\dots,f^{\ell-1}(A)\\}\cap U_{0}=\emptyset,\\\ \
&\mathcal{O}_{f}(A)\cap V\neq\emptyset.\end{split}$

\begin{overpic}[scale={0.1}]{relaciones.jpg} \put(17.0,85.0){\Large{$U$}}
\put(47.0,60.0){\large{$P$}} \put(14.0,160.0){\Large{$U_{Y}$}}
\put(45.0,144.0){\large{$Y$}} \put(100.0,75.0){\Large{$U_{X}$}}
\put(131.0,60.0){\large{$X$}} \put(100.0,168.0){\Large{$V$}}
\put(133.0,144.0){\large{$P^{\prime}$}} \put(223.0,117.0){\large{$A$}}
\put(250.0,140.0){\large{$f^{k}(Q_{k})$} } \put(215.0,147.0){\Large{$U_{A}$}}
\put(297.0,39.0){\large{$f^{\ell}(A)$} } \put(325.0,67.0){\large{$Q_{k}$} }
\put(310.0,127.0){\Large{$\mathfrak{F}$} }
\put(260.0,80.0){\Large{$\mathfrak{L}^{k}$} } \put(217.0,32.0){\Large{$P$} }
\end{overpic} Figure 3. Generation of saddles with prescribed itineraries

Take a small neighbourhood $U_{A}\subset U_{0}$ of $A$ such that
$f^{\ell}(U_{A})\subset U_{0}$, $f^{i}(U_{A})\subset\widehat{U}$ for every
$i\in\\{0,\dots,\ell\\}$, and the sets $f(U_{A}),\dots,f^{\ell-1}(U_{A})$ are
pairwise disjoint and disjoint from $U_{0}$. Let $\mathfrak{F}$ be the
restriction of $f^{\ell}$ to $U_{A}$, that is, the transition map from $U_{A}$
to $U_{0}$ along the orbit of $A$, and $\mathfrak{L}$ the restriction of $f$
to $U_{0}$ (a linear map). See Figure 3. Observe that there is $k_{0}$ such
that for every $k\geqslant k_{0}$ there is a periodic point $Q_{k}$ of period
$k+\ell$ and $u$-index $n$ such that

  * •

$Q_{k}\in f^{\ell}(U_{A})$,

  * •

$f^{i}(Q_{k})\in U_{0}$ for every $i=0,\dots,k$ and $f^{k}(Q_{k})\in U_{A}$,

  * •

the eigenvalues of $Df^{k+\ell}(Q_{k})$ satisfies (following the notation in
(1.1))

$|\lambda^{(k)}_{3}|<|\lambda^{(k)}_{2}|\leqslant|\lambda^{(k)}_{1}|<1<|\gamma^{(k)}_{1}|\leqslant|\gamma^{(k)}_{2}|<|\gamma^{(k)}_{3}|,$

  * •

$Q_{k}$ is homoclinically related to $P$.

This implies that

$Q_{k}=f^{k+\ell}(Q_{k})=\mathfrak{F}\circ\mathfrak{L}^{k}(Q_{k})\in\Lambda$

and that the orbit of $Q_{k}$ intersects both $U_{0}\subset U$ and $V$.

We first consider the case when $P$ is of type $(2,1)$. If there are some
large $k$ such that $\lambda^{(k)}_{2}$ and $\lambda^{(k)}_{1}$ are nonreal or
equal, we are done. Thus, in what follows, we assume that $\lambda^{(k)}_{2}$
and $\lambda^{(k)}_{1}$ are both real and consider the two-dimensional central
stable bundle $E^{cs}(Q_{k})$ associated to them provided by Remark 3.4.

We write the linearising coordinates of $P$ and the matrix $Df(P)$ in the form

$(x_{m-2},x_{c},x_{n})\in\mathbb{R}^{m-2}\times\mathbb{R}^{2}\times\mathbb{R}^{n},\qquad
Df(Q)=\mathfrak{L}=(L_{m-2},\lambda\,R_{\theta},L_{n}),$

where $\lambda\in(0,1)$, $R_{\theta}$ is as in (3.1), and $L_{m-2}$ and
$L_{n}$ are linear maps. We consider linear maps of the form

(3.3)
$\mathfrak{L}_{\alpha}=(L_{m-2},R_{\alpha}\circ(\lambda\,R_{\theta}),L_{n})=(L_{m-2},\lambda
R_{\alpha+\theta},L_{n})$

and the planes

$\pi\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\\{(0^{m-2},v_{c},0^{n}),v_{c}\in\mathbb{R}^{2}\\},\quad\widetilde{\pi}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}D\mathfrak{F}_{\mathfrak{F}(A)}^{-1}(\pi).$

We first claim that we can assume that the determinant of
$D\mathfrak{F}{{}_{A}}|_{\widetilde{\pi}}$ is positive and
$D\mathfrak{F}{{}_{A}}|_{\widetilde{\pi}}$ preserves the orientation.
Otherwise, we consider new homoclinic points corresponding to two “loops” of
$A$, that is, a new transverse homoclinic point $A^{\prime}$ of $P$ nearby $A$
whose orbit is as follows: there is large even $i_{0}$ such that

$\begin{split}&A^{\prime}\in W^{u}_{U_{0},\mathrm{loc}}(P),\quad
f^{\ell+i_{0}+\ell}(A^{\prime})\in W^{s}_{U_{0},\mathrm{loc}}(P),\\\
&f^{\ell}(A^{\prime}),f^{\ell+i_{0}}(A^{\prime})\in U_{A},\\\
&f^{\ell+i}(A^{\prime})\in U_{0}\quad\mbox{for every $i\in\\{0,\dots
i_{0}\\}$}.\end{split}$

As for large $i_{0}$ the transition map $\mathfrak{F}^{\prime}$ associated to
$f^{2\ell+i_{0}}$ at $A^{\prime}$ is close to
$\mathfrak{F}\circ\mathfrak{L}^{i_{0}}\circ\mathfrak{F}$ our claim follows.
Note that for this new point $A^{\prime}$ one must shrink the linearising
neighbourhood in the next steps, but this just adds additional iterations by a
linear map that does not change the central determinant.

To conclude the proof, consider local perturbations $f_{\alpha}$ of $f$ whose
derivative in $U_{0}$ is $\mathfrak{L}_{\alpha}$. For each large $k$ and
$\alpha\in[0,2\pi]$ the diffeomorphism $f_{\alpha}$ has a periodic point
$Q_{k,\alpha}$ that is the continuation of $Q_{k}$. Note that if there is
small $\alpha$ and large $k$ such that

$D(\mathfrak{L}^{k}_{\alpha}\circ\mathfrak{F})_{\mathfrak{F}^{-1}(Q_{k,\alpha})}=\mathfrak{L}^{k}_{\alpha}\circ\mathfrak{F}_{{k,\alpha}},\quad\mathfrak{F}_{{k,\alpha}}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}D\mathfrak{F}_{\mathfrak{F}^{-1}(Q_{k,\alpha})}$

has a nonreal central eigenvalue, we are done. Note that, independently of
$\alpha$,

(3.4) $\widehat{Q}_{k,\alpha}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathfrak{F}^{-1}(Q_{k,\alpha})\to
A\quad\mbox{and}\quad\mathfrak{F}_{{k,\alpha}}\to
D\mathfrak{F}_{A},\quad\mbox{as $k\to\infty$.}$

We also have

$\mathfrak{L}^{k}_{\alpha}=(L_{m-2}^{k},R_{k\alpha}\lambda^{k}R_{k\theta},L_{n}^{k})=(\mathrm{Id}_{m-2},R_{k\alpha},\mathrm{Id}_{n})\circ\mathfrak{L}^{k},$

where $\mathrm{Id}_{r}$ denotes the identity map in $\mathbb{R}^{r}$. Fix
large $k$ and the center eigenvectors of $v^{k}_{\alpha}$ and $w^{k}_{\alpha}$
of $Df^{k+\ell}(\widehat{Q}_{k,0})=\mathfrak{L}^{k}\circ\mathfrak{F}_{k,0}$.
For the parameter $\alpha=0$, we can chose these vectors such that

$u^{k}_{0}=(\tau^{s}_{k}(u^{k}_{c}),u^{k}_{c},\tau^{u}_{k}(u^{k}_{c}))\in\mathbb{R}^{m-2}\times\mathbb{R}^{2}\times\mathbb{R}^{n},\quad
u=v,w,$

where $\tau^{s}_{k}\colon\mathbb{R}^{2}\to\mathbb{R}^{m-2}$ and
$\tau^{u}_{k}\colon\mathbb{R}^{2}\to\mathbb{R}^{n}$ go to the zero map as
$k\to\infty$ and $v^{k}_{c}$ and $w_{c}^{k}$ are unitary eigenvectors in
$\mathbb{R}^{2}$ as in the proof of Claim 3.3. For each $\alpha\in[0,2\pi]$ we
consider

$\widehat{u}^{k}_{\alpha}=D\mathfrak{F}^{-1}_{Q_{k,\alpha}}(u^{k})=(\widehat{\tau}^{s}_{k}(\widehat{u}^{k}_{c,\alpha}),\widehat{u}^{k}_{c,\alpha},\widehat{\tau}^{u}_{k}(\widehat{u}_{c,\alpha})),\quad\quad
u=v,w,$

where the maps $\widehat{\tau}^{s,u}_{k}$ are defined similarly as above.
Observe that, by (3.4), for each fixed $\alpha$ the variation of these vectors
goes to zero as $k\to\infty$.

We now observe that, for small $\alpha$ and big $k$, the images of the vectors
$v^{k}_{0}$ and $w^{k}_{0}$ by $\mathfrak{L}_{\alpha}^{k}$ are the rotation by
the angle $k\alpha$ (that is big if $k$ is big) in the central direction of
their images by $\mathfrak{L}^{k}$ (which are parallel to
$\widehat{v}^{k}_{0}$ and $\widehat{w}^{k}_{0}$, respectively). Thus these
vectors move as in the proof of Claim 3.3. This displacement is not altered by
$D\mathfrak{F}$ since this map preserves the orientation in the central plane.
Now the conclusion follows as in Claim 3.3, implying that $Q_{k,\alpha}$ is of
type $(2,\ast)$, $\ast\in\\{1,2\\}$. The fact that if $P$ is of type $(2,1)$
then $Q_{k,\alpha}$ is of type $(2,1)$ follows from domination arguments.

In the case when $P$ is of type $(2,2)$ the argument is analogous. The only
difference is that we need to add a rotation-like term in the center unstable
direction. Equation (3.3) now reads as follows

$\begin{split}\mathfrak{L}_{\alpha,\beta}&=(L_{m-2},R_{\alpha}\circ(\lambda\,R_{\theta}),R_{\beta}\circ(\gamma\,R_{\varphi}),L_{n-2})\\\
&=(L_{m-2},\lambda
R_{\alpha+\theta},\gamma\,R_{\beta+\varphi}L_{n-2}),\end{split}$

where $\gamma>1$. This leads to a family depending on two independent
parameters $\alpha$ and $\beta$. This allows us to deal with each two-
dimensional central direction independently, simultaneously getting nonreal
eigenvalues.

Finally, to get the property of the leading Jacobians, note that the number of
iterates of the points $Q_{k,\alpha}$ in the neighbourhood $U_{0}$ can be
chosen arbitrarily large compared to the length of the orbit. The assertion
now follows using the domination splitting. This ends the proof of the lemma.
∎

######  Remark 3.5 (The case $n=1$).

In our construction, the orbit of the resulting saddle $Q$ has most of its
iterates in the linearising neighbourhood $U$ of $P$. Therefore, its local
manifolds are large. This implies that $Q$ is homoclinically related to $P$
(and hence to $P^{\prime}$). Consider now the case $n=1$. As the saddle $P$
has transverse homoclinic points, the property of $Q$ having large unstable
manifolds implies that both components of
$W^{u}_{U,\mathrm{loc}}(Q)\setminus\\{Q\\}$ transversely intersects the stable
manifold of $P$ (and hence the stable manifold of $Q$, since $P$ and $Q$ are
homoclinically related). This implies that $Q$ is biaccumulated by its
transverse homoclinic points.

######  Remark 3.6.

Assume that the saddle point $P$ in Lemma 3.2 has a homoclinic tangency. Then
the “local” perturbation $g$ of $f$ can be taken preserving that tangency.

###  3.1. Proof of Proposition 3.1

We need to consider the following types $(2,1)$, $(2,2)$, and $(1,2)$ for $P$.
We prove the proposition when $m_{s}=2$ (in the case $(1,2)$ just consider
$f^{-1}$). The existence of the saddle $Q$ of type $(2,n_{u})$ follows from
Lemma 3.2. The homoclinic relations and the biaccumulation property (when
$n=1$) follow from Remark 3.5. Remark 3.6 implies the preservation of the
tangency of $P$. This ends the proof of the proposition.

