Robust degenerate unfoldings of cycles and tangencies

TL;DR

This paper constructs open sets of degenerate unfoldings of heterodimensional cycles and homoclinic tangencies of arbitrary codimension, revealing complex phenomena like coexistence of infinitely many attractors in dynamical systems.

Contribution

It introduces new methods to create robust degeneracies in dynamical systems, including tangencies of large codimension outside strong hyperbolic sets.

Findings

Support for coexistence of infinitely many attractors

Construction of robust homoclinic tangencies of large codimension

Identification of phenomena outside strong hyperbolic sets

Abstract

We construct open sets of degenerate unfoldings of heterodimensional cycles of any co-index $c>0$ and homoclinic tangencies of arbitrary codimension $c>0$. These sets are known to be the support of unexpected phenomena in families of diffeomorphisms, such as the Kolmogorov typical co-existence of infinitely many attractors. As a prerequisite we also construct robust homoclinic tangencies of large codimension which cannot be inside a strong partially hyperbolic set.

Equations

$$d=d_x(\mathcal{L},\mathcal{S})\stackrel{\mathrm{def}}{=}\dim T_x\mathcal{L}\cap T_x\mathcal{S}.$$

$$c=c_x(\mathcal{L},\mathcal{S})\stackrel{\mathrm{def}}{=}\dim \mathcal{M}-\dim \left(T_x\mathcal{L}+T_x\mathcal{S}\right).$$

$$c_x(\mathcal{L},\mathcal{S})=\dim \mathcal{M}-(\dim T_x\mathcal{L}+\dim T_x\mathcal{S}-\dim T_x\mathcal{L}\cap T_x\mathcal{S})=d_x(\mathcal{L},\mathcal{S}).$$

$$Y\in W^u(P)\cap W^s(Q)\text{and}c=c_Y(W^u(P),W^s(Q)).$$

$$c=d_Y(W^u(P),W^s(Q)).$$

$$d_Y(W^s(P),W^u(Q))=c\text{and}{c}_Y(W^s(P),W^u(Q))=0.$$

$$d_Y(W^u(P),W^s(Q))=0\text{and}{c}_Y(W^u(P),W^s(Q))=c.$$

$$\frac{d\delta }{da}(0)\ne 0\text{where}\delta (a)=\min \{d(x,y):x\in W^u(P_a)\cap U,y\in W^s(Q_a)\cap U\}.$$

$$p_a\in W_{loc}^u(P_a)\text{and}{q}_a\in W^s(Q_a)\text{so that}d(p_a,q_a)=o(\Vert a{\Vert }^d)\text{at a=0}$$

$$d(p_a,q_a)=o(\Vert a{\Vert }^d)\text{and}d(E_a,F_a)=o(\Vert a{\Vert }^d)\text{at a=0}.$$

$${\partial }_a^i{\partial }_x^j{f}_a(x)\text{exists continuously for all 0≤i≤d, 0≤i+j≤r and (a,x)∈Ik×M.}$$

$$\Vert f{\Vert }_{C^{d,r}}=\max \{\sup \Vert {\partial }_a^i{\partial }_x^j{f}_a(x):0\le i\le d,0\le i+j\le r\}\text{where f=(fa)a∈Cd,r(Ik,M).}$$

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

$$W_{loc}^u({\Gamma }_g)\cap \mathcal{D}\ne \varnothing \text{for all D∈D and g∈U}$$

$$T_x{W}_{loc}^u(z)+T_x\mathcal{D}=T_x{W}_{loc}^u(z)\oplus T_x\mathcal{D}.$$

$$c_x(W_{loc}^u(z),\mathcal{D})=\dim \mathcal{M}-\dim (T_x{W}_{loc}^u(z)+T_x\mathcal{D})=\dim \mathcal{M}-(d_u+d_{ss})=c\ge 1.$$

$$m(DF(x))\le \nu <1<{\nu }^{-1}\le \Vert DF(x)\Vert \text{for all x∈Λ.}$$

$$D{F}^{-1}(x){\mathcal{C}}_{\alpha }^{ss}\subset {\mathcal{C}}_{{\nu }^2\alpha }^{ss}\text{for all x∈R.}$$

$${\mathcal{C}}_{\theta }^{ss}\stackrel{\mathrm{def}}{=}\{(u,v)\in {\mathbb{R}}^{d_{ss}}\oplus {\mathbb{R}}^{d_u}:\Vert v\Vert <\theta \Vert u\Vert \}\cup \left\{0\right\}.$$

$${\varphi }_{\ell }(\overline{D})\subset D\text{and}\lambda <m(D{\varphi }_{\ell }(y))<\Vert D{\varphi }_{\ell }(y)\Vert <\beta <1\text{for all y∈D and ℓ=1,…,κ.}$$

$$\Phi =F⋉({\varphi }_1,\dots ,{\varphi }_{\kappa })\text{on}\mathcal{U}=({\mathsf{R}}_1\times D)\cup \cdots \cup ({\mathsf{R}}_{\kappa }\times D)$$

$$\Phi (x,y)=(F(x),\varphi (x,y))\text{with ϕ(x,y)=ϕℓ(y) if (x,y)∈Rℓ×D.}$$

$$\Psi =G⋉({\psi }_1,\dots ,{\psi }_{\kappa })\text{on}\mathcal{V}={\mathcal{V}}_1\cup \cdots \cup {\mathcal{V}}_{\kappa }$$

$$\mathcal{D}(\xi )=(\xi ,g(\xi ),h(\xi ))\in [-2,2{]}^{d_{ss}}\times I_{\ell }\times B\text{for}\xi \in [-2,2{]}^{d_{ss}}\text{and some}\ell \in \{1,\dots ,\kappa \}.$$

$$d(h(\xi ),h({\xi }^{\prime }))\le Cd(\xi ,{\xi }^{\prime })\text{for all ξ,ξ′∈[−2,2]dss.}$$

$${\mathsf{R}}_{\ell }=(-2,2{)}^{ss}\times I_{\ell },\ell =1,\dots ,\kappa$$

$$DF=\left(\begin{array}{cc}{S}_{\ell }& 0\\ 0& U_{\ell }\end{array}\right)\text{on Rℓ where ∥Sℓ∥, ∥Uℓ−1∥<ν for ℓ=1,…,κ.}$$

$$D{F}^{-1}(x){\mathcal{C}}_{\alpha }^{ss}\subset {\mathcal{C}}_{{\nu }^2\alpha }^{ss}\text{for all x∈R=R1∪⋯∪Rκ and α>0,}$$

$$D{\varphi }_{\ell }(y)=T_{\ell }$$

$$\overline{B}\subset {\varphi }_1(B)\cup \cdots \cup {\varphi }_{\kappa }(B).$$

Full text

1. Introduction

Robust homoclinic tangencies and robust heterodimensional cycles are, in general, prerequisites for obtaining abundant complicated dynamical systems [New70, GŠ72, New74, BD08, BD12]. Both configurations imply the existence of a non-transversal intersection between the stable and unstable manifolds of points in the same or in different transitive hyperbolic sets. A priori, the non-transverse intersection could be destroyed by a small perturbation. But since it is robust, this means that a new non-transverse intersection is created between the manifolds of the continuation of the hyperbolic sets. The unfolding of these bifurcations yields a great number of changes in the dynamics. For instance, infinitely many saddle periodic points and sinks appear in the unfolding of homoclinic tangencies. Hence, the persistence of these bifurcations allowed to get a generic coexistence of infinitely many periodic attractors [New79, GTS93, PV94, GST08].