##  4\. Return maps at the homoclinic tangency

Throughout this section, we consider $f\in\mathrm{Diff}^{r}(M^{m+n})$,
$r,m\geqslant 2$ and $n\geqslant 1$, having a homoclinic tangency $Y$
associated to a simple saddle $P$ of type $(m_{s},n_{u})$ and $u$-index $n$.
We first state in Section 4.1 generic conditions (C0)–(C5) at the homoclinic
tangency. Thereafter, we fix a neighbourhood of the tangency and introduce
first return maps to it (Section 4.2). We also see how the expanding leading
Jacobian of the saddle $P$ (see Equation (1.2)) implies expansion properties
of the return maps (Section 4.3).

###  4.1. Generic conditions

For simplicity, in what follows, we assume that the period of $P$ is one. We
also assume that

  1. (C0)

$f$ is $C^{r}$ linearisable at $P$.

By Sternberg linearisation theorem [50], every $C^{r}$ neighbourhood of $f$
contains diffeomorphisms satisfying this condition.

We consider a small neighbourhood $W$ of $P$ with these linearising
coordinates

(4.1)
$(u,x,y,v)\in\mathbb{R}^{m-m_{s}}\times\mathbb{R}^{m_{s}}\times\mathbb{R}^{n_{u}}\times\mathbb{R}^{n-n_{u}}.$

In these coordinates, $P=(0^{m-m_{s}},0^{m_{s}},0^{n_{u}},0^{n-n_{u}})$ and
the map $\mathfrak{T}_{0}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}f|_{W}$ is of the form

(4.2)
$(u,x,y,v)\,\,\mapsto\,\,\mathfrak{T}_{0}(u,x,y,v)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(Au,Bx,Cy,Dv),$

here $A,B,D,C$ are square matrices whose eigenvalues are $\lambda_{m}$,
$\lambda_{m-1},\dots,\lambda_{m_{s}+1}$ (the ones of $A$),
$\lambda_{m_{s}},\dots,\lambda_{1}$ (the ones of $B$),
$\gamma_{1},\dots,\gamma_{n_{u}}$ (the ones of $C$), and
$\gamma_{n_{u}+1},\dots,\gamma_{n}$ (the ones of $D$). In the neighbourhood
$W$ the local invariant sets are (we omit the dependence on $W$)

(4.3) $\begin{split}W^{u}_{\mathrm{loc}}(P,f)&=\\{(u,x,y,v)\colon
u=0^{m-m_{s}},\,x=0^{m_{s}}\\}\subset W,\\\
W^{s}_{\mathrm{loc}}(P,f)&=\\{(u,x,y,v)\colon
y=0^{n_{u}},\,v=0^{n-n_{u}}\\}\subset W,\\\
W^{ss}_{\mathrm{loc}}(P,f)&=\\{(u,x,y,v)\colon
x=0^{m_{s}},\,y=0^{n_{u}},\,v=0^{n-n_{u}}\\}\subset W.\end{split}$

We select points $Y^{\pm}$, with $f^{\ell}(Y^{-})=Y^{+}$ for some $\ell>0$, in
the orbit of the tangency point $Y$ such that

(4.4) $\begin{split}&Y^{-}=(0^{m-m_{s}},0^{m_{s}},y^{-},v^{-})\in
W^{u}_{\mathrm{loc}}(P,f),\\\ &Y^{+}=(u^{+},x^{+},0^{n_{u}},0^{n-n_{u}})\in
W^{s}_{\mathrm{loc}}(P,f),\\\ &\\{f(Y^{-}),\dots,f^{\ell-1}(Y^{-})\\}\cap
W=\emptyset.\end{split}$

This can be obtained by shrinking $W$.

To state the remaining conditions (C1)–(C5), recall that if $m>m_{s}$ there
exists the strong stable foliation $\mathcal{F}^{ss}$ of dimension $m-m_{s}$
defined on $W^{s}(P,f)$. This is the unique $f$ invariant foliation of
dimension $m-m_{s}$ having the strong stable manifold $W^{ss}(P,f)$ of $P$
(tangent to $E^{ss}(P)$) as a leaf. Similarly, if $n>n_{u}$ there is the
strong unstable foliation $\mathcal{F}^{uu}$ of dimension $n-n_{u}$ defined on
$W^{u}(P,f)$. In the coordinates above, the (local) invariant foliations
$\mathcal{F}^{ss}$ on $W^{s}_{\mathrm{loc}}(P,f)$ and $\mathcal{F}^{uu}$ on
$W^{u}_{\mathrm{loc}}(P,f)$ are also straightened and have, respectively, the
form

$\begin{split}&\\{(u,x,y,v)\colon
x=\mathrm{const.},\,y=0^{n_{u}},\,v=0^{n-n_{u}}\\},\\\ &\\{(u,x,y,v)\colon
u=0^{m-m_{s}},\,x=0^{m_{s}},\,y=\mathrm{const.}\\}.\end{split}$

Let $E^{cu}(P)$ be the $Df$-invariant space of dimension $m_{s}+n$
corresponding to the eigenspaces associated to the eigenvalues of modulus
greater or equal than the stable leading multiplier of $P$. Following [52] and
based on [31], an extended center unstable manifold $W^{cu}(P,f)$ of $P$ is an
invariant manifold of dimension $m_{s}+n$ tangent to $E^{cu}(P)$ at $P$
containing $W^{u}(P,f)$. This manifold always exists, but in general, it is
not uniquely defined. Similarly, we can define the extended center stable
manifold $W^{cs}(P,f)$ of $P$ of dimension $m+n_{u}$ tangent to $E^{cu}(P)$ at
$P$ and contains $W^{u}(P,f)$.

Conditions (C1)–(C5) are similar to the ones in [27, 29], translating the
quasi-transverse conditions in [38] to our higher-dimensional context. For
that, consider the subspace

(4.5) $E_{Y}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}T_{Y}W^{s}(P,f)\cap T_{Y}W^{u}(P,f).$

The required conditions are:

  * (C1)

$\dim(E_{Y})=(m+n)-\dim(T_{Y}W^{s}(P,f)+T_{Y}W^{u}(P,f))=1,$

  * (C2)

$E_{Y}$ is not contained in $T_{Y}\mathcal{F}^{ss}(Y)$, where
$\mathcal{F}^{ss}(W)$ denote the leaf of $\mathcal{F}^{ss}$ through $W$,

  * (C3)

the contact between the stable and unstable manifolds at the tangency point
$Y$ is quadratic,

  * (C4)

$W^{cu}(P)$ is transverse to $\mathcal{F}^{ss}(Y^{+})$ at $Y^{+}$, and

  * (C5)

$W^{cs}(P)$ is transverse to $\mathcal{F}^{uu}(Y^{-})$ at $Y^{-}$.

Conditions (C4) and (C5) are intrinsic as they do not depend on the choice of
the extended center manifolds. Moreover, when $n=1$ (and hence $n_{u}=1$)
condition (C5) is vacuous. Finally, our conditions are $C^{r}$ generic.

######  Remark 4.1.

Consider the $f$-invariant compact set $\Lambda$ formed by the orbits of the
saddle $P$ and the tangency $Y$. Under conditions (C1) – (C4), the set
$\Lambda$ has a dominated splitting of the form

(4.6) $T_{\Lambda}M=E^{ss}\oplus
E^{cu}\quad\mbox{with}\quad\dim(E^{cu})=m_{s}+n,$

where $E^{ss}$ is uniformly contracting. We consider cone fields

(4.7) $X\in U\to\mathcal{C}^{ss}(X)\qquad\mbox{and}\qquad X\in
U\to\mathcal{C}^{cu}(X)$

defined on a small neighbourhood $U$ of $\Lambda$ such that
$E^{ss}\subset\mathcal{C}^{ss}$ and $E^{cu}\subset\mathcal{C}^{cu}$ as in
Proposition 9.1. The set $U$ is called a partially hyperbolic neighbourhood of
$\Lambda$.

A similar comment holds when (C4) is replaced by (C5).

######  Remark 4.2.

The strong stable foliation $\mathcal{F}^{ss}$ of $W^{s}(P,f)$ can be smoothly
extended to a $C^{r}$ foliation $\widetilde{\mathcal{F}}^{ss}$ defined on a
neighbourhood of $\Lambda$ (see [52]). In particular, after shrinking the
partially hyperbolic neighbourhood $U$, we can assume that the leaves
$\widetilde{\mathcal{F}}^{ss}(X)$, $X\in U$, are tangent to the cone-field
$\mathcal{C}^{ss}(X)$ and contain disks of uniform size. Finally, the
restriction of this foliation to the set $\Lambda$ is invariant.

Throughout this section, we assume that the saddle $P$ is of type
$(m_{s},n_{u})$ and that $P$ and its homoclinic tangency $Y$ satisfy
conditions (C0)–(C5).

###  4.2. Return maps

We take small closed neighbourhoods $\Pi^{\pm}\subset W$ of $Y^{\pm}$ (called
reference boxes) such that
$f(\Pi^{+})\cap\Pi^{+}=\emptyset=f^{-1}(\Pi^{-})\cap\Pi^{-}$. Following [28,
Corollary 1] (see also [27, 29]), under conditions (C1)–(C5), the Taylor
expansion of the transition map $\mathfrak{T}_{1}$ from $Y^{-}$ to $Y^{+}$ can
be written as follows:

$\mathfrak{T}_{1}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}f^{\ell}|_{\Pi^{-}}\colon\Pi^{-}\to\Pi^{+},\qquad\mathfrak{T}_{1}(u,x,y,v)=(\bar{u},\bar{x},\bar{y},\bar{v})$

where

(4.8)
$\begin{split}\bar{u}-u^{+}&=A_{1}\,u+B_{1}\,x+C_{1}\,(y-y^{-})+D_{1}\,\bar{v}+\dots,\\\
\bar{x}-x^{+}&=A_{2}\,u+B_{2}\,x+C_{2}\,(y-y^{-})+D_{2}\,\bar{v}+\dots,\\\
\bar{y}&=A_{3}\,u+B_{3}\,x+C_{3}\,(y-y^{-})^{2}+D_{3}\,\bar{v}+\dots,\\\
v-v^{-}&=A_{4}\,u+B_{4}\,x+C_{4}\,(y-y^{-})+D_{4}\,\bar{v}+\dots,\end{split}$

where the notation $(\dots)$ stands for higher order terms and
$A_{i},B_{i},C_{i},D_{i}$ are matrices of the appropriate dimensions. Here
$B_{3}$, $C_{2}$ and $C_{3}$ are nonzero matrices.

To study the returns associated to $\mathfrak{T}_{1}$, consider for
sufficiently large $k$, say $k>k_{0}$, the strips

(4.9) $\begin{split}\Pi_{k}^{+}&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathfrak{T}_{0}^{-k}(\Pi^{-})\cap\Pi^{+}\\\
&=\\{x\in\Pi^{+}\colon\mathfrak{T}_{0}^{i}(x)\in W\mbox{ for all $i=0,\dots,k$
and $\mathfrak{T}_{0}^{k}(x)\in\Pi^{-}$}\\},\\\
\Pi_{k}^{-}&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathfrak{T}_{0}^{k}(\Pi_{k}^{+})\subset\Pi^{-}.\end{split}$

The return map at the tangency point $Y^{+}$ is defined in a subset
${\Sigma}^{+}\subset\Pi^{+}$ by

(4.10)
$\mathfrak{R}\colon{\Sigma}^{+}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\bigcup_{k>k_{0}}\Pi^{+}_{k}\to\Pi^{+},\quad\mathfrak{R}|_{\Pi^{+}_{k}}=\mathfrak{R}_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathfrak{T}_{1}\circ\mathfrak{T}_{0}^{k}.$

We start analysing suitable “first return maps” defined on a neighbourhood of
the homoclinic tangency $Y^{+}$.

###  4.3. Expanding returns along center unstable cone fields

We now obtain“expansion of volume” properties (along the leading directions)
derived from the expanding leading Jacobian of the saddle $P$, see Section
4.3.1. Additionally, in the codimension one case, we get expansion of suitable
diameters, see Section 4.3.2.

####  4.3.1. Expansion of volume

Next lemma about expansion of volume of local submanifolds tangent to the cone
field $\mathcal{C}^{cu}$ in Remark 4.1 by the return maps $\mathfrak{R}_{k}$
is a variation of [35, Lemma 5]. To state it recall the definition of
$J_{P}=J_{P}(f)$ in (1.2) and that a submanifold $\mathcal{S}$ is tangent to
$\mathcal{C}^{cu}$ if $T_{X}\mathcal{S}\subset\mathcal{C}^{cu}(X)$ for every
$X\in\mathcal{S}$.