The construction of robust tangencies in lower dimension is based on the creation of thick horseshoes involving distortion estimates which are typically $C^{2}$. However, in higher dimensions it was possible to construct robust homoclinic tangencies in the $C^{1}$-topology using blenders [Asa08, BD12]. Blenders are hyperbolic sets having a thicker invariant manifold than initially expected. They were discovered by Bonatti and Diaz [BD96] and now are essential objects in the study of non-hyperbolic dynamics. On the other hand, all of the above mentioned constructions are of codimension one. That is, the dimension of the coincidence of the tangent spaces at the tangent point. Recently in [BR17], the authors gave the first examples of $C^{2}$-robust tangencies of large codimension. The novelty in the construction was the use of the blender for the dynamics induced in the tangent bundle.

A different approach, when compared to the generic results mentioned above, is to look for bifurcations of homoclinic tangencies in parametric families of diffeomorphisms. For decades it was thought that the coexistence of infinitely many hyperbolic attractors was meager in families of dynamical systems [PS96]. However, recently and far from intuition, Berger showed in [Ber16] that actually these phenomena form a residual set. Behind this result was the construction of open sets of families of endomorphisms with robust homoclinic tangencies and with an extra property: the family unfolds degenerately a tangency. This means the unfolding is slow in the sense of the zeroing of the first terms in a certain Taylor polynomial describing the local separation between the manifolds. Although a degenerate unfolding of a tangency could be destroyed by a small perturbation of the family, this perturbation has another tangency which unfolds also degenerately. The mechanism involved in these constructions of Berger [Ber16] is also the blender, but now constructed for the dynamics induced in the space of jets (the space of velocities). See also [BCP16, Ber17a].

The objective of the present work is to unite the construction of [BR17] and [Ber16] to obtain robust degenerate unfoldings of homoclinic tangencies of large codimension for families of diffeomorphisms. Only this will not be done by merely combining the two previous results. Here we present a new method of construction of robust tangencies of large codimension, different from the one in [BR17]. This is a generalization to higher codimension of the construction in [BD12] using folding manifolds. It is expected that the unfolding of these robust degenerated tangencies gives new dynamical consequences. For example, the existence of residual sets with infinitely many attracting invariant tori of large dimension.

1.1. Dimension and codimension of an intersection

Let us begin with some definitions on the intersections of submanifolds. Let $\mathcal{L}$ and $\mathcal{S}$ be submanifolds of $\mathcal{M}$. We say that $\mathcal{L}$ and $\mathcal{S}$ have an intersection of dimension $d\geq 0$ at $x\in\mathcal{L}\cap\mathcal{S}$, if

[TABLE]

Notice that $d$ is the maximum number of common linearly independent tangent directions in $T_{x}\mathcal{M}$. However, this number is not enough to measure how far the intersection is from being transverse. In order to quantify this we say that $\mathcal{L}$ and $\mathcal{S}$ have an intersection of codimension $c\geq 0$ if

[TABLE]

An intersection of codimension zero is called transverse and is said to be tangencial otherwise. Observe that the definition of a tangency ($c>0$) includes the case $d=0$ in which $T_{x}\mathcal{L}+T_{x}\mathcal{S}=T_{x}\mathcal{L}\oplus T_{x}\mathcal{S}$. In the literature this case is called as a quasi-transverse intersection. On the other hand, the sum of the dimension and the codimension of an intersection between submanifolds is in general not equal to the dimension of $\mathcal{M}$. Moreover, when the codimension of $\mathcal{L}$ coincides with the dimension of $\mathcal{S}$, then if $x\in\mathcal{L}\cap\mathcal{S}$

[TABLE]

1.2. Heterodimensional cycles and homoclinic tangencies

A $C^{r}$-diffeomorphism $f$ of a manifold $\mathcal{M}$ has a homoclinic tangency of codimension $c>0$ if there is a pair of points $P$ and $Q$, in the same transitive hyperbolic set, so that the unstable invariant manifold of $P$ and the stable invariant manifold of $Q$ have an intersection of codimension $c$ at a point $Y$. That is,

[TABLE]

Observe that actually, as the codimension of $W^{u}(P)$ coincides with the dimension of $W^{s}(Q)$ we have that also in this case

[TABLE]

Similarly, $f$ has a heterodimensional cycle of co-index $c>0$ if there exist two transitive hyperbolic sets $\Lambda$ and $\Gamma$ such that their invariant manifolds meet cyclically and $|\mathrm{ind}^{s}(\Lambda)-\mathrm{ind}^{s}(\Gamma)|=c$. Here $\mathrm{ind}^{s}(\cdot)$ denotes the dimension of the stable bundle of the respective set. By means of an arbitrarily small perturbation if necessary, the stable and unstable manifolds have a transverse intersection of dimension $c$ and a tangency which is a quasi-transverse intersection of codimension $c$. Indeed, we can assume that $\mathrm{ind}^{s}(\Lambda)-\mathrm{ind}^{s}(\Gamma)=c$ and for $P\in\Lambda$, $Q\in\Gamma$ suppose that $Y$ belongs to $W^{s}(P)\cap W^{u}(Q)$. Then $\dim T_{Y}W^{s}(P)+\dim T_{Y}W^{u}(Q)=\dim\mathcal{M}+c$ and thus, in general,

[TABLE]

On the other hand, if $Y\in W^{u}(P)\cap W^{s}(Q)$ then $\dim T_{Y}W^{u}(P)+\dim T_{Y}W^{s}(Q)=\dim\mathcal{M}-c$ and hence,

[TABLE]

1.3. Robust tangencies of large codimension

By Kupka-Smale’s theorem, $C^{r}$-generically, the stable and unstable manifolds of a pair of saddle hyperbolic periodic points meet transversally. Hence, tangencies associated with saddles occurs in the complement of a residual set of diffeomorphisms and thus are non-generic dynamical configurations. However, the situation is different if instead of the periodic saddles we consider non-trivial hyperbolic sets. It is well-known the existence of open sets of diffeomorphisms displaying non-transverse intersections between the stable and unstable manifolds of points in the continuations of these hyperbolic sets. That is, the so-called $C^{r}$-open sets of diffeomorphisms with robust tangencies or robust heterodimensional cycles. See [New70, GTS93, PV94, BD12] for robust homoclinic tangencies of codimension one and [BD08] and reference therein for robust heterodimensional cycles.