######  Lemma 4.3.

There is a constant $L>0$ such that for every sufficiently large $k$ and every
submanifold $\mathcal{S}\subset{\Pi}_{k}^{+}$ of dimension $m_{s}+n$ tangent
to $\mathcal{C}^{cu}$ it holds

$\operatorname{vol}(\,{\mathfrak{R}}_{k}(\mathcal{S})\,)>L\,J_{P}^{k}\,\operatorname{vol}(\mathcal{S}),$

where $\operatorname{vol}(\cdot)$ stands for the $(m_{s}+n)$-dimensional
volume.

###### Proof.

We use the linearising coordinates $(u,x,y,v)$ of points on the set $W$ in
(4.1). For points $X$ close to $Y$ the cone $\mathcal{C}^{cu}(X)$ is an open
Grassmannian neighbourhood of the subspace
$\\{0^{m-m_{s}}\\}\times\mathbb{R}^{m_{s}}\times\mathbb{R}^{n}$. We can assume
that $\mathcal{C}^{cu}(X)$ does not depend on the point $X$, and hence it is
constant in a neighbourhood of $Y$. Thus, every $(m_{s}+n)$-dimensional
$C^{1}$ manifold $\mathcal{S}\subset{\Pi}_{k}^{+}$ tangent to
$\mathcal{C}^{cu}$ can be seen as the graph of a $C^{1}$ map in the variables
$(x,y,v)$. Moreover, we have that555The notation $f\asymp g$ means that there
are constants $C,D>0$ such that $C\,|g|<|f|<D\,|g|$.

$\operatorname{vol}(\mathcal{S})\asymp\operatorname{vol}(\,\pi(\mathcal{S})\,)$

where $\pi$ is the canonical projection from $W$ on $\\{u=0^{m-m_{s}}\\}$. For
simplicity, we identify $\\{u=0^{m-m_{s}}\\}$ with
$\mathbb{R}^{m_{s}}\times\mathbb{R}^{n_{u}}\times\mathbb{R}^{n-n_{u}}$ and
write $\pi(u,x,y,v)=(x,y,v)$.

Since, by construction, $\mathcal{C}^{cu}$ is $D\mathfrak{R}_{k}$-invariant,
we have that $\mathfrak{R}_{k}(\mathcal{S})$ is also tangent to
$\mathcal{C}^{cu}$. Thus, we also have that

$\operatorname{vol}(\,\mathfrak{R}_{k}(\mathcal{S})\,)\asymp\operatorname{vol}(\,\pi(\mathfrak{R}_{k}(\mathcal{S}))\,).$

Thus, to prove the lemma it is enough to get a constant $\widetilde{K}>0$
(independent of large $k$) such that

(4.11)
$\operatorname{vol}(\,\pi(\mathfrak{R}_{k}(\mathcal{S}))\,)\geqslant\widetilde{K}\,J_{P}^{k}\,\operatorname{vol}(\,\pi(\mathcal{S})\,).$

As $\pi|_{\mathcal{S}}\colon\mathcal{S}\to\pi(\mathcal{S})$ is a $C^{1}$
diffeomorphism, the map

$\widehat{\mathfrak{R}}_{k}\colon\pi(\mathcal{S})\to\pi(\mathfrak{R}_{k}(\mathcal{S})),\quad\widehat{\mathfrak{R}}_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\pi\circ\mathfrak{R}_{k}\circ(\pi|_{\pi(\mathcal{S})})^{-1}$

is a $C^{1}$ diffeomorphism satisfying

(4.12)
$\pi\circ\mathfrak{R}_{k}=\widehat{\mathfrak{R}}_{k}\circ\pi\quad\mbox{on}\quad\mathcal{S}.$

Thus, we have that

$\operatorname{vol}(\,\pi(\mathfrak{R}_{k}(\mathcal{S})\,)=\operatorname{vol}(\,\widehat{\mathfrak{R}}_{k}(\pi(\mathcal{S}))\,)=\int_{\pi(\mathcal{S})}\left|\det\frac{\partial\widehat{\mathfrak{R}}_{k}}{\partial(x,y,v)}\right|\,dx\,dy\,dv.$

To obtain the inequality (4.11) it is enough to get a constant
$\widetilde{K}>0$ (independent of $k$) such that

(4.13)
$\left|\det\frac{\partial\widehat{\mathfrak{R}}_{k}}{\partial(x,y,v)}\bigg{|}_{\pi(\mathcal{S})}\right|\geqslant\widetilde{K}\,J_{P}^{k}.$

To get the inequality in (4.13), recall that
$\mathfrak{R}_{k}=\mathfrak{T}_{1}\circ\mathfrak{T}_{0}^{k}$, where

$\begin{split}(u,x,y,v)\in\Pi_{k}^{+}\,\,&\mapsto\,\,\mathfrak{T}_{0}^{k}(u,x,y,v)=(u_{k},x_{k},y_{k},v_{k})\in\Pi^{-},\\\
(u_{k},x_{k},y_{k},v_{k})\in\Pi^{-}\,\,&\mapsto\,\,\mathfrak{T}_{1}(u_{k},x_{k},y_{k},v_{k})=(\bar{u}_{k},\bar{x}_{k},\bar{y}_{k},\bar{v}_{k})\in\Pi^{+}.\end{split}$

are given by the Equations (4.2) and (4.8). In view of Equation (4.12), we
have that
$\widehat{\mathfrak{R}}_{k}=(\pi\mathfrak{T}_{1}\pi^{-1})\circ(\pi\mathfrak{T}_{0}^{k}\pi^{-1})$
(here the maps $\pi^{-1}$ are local inverses in the corresponding
submanifolds) and thus

(4.14)
$\frac{\partial\widehat{\mathfrak{R}}_{k}}{\partial(x,y,v)}\bigg{|}_{\pi(\mathcal{S})}=\frac{\partial(\bar{x}_{k},\bar{y}_{k},\bar{v}_{k})}{\partial(x_{k},y_{k},v_{k})}\bigg{|}_{\pi\mathfrak{T}_{0}^{k}(\mathcal{S})}\,\,\frac{\partial(x_{k},y_{k},v_{k})}{\partial(x,y,v)}\bigg{|}_{\pi(\mathcal{S})}.$

Since, see (4.2),

$x_{k}=B^{k}x,\quad y_{k}=C^{k}y,\quad\text{and}\quad v_{k}=D^{k}v,$

where $B$, $C$, and $D$ are square matrices, whose eigenvalues in absolute
value are all equal to $\lambda$, $\gamma$, and strictly greater than
$\gamma$, respectively. Thus, as

$|\det D|>\gamma^{n-n_{u}}>1$

it follows

(4.15)
$\left|\det\frac{\partial(x_{k},y_{k},v_{k})}{\partial(x,y,v)}\bigg{|}_{\\{u=0^{m-m_{s}}\\}}\right|=J_{P}^{k}\,|\det
D^{k}|\geqslant J_{P}^{k}.$

######  Claim 4.4.

There is $\widetilde{K}>0$ such that for every $k$ large enough it holds

$\left|\det\frac{\partial(\bar{x}_{k},\bar{y}_{k},\bar{v}_{k})}{\partial(x_{k},y_{k},v_{k})}\bigg{|}_{\pi(\mathfrak{T}_{0}^{k}(\mathcal{S}))}\right|>\widetilde{K}.$

###### Proof.

First, observe that $\det D\mathfrak{T}_{1}|_{\\{u=0^{m-m_{s}\\}}}(Y^{-})\neq
0$. Otherwise, the tangent space to $W^{cu}(P)$ at $Y^{+}$ (which is the image
of $\\{u=0^{m-m_{s}}\\}$ by $D\mathfrak{T}_{1}$) cannot be transverse to
$\mathcal{F}^{ss}(Y^{+})$ at $Y^{+}$, contradicting assumption (C4). In
particular, $\det D\mathfrak{T}_{1}|_{\\{u=0^{m-m_{s}}\\}}\not=0$ in a
neighbourhood $V$ of the tangency point $Y^{-}$. Note that
$\mathcal{S}\subset\Pi^{+}_{k}$ and
$\mathfrak{T}_{0}^{k}(\Pi^{+}_{k})\subset\Pi^{-}$ converges to
$\\{u=0^{m-m_{s}}\\}$ as $k\to\infty$. Taking, if necessary, $\Pi^{-}$ small
and $k$ large, we can guarantee that $\mathfrak{T}_{0}^{k}(\Pi^{+}_{k})\subset
V$. Then

$\det\frac{\partial(\bar{x}_{k},\bar{y}_{k},\bar{v}_{k})}{\partial(x_{k},y_{k},v_{k})}\bigg{|}_{\pi(\mathfrak{T}_{0}^{k}(\mathcal{S}))}\not=0.$

This concludes the proof. ∎

Equation (4.13) (and the lemma) follows from (4.14), (4.15), and Claim 4.4. ∎

####  4.3.2. Expansion of diameters

We now restrict our attention to the case when the saddle $P$ of $f$ has
$u$-index one (in particular, $n=n_{u}=1$). Note that in this case, the
variables $v,\bar{v}$ in Equation (4.1) do not appear.

Using the linearising coordinates in (4.1), consider the natural projections

(4.16) $\pi\colon
W\to\mathbb{R}^{m_{s}+1},\quad\pi_{c}\colon\mathbb{R}^{m_{s}+1}\to\mathbb{R}^{m_{s}},\quad\pi_{u}\colon\mathbb{R}^{m_{s}+1}\to\mathbb{R}.$

Given a subset $A\subset(\\{0^{m-m_{s}}\\}\times\mathbb{R}^{m_{s}+1})\cap W$
we define

(4.17) $\operatorname{diam}_{i}(A)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathrm{diameter}(\pi_{i}(\pi(A)))\qquad i=c,\,u.$

######  Lemma 4.5.

There is a constant $K>0$, depending only on the transition
$\mathfrak{T}_{1}$, such that for every sufficiently large $k$ and every local
submanifold $\mathcal{S}\subset{\Pi}_{k}^{+}$ of dimension $m_{s}+n$ tangent
to $\mathcal{C}^{cu}$ it holds

$\operatorname{diam}_{c}(\mathfrak{R}_{k}(\mathcal{S}))<K\,\gamma^{k}\,\operatorname{diam}_{u}(\mathcal{S}),$

where $\gamma>1$ is the unstable leading eigenvalue of $P$.

###### Proof.

Consider any $\mathcal{S}$ as in the lemma and a point
$X=(u,x,y)\in\mathcal{S}\subset{\Pi}_{k}^{+}$. Let

$X_{k}=(u_{k},x_{k},y_{k})\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathfrak{T}_{0}^{k}(X)\in\Pi^{-}\quad\widetilde{X}_{k}=\mathfrak{T}_{1}(\mathfrak{T}_{0}^{k}(X))=\mathfrak{R}_{k}(X)\in\Pi^{+}.$

From (4.2), the coordinates $u_{k}$, $x_{k}$ are at most of order of
$\lambda^{k}$, where $\lambda<1$ is the leading stable eigenvalue of $P$ in
(1.1). Thus, by the formula of $\mathfrak{T}_{1}$ in (4.8) (where now the
coordinate $v$ does not appear) the $x$-coordinate of $\widetilde{X}_{k}$ is
of the form

(4.18) $x^{+}+A_{2}u_{k}+B_{2}x_{k}+C_{2}(y_{k}-y^{-})+\dots,$

where $x^{+}$ is the $x$-coordinate of $Y^{+}$, $y^{-}$ is the $y$-coordinate
of $Y^{-}$, see Equation 4.4. Note that Equation (4.18) implies that there is
a constant $K>0$, independent of $\mathcal{S}$ and large $k$, such that

(4.19)
$\operatorname{diam}_{c}(\mathfrak{R}_{k}(\mathcal{S}))<K\,\operatorname{diam}_{u}(\mathfrak{T}_{0}^{k}(\mathcal{S}))=K\,\gamma^{k}\,\operatorname{diam}_{u}(\mathcal{S}),$

where the last equality follows from the linearising hypothesis. This
completes the proof of the lemma. ∎

######  Remark 4.6 (Uniform estimates).

The constants $K$ and $L$ in Lemmas 4.3 and 4.5 can be chosen uniform for
perturbations. To see why this is so just note that the estimates in the
proofs of these lemmas only depend on the center unstable cone field (which
persists by perturbations).

###  4.4. Return maps at the homoclinic tangency: the codimension one case

We now refine the constructions in Section 4.2 in the codimension one case,
when the saddle $P$ has $u$-index one and hence $n=n_{u}=1$. Recall the
choices of the tangency points $Y^{-}$ and $\mathfrak{T}_{1}(Y^{-})=Y^{+}$ in
(4.4), the reference boxes $\Pi^{\pm}$, and the strips $\Pi_{k}^{\pm}$ in
(4.9).

Assuming that the saddle $P$ is biaccumulated by its transverse homoclinic
points (recall the definition in the preamble of Section 3) we will “resize”
the sets $\Pi^{+}_{k}$ to get additional dynamical control. We now go to the
details. Our construction is inspired by [28, Section 1.4] (see also previous
constructions in [39, 40]).

######  Lemma 4.7.

Consider $f\in\mathrm{Diff}^{r}(M^{m+1})$ with a saddle $P$ of $u$-index one
that is biaccumulated by its transverse homoclinic points and has a homoclinic
tangency. Assume that conditions (C0) – (C4) hold. Then the reference box
$\Pi^{-}$ can be chosen such that there are sequences of points $(N^{-}_{j})$
and $(N^{+}_{j})$ and of $m$-dimensional submanifolds $(\Theta_{j})$ with

$N^{\pm}_{j}\in W^{s}(P,f)\pitchfork W^{u}_{{{\mathrm{loc}}}}(P,f),\quad
N^{\pm}_{j}\in\Theta_{j}\subset W^{s}(P,f)\cap\Pi^{-},$

such that:

  * •

the open “interval” $(N^{-}_{j},N^{+}_{j})\subset W^{u}_{\mathrm{loc}}(P,f)$
contains $Y^{-}$ and satisfies

$\delta_{j}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\operatorname{diam}([N^{-}_{j},N^{+}_{j}])\to 0\quad\mbox{as}\quad
j\to\infty,$

  * •

each submanifold $\Theta_{j}$ splits the set $\Pi^{-}$ into two connected
components such that the component $C_{j}$ of $(\Pi^{-}\setminus\Theta_{j})$
containing $Y^{-}$ satisfies $C_{j+1}\subset C_{j}$.

###### Proof.

The codimension one hypothesis implies that $W^{s}_{{{\mathrm{loc}}}}(P,f)$
splits the box $\Pi^{+}$ into two connected components. So the same holds for
$\Pi^{-}$ and $\mathfrak{T}_{1}^{-1}(W^{s}_{{{\mathrm{loc}}}}(P,f))$. By the
biaccumulation property, both components contain disks of $W^{s}(P,f)$
accumulating to $\mathfrak{T}^{-1}(W^{s}_{{{\mathrm{loc}}}}(P,f))$. The disks
in one of these components also have two transverse intersections with
$W^{u}_{\mathrm{loc}}(P,f)$ accumulating to $Y^{-}$. This provides the
sequences of submanifolds $\Theta_{j}$ and of transverse homoclinic points
$N_{j}^{\pm}$. ∎

Fix large $j$ and recall the definition of $\Pi_{k}^{\pm}$ in (4.9). For every
$k$ large enough, the set $\Pi^{-}_{k}\setminus\Theta_{j}$ has three connected
components. Denote by ${\Pi}^{-}_{j,k}$ the closure of the connected component
intersecting $C_{j}$ and let

(4.20) $\Pi_{j,k}^{+}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathfrak{T}_{0}^{-k}(\Pi^{-}_{j,k})\cap\Pi^{+}\subset\Pi^{+}_{k}.$

Define the $s$\-  and $u$-boundaries of ${\Pi}_{j,k}^{+}$ by

(4.21) $\partial^{s}\Pi^{+}_{j,k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathfrak{T}_{0}^{-k}(\Theta_{j}\cap\Pi_{k}^{-})\cap\Pi^{+}_{k}\qquad\partial^{u}\Pi^{+}_{j,k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\overline{\partial\Pi^{+}_{j,k}\setminus\partial^{s}\Pi^{+}_{j,k}}.$

Observe that, by construction, $\partial^{s}\Pi^{+}_{j,k}\subset W^{s}(P,f)$.

Recall the definitions of $\gamma>1$ in (1.1) and of the $c$\- and
$u$-diameters in (4.17). Next claim asserts that $c$-diameters of
$\Pi_{k}^{+}$ are “independent” of $k$ while $u$-diameters exponentially
shrink. The proof is straightforward and hence omitted.

######  Claim 4.8 ($c$\- and $u$-diameters of $\Pi_{j,k}^{+}$).

There is $\rho>0$ such that for every sufficiently large $j$ there is
$k_{0}(j)$ such that for every $k\geqslant k_{0}(j)$ it holds

$\operatorname{diam}_{c}(\Pi^{+}_{j,k})>\rho\quad\mbox{and}\quad\operatorname{diam}_{u}({\Pi}^{+}_{j,k})<\gamma^{-k}\,\delta_{j}.$

######  Remark 4.9 (Choice of quantifiers).

Recall the constants $K$, $\delta_{j}$, and $\rho$ in Lemmas 4.5 and 4.7 and
Claim 4.8, respectively. There is large $j_{*}\geqslant 1$ such that

$K\delta_{j_{*}}<\frac{\rho}{10}.$

As in what follows $j_{*}$ remains fixed, for notational simplicity, we write

(4.22) $\widetilde{\Pi}^{\pm}_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\Pi^{\pm}_{j_{*},k},\quad\mbox{where}\quad k>k_{0}(j_{*}).$

##  5\. Unfolding the homoclinic tangency

Throughout this section, we consider $f\in\mathrm{Diff}^{r}(M^{m+n})$,
$r,\,m\geqslant 2$ and $n\geqslant 1$, with a simple saddle $P$ of type
$(m_{s},n_{u})$ and $u$-index $n$ having a homoclinic tangency $Y$ satisfying
(C0)–(C5). We will study how the unfolding of this tangency generates saddles
with $u$-indices different from the one of $P$.

###  5.1. Unfolding families and return maps

We embed the diffeomorphism $f$ into a family $(f_{\mathbf{t}})$, with
$f_{\mathbf{0}}=f$, unfolding the homoclinic tangency. For that, we borrow the
construction from [28, Section 1.2]. The number of parameters of the family
$(f_{\mathbf{t}})$ depends on the type of the saddle $P$ (which is fixed for
every $f_{\mathbf{t}}$). We use the notation in Section 4.2. Note that
transition map
$\mathfrak{T}_{1,\mathbf{t}}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}f_{\mathbf{t}}^{\ell}$, see (4.8), is defined for every small
$\mathbf{t}$. Recall that $\lambda$ and $\gamma$ defined as in (1.1) are the
modulus of the contracting and expanding leading multipliers of $P$.

Using the effective dimension, denoted by $\mathrm{d}_{\mathrm{e}}$,
introduced in [52], there are three classes of simples saddles with leading
Jacobian greater than one:

  * •

$P$ has $\mathrm{d}_{\mathrm{e}}=1$ if it is of type $(1,1)$ or $(1,2)$ and
$\lambda\gamma>1$;

  * •

$P$ has $\mathrm{d}_{\mathrm{e}}=2$ if it is of type $(1,2)$ with
$\lambda\gamma<1$ or of type $(2,1)$ or $(2,2)$ with $\lambda^{2}\gamma>1$;

  * •

$P$ has $\mathrm{d}_{\mathrm{e}}=3$ if it is of type $(2,2)$ and
$\lambda^{2}\gamma<1$.

The effective dimension is the number of parameters considered in the
unfolding of the tangency. Let us roughly explain the choice of these
parameters. When $\mathrm{d}_{\mathrm{e}}=3$ the parameter
$\mathbf{t}=(t,\alpha,\beta)\in[-\varepsilon,\varepsilon]^{3}$ (in cases
$\mathrm{d}_{\mathrm{e}}=2$ and $\mathrm{d}_{\mathrm{e}}=1$ we consider
$\mathbf{t}=(t,\alpha)\in[-\varepsilon,\varepsilon]^{2}$ and
$\mathbf{t}=t\in[-\varepsilon,\varepsilon]$, respectively) is defined as
follows. Let666In this description, we omit the neighbourhood in the
definition of the local manifold.
$\omega^{u}_{\mathbf{t}}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}W^{u}_{\mathrm{loc}}(P,f_{\mathbf{t}})\cap\Pi^{-}$, then the
distance between the folding point of
$\mathfrak{T}_{1,\mathbf{t}}(\omega^{u}_{\mathbf{t}})$ and
$W^{s}_{\mathrm{loc}}(P,f_{\mathbf{t}})$ is $|t|$. To describe $\alpha$ note
that in this case the leading multipliers of $P$ for $f$ are nonreal and equal
to $\lambda\,e^{\pm i\theta}$ and $\gamma\,e^{\pm i\rho}$. The leading
multipliers of $P$ for $f_{\mathbf{t}}$ are $\lambda\,e^{\pm
i(\theta+\alpha)}$ and $\gamma\,e^{\pm i(\rho+\beta)}$. We let
$\mathfrak{T}_{0,\mathbf{t}}=f_{\mathbf{t}}|_{W}$ (here $W$ is the linearising
neighbourhood in Section 4.1) and note that $\mathfrak{T}_{0,\mathbf{t}}$ is a
“double rotation” of $\mathfrak{T}_{0}$ by angles $\alpha$ and $\beta$.

Note that in the previous construction $P_{f_{\mathbf{t}}}=P$ and that the
leading Jacobian of $P$ is not modified:

(5.1) $J_{P}(f_{\mathbf{t}})=J_{P}(f).$

Recall the construction of the return maps
$\mathfrak{R}_{k}=\mathfrak{T}_{1}\circ\mathfrak{T}_{0}^{k}$ in Section 4.2.
We now consider return maps
$\mathfrak{R}_{k,\mathbf{t}}=\mathfrak{T}_{1,\mathbf{t}}\circ\mathfrak{T}_{0,\mathbf{t}}^{k}$
similarly defined. For large $k>0$, as in (4.9), we consider the strips

$\Pi_{k,\mathbf{t}}^{+}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathfrak{T}_{0,\mathbf{t}}^{-k}(\Pi^{-})\cap\Pi^{+}.$

We now consider, for $k_{0}$ large enough, the corresponding return maps

(5.2)
$\mathfrak{R}_{\mathbf{t}}\colon\Sigma^{+}_{\mathbf{t}}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\bigcup_{k\geqslant
k_{0}}\Pi^{+}_{k,\mathbf{t}}\to\Pi^{+},\quad\mathfrak{R}_{\mathbf{t}}|_{\Pi^{+}_{k,\mathbf{t}}}=\mathfrak{R}_{k,\mathbf{t}}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathfrak{T}_{1,\mathbf{t}}\circ\mathfrak{T}_{0,\mathbf{t}}^{k}.$

A saddle of $f_{\mathbf{t}}$ is said of single round type if it is a fixed
point of $\mathfrak{R}_{\mathbf{t}}$.

######  Remark 5.1 (The codimension one case).

Recall the definition of $\widetilde{\Pi}^{+}_{k}=\Pi^{+}_{j_{*},k}$ in (4.20)
and (4.22). Observe that the $s$-boundary of $\widetilde{\Pi}^{+}_{k}$ is not
necessarily contained in the stable manifold of $P$ for $f_{\mathbf{t}}$ when
$\bf{t}\not=0$. However, the stable manifold of $P$ on $\Pi^{+}$ varies
smoothly with $\mathbf{t}$, for $|\mathbf{t}|$ small enough. Then we can
define continuations of
$\widetilde{\Pi}^{+}_{k,{\mathbf{t}}}\subset\Pi^{+}_{k,{\mathbf{t}}}$ for
small $|\mathbf{t}|$ in such a way the $s$-boundaries are still contained
$W^{s}(P,f_{\mathbf{t}})$ and the estimates for the $c$\- and $u$-diameters in
(4.8) hold.

###  5.2. Index variation

Our next goal is to prove the following proposition, which is essentially
borrowed from [47, 28]. For the next proposition see the cone fields
$\mathcal{C}^{cu}$ in Proposition 9.1 and recall the definition of the leading
Jacobian in (1.2).

######  Proposition 5.2.

Consider $f\in\mathrm{Diff}^{r}(M^{m+n})$, $r,m\geqslant 2$ and $n\geqslant
1$, with a simple saddle $P$ of $u$-index $n$ such that $J_{P}=J_{P}(f)>1.$
Assume that $P$ has a homoclinic tangency satisfying conditions (C0) – (C5).

Let $(f_{\mathbf{t}})$ be the unfolding family associated to $f$ in Section
5.1. Then there is a sequence of open sets of parameters
$\Delta_{k}\to\mathbf{0}$ such that for every ${\mathbf{t}}\in\Delta_{k}$
there are a contracting locally normally hyperbolic $C^{r}$ submanifold
$\mathcal{M}_{k,{\mathbf{t}}}$ of dimension $n+\mathrm{d}_{\mathrm{e}}$
tangent to the cone field $\mathcal{C}^{cu}$ satisfying:

  * (i)

If $\mathrm{d}_{\mathrm{e}}=1$: there is a saddle
$R_{k,\mathbf{t}}\in\mathcal{M}_{k,\mathbf{t}}$ of $u$-index $n+1$.