Robust homoclinic tangencies of large codimension in the $C^{2}$-topology were recently discovered in [BR17] inside strong partially hyperbolic sets. That is, invariant sets with a dominated splitting of the form $E^{s}\oplus E^{c}\oplus E^{u}$, where $E^{s}$ and $E^{u}$ are the non-trivial contracting and expanding bundles respectively. Here, we will construct new examples of a different nature from the robust tangencies of large codimension showed in [BR17]. This is because they cannot be embedded inside a strong partially hyperbolic set. At the point of tangency, the splitting is of the form $E^{s}\oplus E^{c}$, where $E^{c}$ cannot be divided into neither contracting nor expanding subbundles. In this case, we say that the tangency is inside a weak partially hyperbolic set.

Theorem A**.**

*Every manifold of dimension $m>c^{2}+c$ admits a diffeomorphism having a $C^{2}$-robust homoclinic tangency of codimension $c>0$ inside a weak partially hyperbolic set. *

Notice that the above theorem gives as a particular case the well-known results [GTS93, PV94, Rom95] about $C^{2}$-robust homoclinic tangencies of codimension one in higher dimensions. Here, we provide a different proof inspired by the construction of $C^{1}$-robust homoclinic tangencies of Bonatti and Díaz in [BD12]. The concepts of folding manifolds and blenders constructed in the tangent bundle allows us to extend their result to large codimension.

1.4. Degenerate unfoldings

A tangency at a point $Y$ between the unstable manifold $W^{u}(P)$ and the stable manifold $W^{s}(Q)$ of a $C^{r}$-diffeomorphism $f$ can be unfolded by considering $C^{d}$-families $(f_{a})_{a}$ of $C^{r}$-diffeomorphisms parameterized by $a\in\mathbb{I}^{k}$ with $f_{0}=f$ and $\mathbb{I}=[-1,1]$. We will suppose $0<d\leq r<\infty$ and $k\geq 1$. Many articles usually impose a generic condition on the velocity of the unfolding. They assume that the distance between the manifolds has positive derivative with respect to the parameter:

[TABLE]

Here $P_{a}$, $Q_{a}$ are the continuations of the hyperbolic saddles $P_{0}=P$ and $Q_{0}=Q$ for $f_{a}$ respectively and $U$ is a small neighborhood of $Y$. However, in this work we are interested in studying unfoldings where this generic assumption fails.

Let us consider first the case that the family $f=(f_{a})_{a}$ unfolds a heterodimensional cycle of co-index $c>0$ at $a=0$. That is, the above points $P_{0}$ and $Q_{0}$ now belong, respectively, to transitive hyperbolic sets $\Lambda_{0}$ and $\Gamma_{0}$ of $f_{0}$ with co-index $c$ and whose stable and unstable manifolds intersect cyclically. Moreover, say that $Y\in W^{u}(P_{0})\cap W^{s}(Q_{0})$ is the tangency of the cycle (i.e, the intersection that in general could be assumed quasi-transverse and of codimension $c$). The unfolding of this heterodimensional cycle is said to be $C^{d}$-degenerate at $a=0$ if there exist

[TABLE]

where $p_{0}=q_{0}=Y$ and $p_{a}$, $q_{a}$ vary $C^{d}$-continuously with respect to the parameter $a\in\mathbb{I}^{k}$.

Now we consider that the family $f=(f_{a})_{a}$ unfolds a homoclinic tangency of codimension $c>0$ at $a=0$. That is, we assume the points $P_{0}$ and $Q_{0}$ belong to the same hyperbolic set $\Lambda_{0}$ of $f_{0}$ and that the homoclininc tangency $Y\in W^{u}(P_{0})\cap W^{s}(Q_{0})$ has codimension $c>0$. The unfolding of this homoclinic tangency of codimension $c>0$ is said to be $C^{d}$-degenerate at $a=0$ if there are points $p_{a}\in W^{u}_{loc}(P_{a})$, $q_{a}\in W^{s}(Q_{a})$ and $c$-dimensional subspaces $E_{a}$ and $F_{a}$ of $T_{p_{a}}W^{u}(P_{a})$ and $T_{q_{a}}W^{s}(Q_{a})$ respectively such that

[TABLE]

Here, $p_{0}=q_{0}=Y$ and $(p_{a},E_{a})$, $(q_{a},F_{a})$ vary $C^{d}$-continuously with respect to the parameter $a\in\mathbb{I}^{k}$. Observe that in this case it is necessary to assume that $d<r$ because the above definition involves the dynamics of the family $(f_{a})_{a}$ in the tangent bundle (in fact, in certain Grassmannian bundles). In [Ber16] $C^{d}$-degenerate unfoldings of homoclinic tangencies were called for short $C^{d}$-paratangencies. The key consequence of having a $C^{d}$-paratangency at $a=0$ is that one can perturb the family and obtain a new family which now has a persistent homoclinic tangency in the sense of [Ber17a]. That is, a tangency point $Y_{a}$ between the stable and unstable manifold which varies $C^{d}$-continuously with respect to $a$ in an open set of parameters $J\subset\mathbb{I}^{k}$ containing $a=0$.

1.5. Open sets of families with degenerate unfoldings

In order to be more precise, we now introduce the following definitions. The exact notion of a $C^{d}$-family of $C^{r}$-diffeomorphisms and the $C^{d,r}$-topology considered is going to be defined in §1.6.

A $k$-parameter $C^{d}$-family $f=(f_{a})_{a}$ of $C^{r}$-diffeomorphisms $f_{a}$ displays a

$C^{d,r}$-robust $C^{d}$-degenerate unfolding of a homoclinic tangency of codimension $c$ at $a=0$ if there are a transitive hyperbolic set $\Lambda_{0}$ of $f_{0}$ and a $C^{d,r}$-neighborhood $\mathscr{U}$ of $f$, such that any $g=(g_{a})_{a}\in\mathscr{U}$ displays a $C^{d}$-degenerate unfolding of a homoclinic tangency of codimension $c>0$ at $a=0$ associated with the continuations of $\Lambda_{0}$ for $g_{0}$. 2. -

$C^{d,r}$-robust $C^{d}$-degenerate unfolding of a heterodimensional cycle of co-index $c$ at $a=0$ if there are transitive hyperbolic sets $\Lambda_{0}$ and $\Gamma_{0}$ of $f_{0}$ with co-index $c$ and a $C^{d,r}$-neighborhood $\mathscr{U}$ of $f$, such that any $g=(g_{a})_{a}\in\mathscr{U}$ displays a $C^{d}$-degenerate unfolding of a heterodimensional cycle of co-index $c>0$ at $a=0$ associated with the continuations of $\Lambda_{0}$ and $\Gamma_{0}$ for $g_{0}$.

For simplicity, we have chosen $a=0$ as the critical parameter of the unfolding. However, degenerate unfoldings can also be introduced at any other parameter $a=a_{0}$ with $a_{0}\in\mathbb{I}^{k}$. We say that $f=(f_{a})_{a}$ displays a $C^{d,r}$-robust $C^{d}$-degenerate unfolding of a heterodimensional cycle (or a tangency) at any parameter when any $g=(g_{a})_{a}\in\mathscr{U}$ has a heterodimensional cycle (or a homoclinic tangency) at $a=a_{0}$ which unfolds $C^{d}$-degenerate for all $a_{0}\in\mathbb{I}^{k}$. Robust degenerate unfoldings at any parameter are involved in unexpected phenomena as the typical coexistence of infinitely many sinks [Ber16, Ber17a], infinitely many non-hyperbolic strange attractors [Roj17] and fast growth of periodic points [Ber17b] among others.

Theorem B**.**

Any manifold of dimension $m>c^{2}+c$ admits a $k$-parameter $C^{d}$-family of $C^{r}$-diffeomorphisms with $0<d<r-1$, which displays a $C^{d,r}$-robust $C^{d}$-degenerate unfolding of a homoclinic tangency of codimension $c>0$ at any parameter.

We will also show the existence of robust degenerate unfoldings of heterodimensional cycles of any co-index at any parameter:

Theorem C**.**

Any manifold of dimension $m>1+c$ admits a $k$-parameter $C^{d}$-family of $C^{r}$-diffeomorphisms with $0<d<r$ which displays a $C^{d,r}$-robust $C^{d}$-degenerate unfolding of a heterodimensional cycle of co-index $c>0$ at any parameter.

The differences in the regularity and dimension that appear in the above theorems come from the nature of the unfolding of the tangency, as we now explain. With respect to the regularity assumption in Theorem C, the unfolding of a heterodimensional cycle only deals with the distance in the ambient manifold. There is a loss of a derivative ($d<r$) in the moment that we pass to study the kinematic of the movement by lifting the family to the space of velocities (jet space). This is because we will need that the induced dynamics in the jet space is a $C^{1}$-diffeomorphism. On the other hand, the unfolding of tangencies in Theorem B requires first to lift the dynamics to the space where the bifurcation is produced, that is to the Grassmannian bundle. After that one needs to perform a similar analysis in the space of velocities to study the corresponding $C^{d}$-degenerate unfolding. This provides a loss of two degrees of regularity ($d<r-1$) as is claimed in Theorem B. Moreover, we will need to use Theorem A to obtain Theorem B, thus explaining the dimension of the manifold, $m>c^{2}+c$, as is related to the codimension of the tangency.

1.6. Topology of families of diffeomorphisms

Set $\mathbb{I}=[-1,1]$. Given $0<d\leq r\leq\infty$, $k\geq 1$ and a manifold $\mathcal{M}$, we denote by $C^{d,r}(\mathbb{I}^{k},\mathcal{M})$ the space of $C^{d}$-families $f=(f_{a})_{a}$ of $C^{r}$-diffeomorphisms $f_{a}$ of $\mathcal{M}$ parameterized by $a\in\mathbb{I}^{k}$ such that

[TABLE]

We endow this space with the topology given by the $C^{d,r}$-norm

[TABLE]

In what follows we restrict our attention to $C^{d}$-families $f=(f_{a})_{a}$ of $C^{r}$-diffeomorphisms $f_{a}$ of a manifold $\mathcal{M}$ of dimension $m\geq 3$.

1.7. Structure of the paper

Section §2 contains the definition of a blender, one of the main tools in this paper. In section §3 we prove Theorem A. After that, we describe formally the notion of degenerate unfoldings in section §4. In §5 we recall and develop the notion of parablenders, the second main tool of the paper. Finally in sections §6 and §7 we prove Theorems C and B respectively.

2. Blenders

We attribute the following definition to Bonatti and Díaz (see [BBD16]). Blenders were initially defined having central dimension $c=1$ (see [BD96, BDV05, BD12]) and blenders with large central dimension were first studied in [NP12, BKR14, BR17]. After that, they also appeared in [BR18, ACW17] and in holomorphic dynamics in [Bie16, Duj17, Taf17].

Definition 2.1**.**

Let $f$ be a $C^{r}$-diffeomorphism of a manifold $\mathcal{M}$. A non-empty compact set $\Gamma\subset\mathcal{M}$ is a $cs$-blender of central dimension $c\geq 1$ if

1. i)

$\Gamma$* is a transitive, maximal invariant hyperbolic set in the closure of a neighborhood $\mathcal{U}$ having a partially hyperbolic splitting*

[TABLE]

where $E^{s}=E^{ss}\oplus E^{c}$ is the stable bundle, ${d_{ss}}=\dim E^{ss}\geq 1$ and $c=\dim E^{c}\geq 1$, 2. ii)

there exists a non-empty open set $\mathscr{D}$ of $C^{1}$-embeddings of $d_{ss}$-dimensional discs into $\mathcal{M}$, and 3. iii)

there exists a $C^{1}$-neighborhood $\mathscr{U}$ of $f$,

such that

[TABLE]

*where $\Gamma_{g}$ is the continuation of $\Gamma$ for $g$ and W^{u}_{loc}(\Gamma_{g})=\{x\in\mathcal{U}:g^{-n}(x)\in\mathcal{U}\ \text{for all n\geq 0}\}. The set $\mathscr{D}$ is called a superposition region of the blender. Finally, a $cu$-blender of central dimension $c$ is $cs$-blender of central dimension $c$ for $f^{-1}$. *

The hyperbolicity of $\Gamma$ implies that given a point $x\in W^{u}_{loc}(\Gamma)\cap\mathcal{D}$, there is a point $z\in\Gamma$ such that $x\in W^{u}_{loc}(z)\cap\mathcal{D}$. Observe that the local unstable manifold of $z$ is a $C^{1}$-embedded disc of dimension $d_{u}=\dim E^{u}$ and $\mathcal{D}$ is a $d_{ss}$-dimensional disc. These two discs are in relative general position if it holds that

[TABLE]

In this case, we have an intersection of codimension

[TABLE]

Thus, $W^{u}_{loc}(z)$ and $\mathcal{D}$ have a tangency of codimension at least $c$, which is in general, a quasi-transverse intersection of codimension exactly $c$.