  * (ii)

If $\mathrm{d}_{\mathrm{e}}=2$:

    * $\bullet$

when $P$ is of type $(1,2)$ and $\lambda\gamma>1$ there are saddles
$R_{k,\mathbf{t}},\,S_{k,\mathbf{t}}\in\mathcal{M}_{k,\mathbf{t}}$ of
$u$-indices $n-1$ and $n+1$, respectively,

    * $\bullet$

when $P$ is of type $(2,1)$ or $(2,2)$ and $\lambda^{2}\gamma>1$ there are
saddles $R_{k,\mathbf{t}},\,S_{k,\mathbf{t}}\in\mathcal{M}_{k,\mathbf{t}}$ of
$u$-indices $n+1$ and $n+2$, respectively,

  * (iii)

If $\mathrm{d}_{\mathrm{e}}=3$: there are saddles
$Q_{k,\mathbf{t}},\,R_{k,\mathbf{t}},\,S_{k,\mathbf{t}}\in\mathcal{M}_{k,\mathbf{t}}$
of $u$-indices $n-1$, $n+1$ and $n+2$, respectively.

Moreover, these saddles are of single round type and have local unstable
manifolds contained in $\mathcal{M}_{k,\mathbf{t}}$.

######  Remark 5.3.

The submanifold $\mathcal{M}_{k,\mathbf{t}}$ is contained in the set
$\Pi^{+}_{k,{\mathbf{t}}}$ in (4.9) for $k$ large enough and
${\mathbf{t}}\in\Delta_{k}$. In the codimension one case, one gets that for
$\mathcal{M}_{k,{\mathbf{t}}}\subset\widetilde{\Pi}^{+}_{k,\mathbf{t}}$ for
every large $k$ and $\mathbf{t}\in\Delta_{k}$.

###### Proof.

By [28, Theorem. 3, Equation (1.2) and Lemma 2] applied to the unfolding
family $(f^{-1}_{{\mathbf{t}}})$ (see also [47, Theorems A and C]), for
sufficiently large $k$, there is a set $\Delta_{k}$ of parameters
${\mathbf{t}}$, with $\Delta_{k}\to\mathbf{0}$ as $k\to\infty$, such that the
return maps $\mathfrak{R}_{k,{\mathbf{t}}}$ have a saddle $R_{k,{\mathbf{t}}}$
of $u$-index $n+1$. Besides, there are also saddles $S_{k,{\mathbf{t}}}$ of
$u$-index $n+2$ (in cases (ii) and (iii)) and $Q_{k,{\mathbf{t}}}$ of
$u$-index $n-1$ (in case (iii)). These saddles are of single round type.

Moreover, for the parameters ${\mathbf{t}}\in\Delta_{k}$, there is also a
contracting normally hyperbolic $C^{r}$ manifold
$\mathcal{M}_{k,{\mathbf{t}}}\subset\Pi^{+}_{k,{\mathbf{t}}}$ (resp.
$\widetilde{\Pi}^{+}_{k,\mathbf{t}}$) which is locally $f_{{\mathbf{t}}}$
invariant, tangent to the cone field $\mathcal{C}^{cu}$, and contains local
unstable manifolds of $R_{k,{\mathbf{t}}}$, $S_{k,{\mathbf{t}}}$,
$Q_{k,{\mathbf{t}}}$ according to the cases. The dimension of
$\mathcal{M}_{k,{\mathbf{t}}}$ is $n+\mathrm{d}_{\mathrm{e}}$. This completes
the proof. ∎

In the codimension one case we can obtain additional conclusions, compare with
[35, Lemmas 11 and 12].

######  Lemma 5.4.

Under the assumption of Proposition 5.2, if the saddle $P$ has $u$-index one
and it is biaccumulated by its transverse homoclinic points, then for every
large $k$ and every ${\mathbf{t}}\in\Delta_{k}$ it holds:

  * •

$W^{u}(R_{k,{\mathbf{t}}},f_{\mathbf{t}})\pitchfork\partial^{s}\widetilde{\Pi}^{+}_{k,\mathbf{t}}\neq\emptyset$,
in cases (i) and (ii) of Proposition 5.2,

  * •

$W^{u}(S_{k,{\mathbf{t}}},f_{\mathbf{t}})\pitchfork\partial^{s}\widetilde{\Pi}^{+}_{k,\mathbf{t}}\neq\emptyset$,
in case (ii) of Proposition 5.2.

In particular, as $\partial^{s}\widetilde{\Pi}^{+}_{k,\mathbf{t}}\subset
W^{s}(P,f_{\mathbf{t}})$, it holds

$W^{u}(R_{k,\mathbf{t}},f_{\mathbf{t}})\pitchfork
W^{s}(P,f_{\mathbf{t}})\neq\emptyset\quad\text{and}\quad
W^{u}(S_{k,\mathbf{t}},f_{\mathbf{t}})\pitchfork
W^{s}(P,f_{\mathbf{t}})\neq\emptyset.$

###### Proof.

Note that $n=n_{u}=1$. Fix large $k$ and ${\mathbf{t}}\in\Delta_{k}$. Let
$\mathcal{S}$ be an $(m_{s}+1)$-dimensional submanifold contained in
$\widetilde{\Pi}^{+}_{k,\mathbf{t}}$ tangent to the cone field
$\mathcal{C}^{cu}$. This manifold is transverse to the foliation
$\widetilde{\mathcal{F}}^{ss}$ in Remark 4.2 and the angle between them is
uniformly bounded. Recall the linearising neighbourhood $W$ of $f$ and the
linearising coordinates $(u,x,y)$, see (4.1). Consider the projection
$\pi^{ss}\colon W\to\\{u=0\\}\subset W$ along the leaves of
$\widetilde{\mathcal{F}}^{ss}$. By the $C^{2}$ regularity of $f$, the
foliation $\widetilde{\mathcal{F}}^{ss}$ is absolutely continuous. Thus there
are constants $c_{1},c_{2}>0$ (independent of the submanifold $\mathcal{S}$)
such that

(5.3)
$c_{1}\,\mathrm{vol}(\mathcal{S})<\mathrm{vol}(\pi^{ss}(\mathcal{S}))<c_{2}\,\mathrm{vol}(\mathcal{S})$

where $\mathrm{vol}(\cdot)$ denotes $(m_{s}+1)$-volume.

Recall that the unfolding family $(f_{{\mathbf{t}}})$ preserves the leading
Jacobian of $P$. Using Lemma 4.3 and Remark 4.6, we get a constant $L>0$ such
that for every large $k\geqslant 1$ it holds

(5.4)
$\mathrm{vol}(\mathfrak{R}_{k,{\mathbf{t}}}(\mathcal{S}))>L\,J_{P}^{k}\,\mathrm{vol}(\mathcal{S}).$

Thus, from (5.3) and (5.4), it hold that for every $k$ large enough

(5.5)
$\mathrm{vol}(\pi^{ss}(\mathfrak{R}_{k,{\mathbf{t}}}(\mathcal{S})))>c_{1}\,c_{2}^{-1}\,L\,J_{P}^{k}\,\mathrm{vol}(\pi^{ss}(\mathcal{S}))=\varrho_{k}\,\mathrm{vol}(\pi^{ss}(\mathcal{S}))$

where $\varrho_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}c_{1}\,c_{2}^{-1}\,L\,J_{P}^{k}>1.$ As $J_{P}>1$ we get that
$\varrho_{k}>1$ for every large $k$. Arguing inductively, if
$\mathfrak{R}_{k,{\mathbf{t}}}^{i}(\mathcal{S})\subset\widetilde{\Pi}^{+}_{k,{\mathbf{t}}}$
for every $i=0,\dots,n-1$ we get

(5.6)
$\mathrm{vol}(\pi^{ss}(\mathfrak{R}^{n}_{k,{\mathbf{t}}}(\mathcal{S})))>\varrho_{k}^{n}\,\mathrm{vol}(\pi^{ss}(\mathcal{\mathcal{S}})).$

We now focus on the saddles $R_{k,{\mathbf{t}}}$. We have the following facts
about the sets $\widetilde{\Pi}_{k,{\mathbf{t}}}^{+}$. First recall Claim 4.8
about the “unstable size” of the $\widetilde{\Pi}_{k}^{+}$ and the definitions
of $\partial^{u}\widetilde{\Pi}_{k}^{+}$ and $\operatorname{diam}_{c}$, in
(4.21) and (4.17), respectively.

######  Remark 5.5.

. Let $\rho>0$ be the constant in Claim 4.8. Adjusting the “unstable sides” of
$\widetilde{\Pi}^{+}_{k,{\mathbf{t}}}$, we can assume that the saddle
$R_{k,{\mathbf{t}}}$ is $(\frac{\rho}{10})$-_centered_ for every $k$
sufficiently large: for every submanifold $\mathcal{S}$ of dimension $m_{s}+1$
tangent to $\mathcal{C}^{cu}$, containing $R_{k,{\mathbf{t}}}$, and
intersecting $\partial^{u}\widetilde{\Pi}_{k,{\mathbf{t}}}^{+}$ it holds

$\mathrm{diam}_{c}(\mathcal{S})>\frac{\rho}{10}.$

The lemma (for $R_{k,{\mathbf{t}}}$) follows immediately from the next claim:

######  Claim 5.6.

Consider a small disk $W^{u}\subset
W^{u}(R_{k,{\mathbf{t}}},f_{{\mathbf{t}}})\cap\mathcal{M}_{k,{\mathbf{t}}}$ of
dimension $(m_{s}+1)$ centered at $R_{k,{\mathbf{t}}}$. Then there is $m_{0}$
such that
$\mathfrak{R}_{k,{\mathbf{t}}}^{m_{0}}(W^{u})\pitchfork\partial^{s}\widetilde{\Pi}^{+}_{k,{\mathbf{t}}}\not=\emptyset.$

###### Proof.

By Equation (5.6), there is a first $n_{0}$ such that
$\mathfrak{R}_{k,{\mathbf{t}}}^{n_{0}+1}(W^{u})\cap\partial\widetilde{\Pi}^{+}_{k,{\mathbf{t}}}\not=\emptyset$.
If this intersection occurs in
$\partial^{s}\widetilde{\Pi}^{+}_{k,{\mathbf{t}}}$ taking $m_{0}=n_{0}+1$ we
are done. Otherwise, the intersection occurs in
$\partial^{u}\widetilde{\Pi}^{+}_{k,{\mathbf{t}}}$. We see that this gives a
contradiction. Define

$W_{0}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}C(R_{k,{\mathbf{t}}},\mathfrak{R}_{k,{\mathbf{t}}}^{n_{0}}(W^{u})\cap\widetilde{\Pi}^{+}_{k,{\mathbf{t}}}),\quad
W_{1}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathfrak{R}_{k,{\mathbf{t}}}(W_{0}),$

where $C(A,\Upsilon)$ denotes the connected component of the set $\Upsilon$
containing the point $A\in\Upsilon$. Since $W_{1}$ intersects
$\partial^{u}\widetilde{\Pi}^{+}_{k,{\mathbf{t}}}$, $R_{k,{\mathbf{t}}}\in
W_{1}$ (recall that the saddle is single round), and $R_{k,{\mathbf{t}}}$ is
$(\frac{\rho}{10})$-centered, Remark 5.5 implies that

$\operatorname{diam}_{c}(W_{1})>\frac{\rho}{10}.$

On the other hand, since $W_{0}$ is contained in
$\widetilde{\Pi}^{+}_{k,{\mathbf{t}}}$, by item (b) in Claim 4.8 and recalling
(4.22), it follows

$\operatorname{diam}_{u}(W_{0})<\gamma^{-k}\,\delta_{j_{*}}.$

Using Lemma 4.5 and Remark 4.6, the previous inequalities, and the choice of
quantifiers in Remark 4.9 we get

$\frac{\rho}{10}<\operatorname{diam}_{c}(W_{1})=\operatorname{diam}_{c}(\mathfrak{R}_{k,{\mathbf{t}}}(W_{0}))<K\,\gamma^{k}\,\operatorname{diam}_{u}(W_{0})<K\,\delta_{j*}<\frac{\rho}{10}.$

This contradiction implies the claim. ∎

The proof of Lemma 5.4 for the saddle $R_{k,{\mathbf{t}}}$ is now complete.
The proof of the lemma for the saddle $S_{k,{\mathbf{t}}}$ follows similarly
and will be omitted. ∎

##  6\. Proof of Theorems A and B

Consider $f\in\mathrm{Diff}^{r}(M^{m+n})$ with a homoclinic tangency
associated to a saddle $P$ of $u$-index $n$ and a blender $\Gamma$ as in
Theorems A or B. This means that

  * •

$P$ is simple of type $(m_{s},n_{u})$, with $m_{s},n_{u}\in\\{1,2\\}$,

  * •

$J_{P}(f)>1$, and

  * •

$P$ is homoclinically related to a blender $\Gamma$.

In Theorem A we have that $n=n_{u}=1$ and that $\Gamma$ is a $cs$-blender of
central dimension $m_{s}$. In Theorem B we have that $\Gamma$ is a double
blender of central dimensions $(m_{s},n_{u})$.

Recalling Terminology 1.8, from now on we use the following nomenclature:

######  Terminology 6.1 ($C^{r}$ perturbation).

A $C^{r}$ perturbation of $f$ is a $C^{\infty}$ diffeomorphism $g$ that can be
obtained arbitrarily $C^{r}$ close to $f$.

###  6.1. Proof of Theorem A

The main step to prove Theorem A is the following proposition which summarises
the previous constructions.

######  Proposition 6.2.

Consider $f$ with a saddle $P$ and a $cs$-blender $\Gamma$ as in Theorem A.
There is a $C^{r}$ perturbation ${g}$ of $f$, $r\geqslant 1$, having a saddle
$Q$ such that

  * (a)

$Q$ is simple of type $(m_{s},1)$ and $J_{Q}(g)>1$,

  * (b)

$Q$ is homoclinically related to the blender $\Gamma_{g}$ and to $P_{g}$,

  * (c)

$Q$ is biaccumulated by its transverse homoclinic points,

  * (d)

$Q$ has a homoclinic tangency in the superposition domain of $\Gamma_{g}$, and

  * (e)

the superposition region of the blender $\Gamma_{{g}}$ contains a local strong
stable manifold $W^{ss}_{\mathrm{loc}}(Q,{g})$.

###### Proof.

The proof of the proposition follows from Lemma 3.2 taking $P$ and
$P^{\prime}=Q^{\ast}$ a distinctive point of the blender $\Gamma$ (which are
homoclinically related to $P$) and a small neighbourhood $V$ of $Q^{\ast}$
contained in the superposition domain of $\Gamma$. Let $g$ and $Q$ be the
diffeomorphism and the saddle provided by Lemma 3.2. Note that the saddle $Q$
satisfies (a)–(b) and its orbit intersects $V$ (thus, after replacing by some
iterate, we can assume that $Q\in V$). Since we have $n=1$, Remark 3.5 implies
the biaccumulation property in (c). Moreover, by Remark 3.6, the saddle $Q$
can be taken with a homoclinic tangency, obtaining (d). Finally, item (e) also
holds provided $V$ is sufficiently small (so that $W^{s}_{\mathrm{loc}}(Q,g)$
and $W^{s}_{\mathrm{loc}}(Q^{*}_{g},g)$ are close enough) and the fact that
the disks of the superposition domain form an open set. This completes the
proof of the proposition. ∎

We are now ready to conclude the othe proof of Theorem A. Without lost of
generality, after a $C^{r}$ perturbation, we can assume that the
diffeomorphism $g$, the saddle $Q$, and its homoclinic tangency provided by
Proposition 6.2 satisfy conditions (C0)–(C4). Using Proposition 5.2, we embed
$g$ into an unfolding family $(g_{\mathbf{t}})$ (with one parameter if
$m_{s}=1$ and two parameters if $m_{s}=2$), getting open sets of parameters
$\Delta_{k}\to\mathbf{0}$, submanifolds $\mathcal{M}^{k}_{{\mathbf{t}}}$, and
saddles $R^{k}_{\mathbf{t}}$ of $u$-index two and $S^{k}_{\mathbf{t}}$ (if
$m_{s}=2$) of $u$-index three, ${\mathbf{t}}\in\Delta_{k}$, as in Proposition
5.2 and Remark 5.3. Fix $k$ large enough (so Lemma 5.4 is satisfied) and
${\mathbf{t}}\in\Delta_{k}$. We will show that there are $C^{r}$ robust
heterodimensional cycles between each saddle $R^{k}_{{\mathbf{t}}}$ and
$S^{k}_{{\mathbf{t}}}$ and the blender
$\Gamma_{{\mathbf{t}}}=\Gamma_{g_{{\mathbf{t}}}}$, see Lemmas 6.3 and 6.4
below.

######  Lemma 6.3 (Robust transverse intersections).

Let ${\mathbf{t}}\in\Delta_{k}$. The unstable manifolds
$W^{u}(R^{k}_{\mathbf{t}},g_{\mathbf{t}})$ and
$W^{u}(S^{k}_{\mathbf{t}},g_{\mathbf{t}})$ (if $m_{s}=2$) transversely
intersect the stable set $W^{s}(\Gamma_{\mathbf{t}},g_{\mathbf{t}})$.
Therefore these intersections are $C^{r}$ robust.

###### Proof.

As the $u$-index of $Q$ is one ($n=n_{u}=1$) it is either of type $(1,1)$ or
$(2,1)$. First, if $Q$ is of type $(1,1)$ then, by Lemma 5.4, for every
${\mathbf{t}}\in\Delta_{k}$ it holds

$W^{u}(R^{k}_{\mathbf{t}},g_{\mathbf{t}})\pitchfork
W^{s}(Q_{{\mathbf{t}}},g_{\mathbf{t}})\neq\emptyset.$

Second, if $Q$ is of type $(2,1)$, again by Lemma 5.4, then for every
${\mathbf{t}}\in\Delta_{k}$ it holds

$W^{u}(R^{k}_{\mathbf{t}},g_{\mathbf{t}})\pitchfork
W^{s}(Q_{{\mathbf{t}}},g_{\mathbf{t}})\neq\emptyset\quad\mbox{and}\quad
W^{u}(S^{k}_{\mathbf{t}},g_{\mathbf{t}})\pitchfork
W^{s}(Q_{{\mathbf{t}}},g_{\mathbf{t}})\neq\emptyset.$

In both cases, as $Q_{{\mathbf{t}}}$ and the blender $\Gamma_{{\mathbf{t}}}$
are homoclinically related, we get the corresponding intersections between the
unstable manifolds of the saddles $R^{k}_{{\mathbf{t}}}$ and
$S^{k}_{{\mathbf{t}}}$ and the stable set of $\Gamma_{\mathbf{t}}$. ∎

######  Lemma 6.4 (Robust quasi-transverse intersections).

Let ${\mathbf{t}}\in\Delta_{k}$. There is a $C^{r}$ neighbourhood
$\mathcal{U}$ of $g_{\mathbf{t}}$ such that for every $h\in\mathcal{U}$ the
manifolds and $W^{ss}(R^{k}_{h},h)$ and $W^{s}(S^{k}_{h},h)$ (if $m_{s}=2$)
contain disks in the superposition region of the blender $\Gamma_{h}$. As a
consequence, for every $h\in\mathcal{U}$ it holds

$W^{s}(S^{k}_{h},h)\cap
W^{u}_{\mathrm{loc}}(\Gamma_{h},h)\neq\emptyset\quad\mbox{and}\quad
W^{ss}(R^{k}_{h},h)\cap W^{u}_{\mathrm{loc}}(\Gamma_{h},h)\neq\emptyset.$

###### Proof.

By Remark 4.1, the set $\Lambda_{g}$ consisting of the orbits of $Q$ and the
tangency point $Y$ is partially hyperbolic with a splitting of the form
$E^{ss}\oplus E^{cu}$, where $\dim(E^{ss})=m-m_{s}$. Remark 4.2 provides the
foliation $\widetilde{\mathcal{F}}^{ss}_{g}$ defined on a neighbourhood of
$\Lambda_{g}$ and tangent to the cone-field $\mathcal{C}^{ss}$. The leaves of
this foliation have uniform size (depending only on the neighbourhood of $g$)
and restricted to the set $\Lambda_{g}$ is invariant. This, in particular,
implies that $\widetilde{\mathcal{F}}^{ss}_{\mathrm{loc}}(Q)$ is a local
strong stable manifold $W^{ss}_{\mathrm{loc}}(Q,g)$ of $Q$. Moreover, by item
(e) of Proposition 6.2, the set $W^{ss}_{\mathrm{loc}}(Q,g)$ is a disk in the
region of superposition $\mathscr{D}^{ss}$ of the blender $\Gamma_{g}$.

Note that for $k$ large enough the orbit of $R^{k}_{\mathbf{t}}$ can be chosen
with iterates arbitrarily close to the orbit of $Y$, in particular, close to
$Q$. Thus, after replacing by some iterate, we can assume that
$R^{k}_{\mathbf{t}}$ is close enough to $Q$ so that its local strong stable
manifold
$W^{ss}_{\mathrm{loc}}(R_{\mathbf{t}}^{k},g_{\mathbf{t}})=\widetilde{\mathcal{F}}^{ss}_{\mathrm{loc}}(R^{k}_{\mathbf{t}})$
is close to $W^{ss}_{\mathrm{loc}}(Q_{g},g)$. Since the superposition domain
$B^{ss}$ of the blender $\Gamma_{g}$ is an open set and the (local) strong
stable manifold of $R_{{\mathbf{t}}}^{k}$ varies $C^{1}$ continuously in a
neighbourhood of $g_{\mathbf{t}}$, we have that (provided that $\mathcal{U}$
is small) $W^{ss}_{\mathrm{loc}}(R_{h}^{k},h)$ is also a disk in
$\mathscr{D}^{ss}$. By item (1) in Remark 2.3 about continuations of blenders,
the superposition domain ${B}^{ss}$ is also a superposition domain of the
continuation $\Gamma_{h}$ of $\Gamma_{{\mathbf{t}}}$. The robust intersection
property of blenders (item (c) (ii) in Definition 2.1) implies that
$W^{ss}_{\mathrm{loc}}(R^{k}_{h},h)\cap
W^{u}_{\mathrm{loc}}(\Gamma_{h},h)\neq\emptyset$, proving the lemma for the
saddle $R_{h}^{k}$. The proof for the saddle $S_{\mathbf{t}}^{k}$ is analogous
and omitted. ∎

Note that if $m_{2}=2$, we get robust cycles of coindex one (associated to
$R^{k}_{h}$ and $\Gamma_{h}$) and two (associated to $S^{k}_{h}$ and
$\Gamma_{h}$). This ends the proof of Theorem A. ∎

###  6.2. Proof of Theorem B

The proof of Theorem B is similar to the one of Theorem A. The robust quasi-
transverse intersections are obtained exactly as in Theorem A (see Lemma 6.7).
However, since the biaccumulation property is not available in the general
case, the proof of the existence of transverse intersections must be
different. For that, we use the geometrical properties of double blenders (see
Lemma 6.6).

The main step to prove Theorem B is the following proposition (analogous to
Proposition 6.2) summarising the constructions in previous sections when
$n>1$.

######  Proposition 6.5.

Consider $f$ with a saddle $P$ and a double blender $\Gamma$ as in Theorem B.
There is a $C^{r}$ perturbation ${g}$ of $f$, $r\geqslant 1$, having a saddle
$Q$ such that

  * (a)

$Q$ is simple of type $(m_{s},n_{u})$ and $J_{Q}(g)>1$,

  * (b)

$Q$ is homoclinically related to the double blender $\Gamma_{g}$ and to
$P_{g}$,

  * (c)

$Q$ has a homoclinic tangency in the $ss$-superposition domain of
$\Gamma_{g}$, and

  * (d)

the $uu$-superposition region of the double blender $\Gamma_{{g}}$ contains a
local strong unstable manifold of some point of the orbit of $Q$.

###### Proof.

Recall that the double blender $\Gamma$ has a pair of distinctive saddles
$Q^{\ast}_{cs}$ and $Q^{\ast}_{cu}$ and $ss$\- and $uu$-superposition regions
($\mathscr{D}^{ss}$ and $\mathscr{D}^{uu}$) and domains ($B^{ss}$ and
$B^{uu}$), see Definition 2.1.

Let $U$ and $V$ be neighbourhoods of $Q^{\ast}_{cs}$ and $Q^{\ast}_{cu}$ such
that $U\subset B^{ss}$ and $V\subset B^{uu}$. By hypothesis, $Q^{\ast}_{cu}$
is homoclinically related to $Q^{\ast}_{cs}$ and $P$. Having this in mind, a
slight variation of Lemma 3.2 applied $P$ and $P^{\prime}=Q^{\ast}_{cs}$, we
get a diffeomorphism $g$ with a saddle $Q$ satisfying (a)–(b) such that $Q\in
U$ and has some iterate $Q^{\prime}\in V$. Taking $V$ sufficiently small, the
continuous dependence of the local invariant manifolds (with respect to the
point and the diffeomorphism) implies that $W^{ss}(Q,g)$ contains a disk in
$\mathscr{D}^{ss}$ and $W^{uu}(Q^{\prime},g)$ contains a disk in
$\mathscr{D}^{uu}$. Finally, by Remark 3.6, the saddle $Q$ can be taken with a
homoclinic tangency, obtaining (c)-(d). This completes the proof of the
proposition. ∎

We are now ready to complete the proof of Theorem B. Without loss of
generality, after a $C^{r}$ perturbation, we can assume that the saddle $Q$
and its homoclinic tangency provided by Proposition 6.5 satisfy conditions
(C0)–(C5). As in the proof of Theorem A, we apply Proposition 5.2 to $Q$ and
$g$ to get an unfolding family $(g_{\mathbf{t}})$ and sets of parameters
$\Delta_{k}\to\mathbf{0}$ such that for each $\mathbf{t}\in\Delta_{k}$

  * i)

if $m_{s}=1$ then there exists a saddle $R^{k}_{\mathbf{t}}$ of $u$-index
$n+1$, and

  * ii)

if $m_{s}=2$ then there exist two saddles $R^{k}_{\mathbf{t}}$ and
$S^{k}_{\mathbf{t}}$ of $u$-indices $n+1$ and $n+2$, respectively.