2.1. Covering criterium

In [BKR14, BR17] blenders of large central dimension were constructed by using the covering criterium. Namely, we consider $C^{1}$-diffeomorphisms which are locally defined as a skew-product as explained below.

First, we consider a $C^{1}$-diffeomorphism $F$ of a manifold $\mathsf{N}$ having a horseshoe $\mathsf{\Lambda}$ contained in a local chart which is the maximal $F$-invariant set in the closure of some bounded open set $\mathsf{R}$ of $\mathsf{N}$. The horseshoe has stable index (dimension of the stable bundle) equal to $d_{ss}=\mathrm{ind}^{s}(\mathsf{\Lambda})>0$ and satisfies that

1. i)

$F|_{\Lambda}$ is conjugate to a shift of $\kappa$-symbols and 2. ii)

there exists $0<\nu<1$ such that

[TABLE]

Here $m(T)=\|T^{-1}\|^{-1}$ denotes the co-norm of a linear operator $T$. Let $\{\mathsf{R}_{1},\dots,\mathsf{R}_{\kappa}\}$ be an open covering of $\Lambda$, whose intersection with $\Lambda$ is a Markov partition. There is no loss of generality in assuming that $\mathsf{R}=\mathsf{R}_{1}\cup\dots\cup\mathsf{R}_{\kappa}$ with $\mathsf{R}_{\ell}=(-2,2)^{d_{ss}}\times I_{\ell}$, where $I_{\ell}$ is a product of $d_{u}$ open intervals in $[-2,2]$ with $\dim\mathsf{N}=d_{ss}+{d_{u}}$ for $\ell=1,\dots,\kappa$. Moreover, from the hyperbolicity of $\Lambda$, we can assume that there is a $DF^{-1}$-invariant cone-field on $\mathsf{R}$:

1. iii)

there exist $\alpha>0$ such that

[TABLE]

Here, for a given $\theta>0$, we denote

[TABLE]

We will call $\mathcal{C}^{ss}_{\alpha}$ as a stable cone-field on $\mathsf{R}$ of $F$ and refer to the parameter $\alpha$ as the width of the cone.

Now take $C^{1}$-diffeomorphisms $\phi_{1},\dots,\phi_{\kappa}$ of another manifold $M$ of dimension $c>0$, which are local $(\lambda,\beta)$-contractions in a bounded open set $D\subset M$, with $0<\lambda<\beta<1$:

[TABLE]

Finally, we consider a $C^{1}$-diffeomorphism $\Phi$ of $\mathcal{M}=\mathsf{N}\times M$ locally defined as a skew-product

[TABLE]

so that

[TABLE]

Notation**.**

In the rest of the paper, we will use the notation

[TABLE]

to define the skew-product map $\Psi(x,y)=(G(x),\psi_{\ell}(y))$ with $(x,y)\in\mathcal{V}_{\ell}$ for $\ell=1,\dots,\kappa$, where $\mathcal{V}_{1},\dots,\mathcal{V}_{\kappa}$ are pairwise disjoint sets.

The following theorem from [BKR14, Thm. C] and [BR17, Thm. 3.8] shows that under the assumption of domination and the covering criterium, the map $\Phi$ has a $cs$-blender of central dimension $c\geq 1$.

Theorem 2.2**.**

Let $\Phi$ be a $C^{1}$-diffeomorphism of a manifold $\mathcal{M}$ locally defined as a skew-product $\Phi=F\ltimes(\phi_{1},\dots,\phi_{\kappa})$ on $\mathcal{U}$ as above. Assume that

1. i)

the hyperbolic base $F|_{\mathsf{\Lambda}}$ dominates the fiber dynamics $\phi_{\ell}$, i.e, it holds that $\nu<\lambda$, 2. ii)

there exists an open set $B\subset D$ such that $\overline{B}\subset\phi_{1}(B)\cup\dots\cup\phi_{\kappa}(B).$

Then the maximal invariant set $\Gamma$ of $\Phi$ in $\overline{\mathcal{U}}$ is a $cs$-blender of central dimension $c$. The superposition region of the blender is the family of $({\alpha,}\nu,\delta)$-horizontal, $d_{ss}$-dimensional $C^{1}$-discs into $\mathcal{B}=\mathsf{R}\times B$, where $0<\delta<\lambda L/2$ and $L>0$ is the Lebesgue number of the cover of $B$ in (ii).

The open set $\mathcal{B}=\mathsf{R}\times B$ of $\mathcal{M}$ is called a superposition domain. Also, in the above theorem appears the notion of a family of $({\alpha,}\nu,\delta)$-horizontal discs in $\mathcal{B}$ that we define as follows.

A proper $C^{r}$-embedded $d_{ss}$-dimensional disc $\mathcal{D}$ into $\mathcal{B}=\mathsf{R}\times B$ (or a $d_{ss}$-dimensional $C^{r}$-disc in $\mathcal{B}$ for short) will be an injective $C^{r}$-immersion $\mathcal{D}:[-2,2]^{d_{ss}}\to\overline{\mathcal{B}}$ of the form,

[TABLE]

As usual, we will identify the embedding $\mathcal{D}$ with its image $\mathcal{D}([-2,2]^{d_{ss}})$.

Definition 2.3**.**

We say that a $d_{ss}$-dimensional $C^{1}$-disc $\mathcal{D}$ in $\mathcal{B}=\mathsf{R}\times B$ is $({\alpha,}\nu,\delta)$-horizontal if

1. i)

$\|Dg\|_{\infty}\leq\alpha$, 2. ii)

*there is a point $y\in B$ such that $d(y,h(\xi))<\delta$ for all $\xi\in[-2,2]^{d_{ss}}$, * 3. iii)

${C\cdot\nu}<\delta$* where $C\geq 0$ is a Lipschitz constant of $h$, i.e.,*

[TABLE]

Since $\mathcal{D}$ is a $C^{1}$-disc notice that $C$ is any positive constant satisfying $\|Dh\|_{\infty}\leq C$. If $C=0$ we say that $\mathcal{D}$ is horizontal. Notice that by condition (i), the disc $\mathcal{D}$ is tangent to the stable cone-field defined on $\mathcal{B}$. Moreover, from condition (ii), $\mathcal{D}$ is $C^{0}$-close to an horizontal disc. On the other hand, although $\mathcal{D}$ may not be $C^{1}$-close to a horizontal disc, condition (iii) asks that we still have a good control of the distortion. Finally, for a fixed $\alpha$, $\nu$ and $\delta$ under the conditions in Theorem 2.2, the set of $({\alpha,}\nu,\delta)$-horizontal discs in $\mathcal{B}$ is said to be, for short, the family of almost-horizontal discs.

In the next sections we construct diffeomorphisms having robust tangencies in any manifold of dimension $m\geq 3$. Our constructions will use the following particular class of blenders obtained from the covering criterium.

2.2. Affine blender

We will introduce a class of $C^{r}$-diffeomorphims $f$ of $\mathbb{R}^{m}=\mathbb{R}^{n}\times\mathbb{R}^{c}$ with $r\geq 1$, $n=ss+u\geq 2$ and $c\geq 1$. To do this, consider first a $C^{r}$-diffeomorphism $F$ of $\mathbb{R}^{n}$ having a horseshoe $\mathsf{\Lambda}$ in the the open cube $\mathsf{V}=(-2,2)^{n}$. The horseshoe has stable index $ss=\mathrm{ind}^{s}(\mathsf{\Lambda})>0$ and $F|_{\Lambda}$ is conjugate to a full shift of a large number $\kappa$ of symbols to be specified later. We notice that this number will depend only on the dimension $c$. For simplicity, assume that

[TABLE]

is a Markov partition of $\mathsf{\Lambda}$ where $I_{\ell}$ is an open disc in $[-2,2]^{u}$ and $F$ is affine on each rectangle $\mathsf{R}_{\ell}$. More precisely, there are $0<\nu<1$ and linear maps $S_{\ell}:\mathbb{R}^{ss}\to\mathbb{R}^{ss}$ and $U_{\ell}:\mathbb{R}^{u}\to\mathbb{R}^{u}$ such that

[TABLE]

Notice that,

[TABLE]

where the cones $\mathcal{C}^{ss}_{\alpha}$ and $\mathcal{C}^{ss}_{\nu^{2}\alpha}$ are defined as in (2). In particular, the cone-field $\mathcal{C}^{ss}_{\alpha}$ is $DF^{-1}$-invariant.

Take affine $(\lambda,\beta)$-contractions $\phi_{1},\dots,\phi_{\kappa}$ on $D=(-2,2)^{c}$ with $\nu<\lambda<\beta<1$. That is, $C^{r}$-diffeomorphisms $\phi_{\ell}$ of $\mathbb{R}^{c}$ such that $\phi_{\ell}(\overline{D})\subset D$ and there are linear maps $T_{\ell}:\mathbb{R}^{c}\to\mathbb{R}^{c}$ so that

[TABLE]

Moreover, we ask that there is an open set $B\subset D$ containing the origin such that

[TABLE]

Example 2.4**.**

Take $\phi_{\pm}(t)=\lambda t\pm(1-\lambda)$ for $t\in[-2,2]$ with $1/2<\lambda<1$ and consider

[TABLE]

Observe that here $\kappa=2^{c}$. It is not difficult to see that $B=(-1,1)^{c}$ satisfies (3).

Finally we consider a $C^{r}$-diffeomorphism $\Phi$ of $\mathbb{R}^{m}$ locally defined as the skew-product

[TABLE]

According to Theorem 2.2, the maximal invariant set $\Gamma$ in $\overline{\mathcal{U}}$ is a $cs$-blender of central dimension $c>0$. Moreover, the superposition region is the family of almost-horizontal $d_{ss}$-dimensional $C^{1}$-discs in $\mathcal{B}=\mathsf{R}\times B$, where $\mathsf{R}=\mathsf{R}_{1}\cup\dots\cup\mathsf{R}_{\kappa}$.

3. Robust homoclinic tangencies

In this section we prove Theorem A. We provide the existence of $C^{r}$-diffeomorphisms with $r\geq 2$ having $C^{2}$-robust homoclinic tangencies of large codimension by constructing these objets in local coordinates. Thus, we may consider $\mathbb{R}^{m}=\mathbb{R}^{n}\times\mathbb{R}^{c}$ with $n\geq 2$ and $c\geq 1$. Throughout this section, we ask that $n=ss+u$ and $c=u^{2}$ but we keep the notation $u$, $c$ in order to distinguish coordinates. We divide the proof into several parts and for the convenience of the reader will explain next the ideas involved.

We study the homoclinic tangencies of $f$ by analyzing the induced map $f^{G}$ on Grassmannian manifolds, and we would like for this induced map to have a blender. The main idea is to obtain robust tangencies for $f$ by means of a robust intersection between the local unstable manifolds of a blender (for the induced dynamics) and a particular disc in the superposition region. Hence, in §3.2 we will construct a class of $C^{r}$-diffeomorphisms of $\mathbb{R}^{m}$ which induces a $cs$-blender $\Gamma^{G}$ on the Grassmannian manifold. This class of diffeomorphisms are the locally defined skew-product maps having an affine blender $\Gamma$ introduced in §2.2 with some additional restrictions. Afterwards, we introduce in §3.3 the notion of a folding manifold $\mathcal{S}$ in $\mathbb{R}^{m}$, having the main property of inducing a disc $\mathcal{S}^{G}$ in the superposition region of $\Gamma^{G}$. Finally, in §3.4 we show how the robust intersection between the local unstable manifolds of $\Gamma^{G}$ and the induced disc $\mathcal{S}^{G}$ provides a robust tangency between the unstable manifolds of $\Gamma$ and the folding manifold $\mathcal{S}$. One can see the folding manifold as a piece of a leaf of the stable manifold of $\Gamma$ and then the proof of Theorem A can be concluded in §3.5.

3.1. Grassmannian manifold

Let $f$ be a $C^{r}$-diffeomorphism of $\mathbb{R}^{m}$. We will consider an induced map by $f$ on the Grassmannian manifold $G_{u}(\mathbb{R}^{m})=\mathbb{R}^{m}\times G(u,m)$ given by

[TABLE]

where $G(u,m)$ is the set of $u$-planes in $\mathbb{R}^{m}$. Notice that $f^{{G}}$ is a $C^{r-1}$-diffeomorphism of ${G}_{u}(\mathbb{R}^{m})$.