Now we are going to verify the existence of robust cycles stated in the
theorem.

######  Lemma 6.6 (Robust transverse intersections).

Let ${\mathbf{t}}\in\Delta_{k}$. The unstable manifolds
$W^{u}(R^{k}_{\mathbf{t}},g_{\mathbf{t}})$ and
$W^{u}(S^{k}_{\mathbf{t}},g_{\mathbf{t}})$ (if $m_{s}=2$) transversely
intersect the stable set $W^{s}(\Gamma_{\mathbf{t}},g_{\mathbf{t}})$.
Therefore these intersections are $C^{r}$ robust.

###### Proof.

By construction, the orbits of $R^{k}_{\mathbf{t}}$ and $S^{k}_{\mathbf{t}}$
follow the orbit of $Q$, thus they intersect the open set $V$. Thus, taking
$V$ small enough, there are points
$\widehat{R}^{k}_{\mathbf{t}}\in\mathcal{O}(R^{k}_{\mathbf{t}})$ and
$\widehat{S}^{k}_{\mathbf{t}}\in\mathcal{O}(S^{k}_{\mathbf{t}})$ such that
both strong unstable manifolds
$W^{uu}(\widehat{R}^{k}_{\mathbf{t}},g_{\mathbf{t}})$ and
$W^{uu}(\widehat{S}^{k}_{\mathbf{t}},g_{\mathbf{t}})$ contain a disk in
$\mathscr{D}^{uu}$. The robust intersection property of the $cu$-blender
$\Gamma_{\mathbf{t}}$, see (c)(ii) in Definition 2.1, now implies that the
intersections

$W^{u}(\widehat{R}^{k}_{\mathbf{t}},g_{\mathbf{t}})\cap
W^{s}(\Gamma_{\mathbf{t}},g_{\mathbf{t}})\neq\emptyset\quad\mbox{and}\quad
W^{u}(\widehat{S}^{k}_{\mathbf{t}},g_{\mathbf{t}})\cap
W^{s}(\Gamma_{\mathbf{t}},g_{\mathbf{t}})\neq\emptyset$

are $C^{r}$-robust. Finally, note that since the $u$-indices of
$\Gamma_{\mathbf{t}}$, $R^{k}_{\mathbf{t}}$, and $S^{k}_{\mathbf{t}}$ (when
$m_{s}=2$) are $n$, $n+1$, and $n+2$, respectively, the intersections above
can be done transverse after a perturbation. ∎

######  Lemma 6.7 (Robust quasi-transverse intersections).

Let ${\mathbf{t}}\in\Delta_{k}$. There is a $C^{r}$ neighbourhood
$\mathcal{U}$ of $g_{\mathbf{t}}$ such that for every $h\in\mathcal{U}$ the
manifolds and $W^{ss}(R^{k}_{h},h)$ and $W^{s}(S^{k}_{h},h)$ (if $m_{s}=2$)
contain disks in the superposition region of the blender $\Gamma_{h}$. As a
consequence, for every $h\in\mathcal{U}$ it holds

$W^{s}(S^{k}_{h},h)\cap
W^{u}_{\mathrm{loc}}(\Gamma_{h},h)\neq\emptyset\quad\mbox{and}\quad
W^{ss}(R^{k}_{h},h)\cap W^{u}_{\mathrm{loc}}(\Gamma_{h},h)\neq\emptyset.$

###### Proof.

As in the proof of Theorem A, the strong stable manifolds
$W^{ss}(R^{k}_{\mathbf{t}},g_{\mathbf{t}})$ (in both cases i) and ii)) and
$W^{ss}(S^{k}_{\mathbf{t}},g_{\mathbf{t}})$ contain a disk in
$\mathscr{D}^{ss}$. The robust intersection property of the $cs$-blender
$\Gamma_{\mathbf{t}}$ provides the $C^{r}$-robust intersections

$W^{s}(R^{k}_{\mathbf{t}},g_{\mathbf{t}})\cap
W^{u}(\Gamma_{\mathbf{t}},g_{\mathbf{t}})\neq\emptyset\quad\mbox{and}\quad
W^{s}(S^{k}_{\mathbf{t}},g_{\mathbf{t}})\cap
W^{u}(\Gamma_{\mathbf{t}},g_{\mathbf{t}})\neq\emptyset,$

proving the lemma. ∎

Summarising, the sets $\mathcal{O}(R^{k}_{\mathbf{t}})$ and
$\Gamma_{\mathbf{t}}$ form a $C^{1}$ robust heterodimensional cycles of
coindex one and the sets $\mathcal{O}(S^{k}_{\mathbf{t}})$ and
$\Gamma_{\mathbf{t}}$ form a $C^{r}$-robust heterodimensional cycles of
coindex two. The proof of Theorem B is now complete. ∎

######  Remark 6.8.

When the effective dimension is three, we can also obtain robust cycles of
coindex one associated to the continuation of the double blender and saddles
of $u$-index $n-1$.

##  7\. Simultaneity of robust cycles: Simon-Asaoka’s examples

In this section, we use Theorem A to prove Corollary C, claiming that the
diffeomorphisms with robust homoclinic tangencies in [49, 3] also display
$C^{1}$ robust heterodimensional cycles. We start by introducing the
$cs$-blenders of Plykin type.

###  7.1. $cs$-blenders of Plykin type

Consider a two-dimensional diffeomorphism $h$ with a Plykin repeller $\Sigma$
inside the unitary disk $\mathbb{D}\subset\mathbb{R}^{2}$. For a detailed
description of this repeller, see [45, Chapter 8.6], for instance. The key
property of the repeller $\Sigma$ is that the disk $\mathbb{D}$ is foliated by
local unstable manifolds $W^{u}_{\mathrm{loc}}(p,h)$ of points $p\in\Sigma$.
The repeller $\Sigma$ has a fixed point, say the origin $(0,0)$, such that

(7.1) $Dh(0,0)=\begin{pmatrix}\lambda&0\\\ 0&\sigma\end{pmatrix},\quad
0<\lambda<1<\sigma.$

Consider $U\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(-1,1)^{n-2}\times\mathbb{D}$ and a local diffeomorphisms of the
form

$\varphi\colon U\to\mathbb{R}^{n-2}\times\mathbb{D}\quad\varphi(x,y,z)=(\gamma
x,h(y,z))$

where

$0<\gamma<\inf_{(y,z)\in\mathbb{D}}\|Dh(y,z)\|.$

Thus $\Gamma\stackrel{{\scriptstyle\scriptscriptstyle\rm def}}{{=}}\
\\{0^{n-2}\\}\times\Sigma$ is a hyperbolic set of $\varphi$ and
$P=(0^{n})\in\Gamma$ is a fixed point of $\varphi$ of $u$-index one. Note also
that $\Gamma$ is the maximal invariant set of $\varphi$ in $U$. See Figure 4.

Given now any compact manifold $M^{n}$, $n\geqslant 3$, we can embed the local
diffeomorphism $\varphi$ into a diffeomorphism
$f_{\varphi}=f\in\mathrm{Diff}^{r}(M^{n})$. With a slight abuse of notation,
we denote by $\Gamma$ and $P$ the corresponding hyperbolic set and saddle of
$f$.

\begin{overpic}[scale={0.04}]{plykin.jpg} \put(192.0,87.0){\Large{$P$}}
\end{overpic} Figure 4. A $cs$-blender of Plykin type

######  Remark 7.1.

By construction, the surface
$\mathcal{S}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\\{0^{n-2}\\}\times\mathbb{D}$ is normally hyperbolic. Thus there is
a $C^{r}$ neighbourhood $\mathscr{U}$ of $f$ such that for every
$g\in\mathscr{U}$ there are well defined continuations $\Gamma_{g}$ and
$\mathcal{S}_{g}$ of $\Gamma$ and $\mathcal{S}$, respectively, with
$\Gamma_{g}\subset\mathcal{S}_{g}$. As $\mathbb{D}\subset\mathbb{R}^{2}$ is
foliated by the (local) unstable manifolds
$W^{u}_{\mathbb{D},\mathrm{loc}}(p,h)$, $p\in\Sigma$, by normal hyperbolicity
of $\mathcal{S}_{g}$, see [31], we have that $\mathcal{S}_{g}$ is also
foliated by local unstable manifolds of $\Gamma_{g}$.

Next lemma is just a reformulation of the discussion in [15, Section 4.2].

######  Lemma 7.2.

The set $\Gamma$ is a $cs$-blender.

We say the blender in Lemma 7.2 is of Plykin type. As a consequence of item
(1) in Remark 2.3, for every $g$ sufficiently $C^{1}$ close to $f$, the
continuation $\Gamma_{g}$ of $\Gamma$ is also $cs$-blender (of Plykin type).

###### Proof.

Since having a $cs$-blender is a $C^{1}$ robust property, it is enough to
prove the lemma for $f$. Properties (a) of local maximality of $\Gamma$ and
(b) partial hyperbolicity of $\Gamma$ follows considering any neighbourhood
$V=(-\delta,\delta)^{n-2}\times\mathbb{D}$ (small $\delta>0$) of $\Gamma$ and
the (dominated) splitting $T_{\Gamma}M^{n}=E^{ss}\oplus E^{cs}\oplus E^{u}$,
where $E^{ss}$ is $\mathbb{R}^{n-2}\times\\{0^{2}\\}$ and
$T_{\Sigma}\mathbb{R}^{2}=E^{cs}\oplus E^{u}$ is the hyperbolic splitting of
$\Sigma$.

To define the family of disks $\mathscr{D}^{ss}$ of the superposition region
of $\Gamma$, consider first a thin cone field $\mathcal{C}^{ss}$ associated to
the strong stable direction $E^{ss}$ of $\Gamma$. We extend $\mathcal{C}^{ss}$
to a small neighbourhood $W$ of $V$ (maybe after shrinking $\delta$) and, for
simplicity, denote this extension also by $\mathcal{C}^{ss}$. Recall that a
$(n-2)$-dimensional disk $\Delta\subset V$ is tangent to the cone field
$\mathcal{C}^{ss}$ if $T_{p}\Delta\subset\mathcal{C}^{ss}(p)$ for every
$p\in\Delta$. The family $\mathscr{D}^{ss}$ consists of all
$(n-2)$-dimensional disks $\Delta$ containing in $W$ and tangent to
$\mathcal{C}^{ss}$ whose intersection with the boundary of $V$ has two
connected components. This defines an open family of disks.

Given any saddle $Q\in\Gamma$, consider its local strong stable manifold
relative to $U$ $W^{ss}_{U,\mathrm{loc}}(Q,f)$. This set contains a disk of
$\mathscr{D}^{ss}$ containing $Q$ in its interior. Thus, any periodic point of
$\Gamma$ can be chosen as a distinctive saddle of the blender, proving
property (c-i) in the definition of a $cs$-blender.

To get property (c-ii) about robust intersections, just observe that
$W^{u}_{U,\mathrm{loc}}(\Gamma,f)=\mathcal{S}$ and that for every disk $D$ of
$\mathscr{D}^{ss}$ it holds $\mathcal{S}\pitchfork D\neq\emptyset$. The proof
of the lemma is now complete. ∎

###  7.2. Local deformations: generation of robust tangencies

To generate a robust homoclinic tangency associated to the blender $\Gamma$ we
argue as in [49, 3]. Let us sketch the main arguments there. Recall that $P$
is the fixed point of $\Gamma$. Consider a small $(n-1)$-dimensional disk
$\Delta\subset W^{s}(P,f)$ disjoint from the closure of $V$, hence disjoint
from $\Gamma$. Through a smooth isotopy supported outside the closure of $V$,
we carry $\Delta$ close to $\Gamma$ in such a way that the following
properties hold:

  * (i)

$\mathcal{T}\stackrel{{\scriptstyle\scriptscriptstyle\rm def}}{{=}}\Delta\cap
V$ is a $ss$-tube: $\mathcal{T}$ is a manifold diffeomorphic to
$(-1,1)^{n-2}\times\mathbb{S}^{1}$, foliated by disks of $\mathscr{D}^{ss}$.

  * (ii)

the transverse intersection between $\mathcal{T}$ and $\mathcal{S}$ is a
simple closed curve $\alpha$,

  * (iii)

the curve $\alpha$ (and thus the manifold $\mathcal{T}$) is tangent to
$W^{\mathrm{u}}_{\mathrm{loc}}(P,f)$.