3.2. Blender induced on the Grassmannian manifold

Fix $r\geq 2$. We will start by considering a $C^{r}$-diffeomorphism $\Phi$ of $\mathbb{R}^{m}$ locally defined as a skew-product $\Phi=F\ltimes(\phi_{1},\dots,\phi_{\kappa})$ and having an affine $cs$-blender $\Gamma$, as in §2.2. Notice that for each $\ell=1,\dots,\kappa$, the differential map $D\Phi(x,y)$ is the same linear map $D\Phi_{\ell}$ for all $(x,y)\in\mathsf{R}_{\ell}\times D$. Moreover, $E^{u}=\{0^{ss}\}\times\mathbb{R}^{u}\times\{0^{c}\}$ is an attracting fixed point of the action of these maps on $G(u,m)$ with eigenvalues less than $\beta\nu<1$. Let $\mathcal{C}^{{G}}$ be an open neighborhood of $E^{u}$ in $G(u,m)$ so that $D\Phi_{\ell}\cdot\mathcal{C}^{{G}}\subset\mathrm{int}(\mathcal{C}^{{G}})$ for all $\ell=1,\dots,\kappa$. The Grassmannian induced map $\Phi^{{G}}$ restricted to $\mathsf{R}_{\ell}\times D\times\mathcal{C}^{{G}}$ is given by

[TABLE]

By a change of coordinates we can write $\Phi^{{G}}$ restricted to $\mathcal{U}^{{G}}=\mathcal{U}\times\mathcal{C}^{{G}}$ as the skew-product

[TABLE]

where

[TABLE]

Moreover, $F^{{G}}$ has a horseshoe ${\mathsf{\Lambda}}^{{G}}=\mathsf{\Lambda}\times\{E^{u}\}$ with stable index

[TABLE]

Observe that shrinking $\mathcal{C}^{G}$ if necessary, the contraction of $D\Phi_{\ell}$ on $\mathcal{C}^{G}$ dominates the contraction of $F$ (that is $\beta\nu<\nu$). Then, the width $\alpha>0$ of the stable cone-field on $\mathsf{R}\times\mathcal{C}^{G}$ of $F^{G}$ is the same width $\alpha$ that we have for the stable cone-field on $\mathsf{R}$ of $F$.

Since $\beta\nu<\nu<\lambda$ then $F^{{G}}|_{{\mathsf{\Lambda}}^{{G}}}$ dominates the fiber dynamics given by $\phi_{1},\dots,\phi_{\kappa}$. By Theorem 2.2, we have that $\Gamma^{{G}}=\Gamma\times\{E^{u}\}$ is a $cs$-blender of central dimension $c>0$ of $\Phi^{{G}}$. Moreover, the family of $({\alpha,}\nu,\delta)$-horizontal, $d^{{G}}_{ss}$-dimensional $C^{1}$-discs in ${\mathcal{B}}^{{G}}=\mathcal{B}\times\mathcal{C}^{{G}}$ is a superposition region of the blender $\Gamma^{{G}}$. Here, as in §2.2, $0<\delta<\lambda L/2$ whereas $L$ is the Lebesgue number of the cover (3).

3.3. Folding manifold with respect to the affine blender

Next we introduce the notion of a folding manifold. To do this, we will consider a submanifold $\mathcal{S}$ of $\mathbb{R}^{m}$ of dimension $ss+c$. In what follows we identify canonically the tangent space $T_{z}\mathcal{S}$ of $\mathcal{S}$ at $x$ with a subspace of $\mathbb{R}^{m}$.

Definition 3.1**.**

We say that $\mathcal{S}$ is a $({\alpha,}\nu,\delta)$-folding $C^{r}$-manifold with respect to $\mathcal{B}^{{G}}=\mathcal{B}\times\mathcal{C}^{{G}}$ if there is $\epsilon>0$ such that

1. i)

$\mathcal{S}$* is parameterized as a $(ss+c)$-dimensional $C^{r}$-embedding $\mathcal{S}:[-2,2]^{ss}\times[-\epsilon,\epsilon]^{c}\to\overline{\mathcal{B}}$, of the form*

[TABLE]

with $x\in[-2,2]^{ss}$ and $t=(t_{1},\dots,t_{u},\dots,t_{c})\in\mathbb{R}^{c}$; 2. ii)

*there is $y\in B$ such that $d(h(x,t),y)<\delta$ for all $(x,t)\in[-2,2]^{ss}\times[-\epsilon,\epsilon]^{c}$; * 3. iii)

for all $x\in[-2,2]^{ss}$ and $E\in{\overline{\mathcal{C}^{G}}}$ there is a unique $t\in[-\epsilon,\epsilon]^{c}$ such that $E$ is a subspace of $T_{z}\mathcal{S}$ with $z=\mathcal{S}(x,t)$. Moreover, $t=t(x,E)$ varies $C^{r-1}$-continuously with $(x,E)$ and

[TABLE]

Let us explain geometrically the above notion of a folding manifold. First define the unstable cone for some small $\theta>0$ as

[TABLE]

Each vector subspace $E$ of dimension $u$ contained in the cone $\mathcal{C}^{u}_{\theta}$ can be identified with an element of $\mathcal{C}^{G}$ and vice-versa. Then condition (iii) implies that for every $x\in[-2,2]^{ss}$,

[TABLE]

In fact, the uniqueness in condition (iii) implies the injectivity of the map $t\mapsto T_{\gamma(t)}\mathcal{S}$ and thus, the parameters $\epsilon$ and $\theta$ can be interpreted as the size of the neighborhood $\mathcal{C}^{G}$ of $E^{u}=\{0^{ss}\}\times\mathbb{R}^{u}\times\{0^{c}\}$ in $G(u,m)$.

Next we will show an example of a $({\alpha,}\nu,\delta)$-folding manifold with respect to $\mathcal{B}^{G}=\mathcal{B}\times\mathcal{C}^{G}$. Recall that $B$ is an open set of $\mathbb{R}^{c}$ containing the origin. Up to a conjugacy with a translation, we can assume that $\mathcal{B}=\mathsf{R}\times B$ contains $(-2,2)^{ss}\times\{0^{u}\}\times\{0^{c}\}$.

Example 3.2**.**

Consider the $(ss+c)$-dimensional embedding given by

[TABLE]

where $x\in\mathbb{R}^{ss}$, $t=(t_{1},\dots,t_{u},t_{u+1},\dots,t_{c})\in[-\epsilon,\epsilon]^{c}$ and ${H}(t)=({H}_{1}(t),\dots,{H_{u}}(t))$ with

[TABLE]

For a fixed $\delta>0$, we will prove that $\mathcal{S}$ is a $({\alpha,}\nu,\delta)$-folding $C^{\infty}$-manifold for any $\alpha>0$ large enough and $\epsilon,\nu>0$ small enough. To do this, we will show that $\mathcal{S}$ satisfies all the conditions of Definition 3.1.

It is straightforward that $\mathcal{S}$ is a $(ss+c)$-dimensional $C^{\infty}$-embedding. Since $\mathcal{B}$ contains $(-2,2)^{ss}\times\{0^{u}\}\times\{0^{c}\}$, then $\mathcal{S}([-2,2]^{ss}\times[-\epsilon,\epsilon]^{c})\subset\mathcal{B}$ for any $\epsilon>0$ small enough, concluding the first condition in Definition 3.1. Observe now that the central coordinate of $\mathcal{S}$, i.e., the map $h(t)=\mathscr{P}\circ\mathcal{S}(x,t)$ does not depend on $x$. Moreover, if $\epsilon>0$ is small enough then $d(h(t),0^{c})<\delta$ for all $(x,t)\in[-2,2]^{ss}\times[-\epsilon,\epsilon]^{c}$, as is required by the second condition in Definition 3.1. To conclude that $\mathcal{S}$ is a folding manifold, it only remains to prove the last condition in Definition 3.1, which is somewhat longer and will be done in the next paragraphs.