This isotopy produces a new diffeomorphism whose restriction to $V$ coincides
with $f$. Hence $\Gamma$ is a $cs$-blender of this new diffeomorphism. With a
slight abuse of notation, we continue calling the resulting diffeomorphism by
$f$. We claim that for every $g$ sufficiently $C^{r}$ close to $f$ the set
$\Gamma_{g}$ has a homoclinic tangency and thus $\Gamma$ has a $C^{r}$ robust
tangency. Indeed, consider the continuations $\mathcal{T}_{g}$ of
$\mathcal{T}$ and $\mathcal{S}_{g}$ of $\mathcal{S}$ for $g$ close to $f$.
Note that $\mathcal{T}_{g}\subset W^{s}(P_{g},g)$ is also a $ss$-tube and
$\alpha_{g}=\mathcal{T}_{g}\pitchfork\mathcal{S}_{g}$ is also a simple closed
curve. On the other hand, as
$\mathcal{S}_{g}=W^{u}_{U,\mathrm{loc}}(\Gamma_{g},g)$, there is
$P_{0}\in\Gamma_{g}$ such that $W^{u}_{\mathrm{loc}}(P_{0},g)$ and
$\alpha_{g}$ are tangent (this is a consequence of Remark 7.1).

We denote by $\mathscr{T}$ the $C^{r}$ neighbourhood of $f$ obtained above.

###  7.3. Proof of Corollary C

The first item of the proposition was obtained above. The second item follows
applying Theorem A to $f$ and $P$. Note that $P$ is a simple saddle of type
$(1,1)$ with leading Jacobian $J_{P}(f)=\lambda\,\sigma>1$ and homoclinically
related to $\Gamma$. After shrinking $\mathscr{T}$, we can assume that these
conditions hold for $P_{g}$ and $\Gamma_{g}$ for every $g\in\mathscr{T}$. By
the second part of Remark 2.4, there is a dense subset $\mathscr{D}$ of
$\mathscr{T}$ of diffeomorphisms $g$ having a homoclinic tangency associated
to $P_{g}$. The diffeomorphisms in $\mathscr{D}$ satisfy the hypotheses of
Theorem A. Thus for each $g\in\mathscr{D}$ there is an open set
$\mathscr{C}_{g}$ whose closure contains $g$ consisting of diffeomorphisms
with robust heterodimensional cycles as above. Taking now

$\mathscr{C}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\bigcup_{g\in\mathscr{D}}\mathscr{C}_{g}.$

the corollary follows. ∎

##  8\. Appendix: Horseshoes with large entropy and blenders.
Proof of Theorem 1.4

In this appendix, we explain how Theorem 1.4 follows from the results in [4].

By the assumption, the hyperbolic measure $\mu$ is such that there is
$\epsilon>0$ with

(8.1)
$h_{\mu}(f)>-\log\mathrm{J}_{\mu}^{cs}(f)+\frac{1}{2r}\chi^{cs}_{\mu}(f)+\epsilon.$

Applying [4, Theorem B’] to the hyperbolic measure $\mu$, given any $\delta>0$
we get $C^{1}$ perturbation $h\in\mathrm{Diff}^{r}(M)$ of $f$ with an affine
horseshoe777A horseshoe $\Lambda$ is said _affine_ if there are a
neighbourhood $U$ of $\Lambda$ and a chart $\varphi\colon U\to\mathbb{R}^{d}$
such that $\varphi\circ f\circ\varphi^{-1}$ is locally affine in $U$. If the
chart $\varphi$ can be chosen such that $D(\varphi\circ
f\circ\varphi^{-1})(x)$ is a matrix $A$ independent of $x$, we say that
$\Lambda$ has _constant linear part $A$_. $\Gamma_{\delta}$ whose hyperbolic
splitting $E_{\delta}^{s}\oplus E_{\delta}^{u}$ satisfies

  * •

$\Gamma_{\delta}$ has a constant linear part $A_{\delta}$ which is a diagonal
matrix with distinct real positive eigenvalues,

  * •

The Lyapunov exponents of $\mu$ are $\delta$-close to the Lyapunov exponents
of $A_{\delta}$. In particular,

$\mathrm{Jac}_{E^{s}_{\delta}}(A_{\delta})\to\mathrm{J}^{s}_{\mu}\quad\mbox{and}\quad\chi^{cs}(A_{\delta})\to\chi^{cs}_{\mu}\quad\mbox{as
$\delta\to 0$},$

where
$\mathrm{Jac}_{E_{\delta}^{s}}(A_{\delta})\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\det A_{\delta}|_{E_{\delta}^{s}}$ and $\chi^{cs}(A_{\delta})$ is
the negative Lyapunov exponent of $A_{\delta}$ closest to zero. From this
convergence, taking $\delta>0$ small enough, we have

(8.2)
$\begin{split}-\log\mathrm{J}^{s}_{\mu}+\frac{\epsilon-\delta}{2}&>-\log\mathrm{Jac}_{E_{\delta}^{s}}(A_{\delta})\quad\mbox{and}\\\
\frac{1}{2r}\chi^{cs}_{\mu}+\frac{\epsilon-\delta}{2}&>\frac{1}{2r}\chi^{cs}(A_{\delta}),\end{split}$

  * •

$\Gamma_{\delta}$ is $\delta$-close to the support of $\mu$ in Hausdorff
distance,

  * •

$h_{\mathrm{top}}({\Gamma_{\delta}},{h_{\delta}})>h_{\mu}(f)-\delta$.

The last inequality implies that

(8.3)
$\begin{split}h_{\mathrm{top}}({\Gamma_{\delta}},{h_{\delta}})&>h_{\mu}(f)-\delta\overset{\eqref{e.byassumption}}{\geqslant}-\log\mathrm{J}^{s}_{\mu}+\frac{1}{2r}\chi^{cs}_{\mu}+\varepsilon-\delta\\\
&\overset{\eqref{e.inparticular}}{>}-\log\mathrm{Jac}_{E_{\delta}^{s}}(A_{\delta})+\frac{1}{2r}\chi^{cs}(A_{\delta}).\end{split}$

Observe that the inequality in (8.3) allows us to apply [4, Theorem C] to the
affine horseshoe $\Gamma_{\delta}$ of $h_{\delta}$, getting a perturbation
$g_{\delta}$ of $h_{\delta}$ supported in a small neighbourhood of
$\Gamma_{\delta}$ such that the continuation $\Gamma_{g_{\delta}}$ of
$\Gamma_{\delta}$ is a $cs$-blender of central dimension $d_{cs}=m-1$ with a
dominated splitting whose bundles are one-dimensional. Taking now a sequence
$(\delta_{k})\to 0$, for each large $k$ we get diffeomorphisms
$h_{k}=h_{\delta_{k}}$ and perturbations $g_{k}$ of $h_{k}$ with blenders
$\Gamma_{k}$ as before. These blenders satisfy conditions (1), (2), and (3) in
the theorem.

To conclude the proof of the theorem, it remains to see that if the saddle $P$
is homoclinically related to $\mu$ then we can chose the perturbations $g_{k}$
such that the continuations $P_{k}$ of $P$ and $\Gamma_{k}$ of $\Gamma$ are
homoclinically related. To see this, we need to review the steps in the proof
of [4, Theorem B’], which is a combination of [4, Theorem B] and Katok’s
approximation theorem [32] and its extensions in [4, Theorem. 3.3] and [22,
Theorem 2.12].

The first step is to apply Katok’s approximation theorem to the hyperbolic
measure $\mu$ of $f$ to obtain a horseshoe $\Lambda$ of $f$ such that:

  1. (a)

$\Lambda$ is homoclinically related to $\mu$,

  2. (b)

$\Lambda$ is close to the support of $\mu$ in the Hausdorff distance,

  3. (c)

the topological entropy of $\Lambda$ is close to $h_{\mu}(f)$,

  4. (d)

the $f$-invariant measures of $\Lambda$ are close to $\mu$ in the weak*
topology, and

  5. (e)

the Lyapunov exponents of any ergodic measure of $\Lambda$ are close to the
Lyapunov exponents of $\mu$.

By hypothesis, the saddle $P$ is homoclinically related to $\mu$. Thus (a) and
the $\lambda$-lemma imply that $P$ and $\Lambda$ are homoclinically related.
We can assume that $P\not\in\Lambda$. Otherwise, if $P\in\Lambda$, we extract
a subhorseshoe $\widetilde{\Lambda}$ of $\Lambda$ such that
$P\not\in\widetilde{\Lambda}$ and $h_{\mathrm{top}}(\widetilde{\Lambda},f)$ is
close to $h_{\mathrm{top}}(\Lambda,f)$. In particular, properties (a)–(e) also
hold for $\tilde{\Lambda}$.

The second step in the proof of [4, Theorem B’] is to apply [4, Theorem B] to
the horseshoe $\Lambda$ to get affine horseshoes with constant linear part.
Since the perturbation in the latter result is done in a small neighbourhood
of $\Lambda$ and $P\not\in\Lambda$, we obtain a $C^{1}$ perturbation
$h\in\mathrm{Diff}^{r}(M)$ of $f$ with $P_{h}=P$ and such that the
continuation $\Lambda_{h}$ of $\Lambda$ contains an affine horseshoe
$\Gamma_{h}$ with constant linear part. In particular, $P_{h}$ is
homoclinically related with $\Gamma_{h}$.

Finally, when $P$ has a homoclinic tangency, the constructions above can be
done preserving that tangency. This concludes the proof of the theorem.

##  9\. Appendix: Dominated splittings

In this appendix, we review the notion of a dominated splitting.

Given an invariant set $\Lambda$ of $f\in\mathrm{Diff}^{1}(M)$, a splitting
$T_{\Lambda}M=E\oplus F$ over $\Lambda$ is called dominated if it is $Df$
invariant and there is $\ell\in\mathbb{N}$ such that for every $x\in\Lambda$
and every pair of unit vectors $u\in E(x)$ and $v\in F(x)$ it holds

$\frac{||Df^{\ell}(x)(u)||}{||Df^{\ell}(x)(v)||}<\frac{1}{2},$

here $E(x)$ and $F(x)$ are the bundles at $x$ and $||\cdot||$ is the norm. In
this case, we say that $F$ dominates $E$.

A $Df$ invariant bundle $T_{\Lambda}M=E_{1}\oplus\cdots\oplus E_{k}$ with $k$
bundles, $k\geqslant 3$, is dominated if the bundles
$T_{\Lambda}M=(E_{1}\oplus\cdots\oplus E_{i})\oplus(E_{i+1}\oplus\cdots\oplus
E_{k})$ are dominated for every $i\in\\{1,\dots,k-1\\}$.

The bundle $E$ over $\Lambda$ is uniformly contracting if there are constants
$C>0$ and $\kappa\in(0,1)$ such that

$||Df^{n}(x)(u)||\leqslant C\,\kappa^{n}||u||$

for every $x\in\Lambda$, $n\geqslant 0$, and every vector $u\in E(x)$.
Similarly, the bundle $F$ is uniformly expanding if it is uniformly
contracting for $f^{-1}$. A dominated splitting $E\oplus F$ over $\Lambda$ is
partially hyperbolic if either $E$ is uniformly contracting or $F$ is
uniformly expanding. When $E$ is uniformly contracting and $F$ is uniformly
expanding the splitting is hyperbolic. An invariant set $\Lambda$ is
(partially) hyperbolic if it has a (partially) hyperbolic splitting.

The following proposition is a well-known result about dominated splittings.
We refer to [21, Chapter B.2] for details (including the definitions of cone
fields).

######  Proposition 9.1.

Let $f\in\mathrm{Diff}^{1}(M)$ and $\Lambda\subset M$ be a compact
$f$-invariant set with a dominating splitting $T_{\Lambda}M=E\oplus F.$ Then
there are neighbourhoods $U$ of $\Lambda$ in $M$ and $\mathscr{N}$ of $f$ in
$\mathrm{Diff}^{1}(M)$ such that for every $g\in\mathscr{N}$ the maximal
invariant set $\Lambda_{g}$ of $g$ in $U$ has a dominated splitting
$T_{\Lambda_{g}}=E_{g}\oplus F_{g}$ with $\dim(E_{g})=\dim(E)$ such that

  1. (1)

the bundles of the dominated splitting depend continuously with the point $x$
and the map $g$,

  2. (2)

there are continuous open cone fields $\mathcal{C}_{E}$ and $\mathcal{C}_{F}$
defined on $U$ with $E(x)\subset\mathcal{C}_{E}(x)$ and
$F(x)\subset\mathcal{C}_{F}(x)$ such that for every $g\in\mathscr{N}$ it
holds:

     * •

$Dg^{-1}(\mathcal{C}_{E}(x))\subset\mathcal{C}_{E}(g^{-1}(x))$ if
$x,g^{-1}(x)\in U$ and

     * •

$Dg(\mathcal{C}_{F}(x))\subset\mathcal{C}_{F}(g(x))$ if $x,g(x)\in U$.

## References