By a direct computation, $T_{z}\mathcal{S}$ at $z=\mathcal{S}(x,t)$ is given by

[TABLE]

where $x^{\prime}\in\mathbb{R}^{ss}$, $t^{\prime}=(t^{\prime}_{1},\dots,t^{\prime}_{u},t^{\prime}_{u+1},\dots,t^{\prime}_{c})\in\mathbb{R}^{c}$ and $g(t^{\prime},t)=(g_{1}(t^{\prime},t),\dots,g_{{u}}(t^{\prime},t))$ with

[TABLE]

We want to prove that for any $x\in[-2,2]$ and $E\in\overline{\mathcal{C}^{G}}$, there is a unique $t\in[-\epsilon,\epsilon]^{c}$ such that $E$ is a subspace of $T_{z}\mathcal{S}$ for $z=\mathcal{S}(x,t)$. Observe that if $E=\langle v_{k}:k=1,\dots,u\rangle$ is generated by linearly independent vectors $v_{k}$, then $E$ is a subspace of $T_{z}\mathcal{S}$ if and only if $v_{k}\in T_{z}\mathcal{S}$ for all $k=1,\dots,u$. Denoting $v_{k}=(a_{k},b_{k},c_{k})\in\mathbb{R}^{ss}\times\mathbb{R}^{u}\times\mathbb{R}^{c}$, the above condition is equivalent to the existence of $t\in[-\epsilon,\epsilon]^{c}$ such that for every $k=1,\dots,u$, there are $x_{k}^{\prime}\in\mathbb{R}^{ss}$, and $t^{\prime}_{k}\in\mathbb{R}^{c}$ satisfying:

[TABLE]

Notice that $g_{i}(t^{\prime}_{k},t)$ can be written as a scalar product of $t$ by a vector $\vec{a}_{ki}\in\mathbb{R}^{c}$ that depends on $t^{\prime}_{k}$. Thus, having into account that $c=u^{2}$, we can write the relation $c_{ki}=g_{i}(t^{\prime}_{k},t)$ for $k=1,\dots,u$ and $i=1,\dots,u$ as a matrix product

[TABLE]

and $A{=[\vec{a}_{11};\dots;\vec{a}_{1u};\dots;\vec{a}_{u1};\dots;\vec{a}_{uu}]}$ is a $c$-by-$c$ matrix that depends on $t^{\prime}_{k}$ for $k=1,\dots,u$. In fact, since $t^{\prime}_{k}$ form part of the coordinates of the vector $v_{k}$, then $A=A(E)$ depends on the vector space $E$. Similarly this holds for $\vec{c}=\vec{c}(E)$. Hence, to find the required $t\in[-\epsilon,\epsilon]^{c}$ we only need to show that the linear system $A(E)\cdot t=\vec{c}(E)$ is uniquely solved.

To do this, we will analyze the determinant of $A$ at $E^{u}=\{0^{ss}\}\times\mathbb{R}^{u}\times\{0^{c}\}$ in which the open set $\mathcal{C}^{G}$ is centered. Observe that $E^{u}$ is generated by the vectors $e^{u}_{k}=(0^{ss},e_{k},0^{c})$ for $k=1,\dots,u$, where $e_{k}$ is the $k$-th canonical vector in $\mathbb{R}^{u}$. Thus, $t^{\prime}_{k}=(e_{k},0^{c-u})$ and then

[TABLE]

where $\delta_{ki}$ is the Kronecker delta. In view of this, $A(E^{u})=\mathrm{Id}+\mathrm{L}$ where $\mathrm{Id}$ is the identity matrix and $\mathrm{L}$ is a matrix whose first column is given by $(e_{1},e_{2},\dots,e_{u})^{T}$ and the rest of the elements are zero. Hence $A(E^{u})$ is a triangular matrix with $\det A(E^{u})=2\not=0$. Thus, we get that $A(E)\cdot t=\vec{c}(E)$ is uniquely solved for any $E$ close enough to $E^{u}$.

This shows the first part of the last condition in Definition 3.1 but still we need to prove that $C\nu<\delta$ where $C=\|Dh\|_{\infty}\cdot\max\{1,\|Dt\|_{\infty}\}$ and $\|Dt\|_{\infty}\leq\alpha$. Since both $h=h(t)$ and $t=t(E)$ are functions of class $C^{{\infty}}$, then $C<\infty$ over $[-2,2]^{ss}\times\mathcal{C}^{G}$ and $\|Dt\|_{\infty}<\infty$. Thus, this condition trivially holds by taking $\nu>0$ small enough and $\alpha$ large enough.

Remark 3.3**.**

In Proposition A.1 in Appendix A we show that actually $\mathcal{S}$ in the above example is a $({\alpha,}\nu,\delta)$-folding ${C^{\infty}}$-manifold for any $\nu<\delta$ and $\alpha>1$.

Remark 3.4**.**

Fixing a small enough $\epsilon>0$, for which $\mathcal{S}$ is a $({\alpha,}\nu,\delta)$-folding manifold, the above example is $C^{2}$-robust in the following sense. Let $\tilde{\mathcal{S}}:[-2,2]^{ss}\times[-\epsilon,\epsilon]^{c}\to\overline{\mathcal{B}}$ be $(ss+c)$-dimensional $C^{2}$-embedding, which is $C^{2}$- sufficiently close to $\mathcal{S}$. Then $\tilde{\mathcal{S}}$ is also a $({\alpha,}\nu,\delta)$-folding manifold. Let us comment on why this is true. What has to be shown is basically condition (iii) of Definition 3.1. The function $\tilde{\mathcal{S}}$ can be written in the form

[TABLE]

Here $H(t)$ comes from the embedding $\mathcal{S}$, and $\kappa(x,t)$ is a small $C^{2}$-perturbation. Following the notation of the previous example, the non-linear equations that now have to be solved take the form

[TABLE]

where $\varkappa_{i}$ are functions which depend on $(x,t,x^{\prime},t^{\prime})$ with small $C^{1}$-derivative. The solution of these equations for $t$ given $x$ and $E$, is then guaranteed by an application of the Implicit Function Theorem.

Let $\mathcal{S}$ be a $({\alpha,}\nu,\delta)$-folding $C^{r}$-manifold with respect to $\mathcal{B}^{{G}}=\mathcal{B}\times\mathcal{C}^{{G}}$. Consider

[TABLE]

One can see $\mathcal{S}^{{G}}$ as a fiber bundle over $\mathcal{S}$ with fibers

[TABLE]

Notice that $(\mathcal{S}^{{G}})_{z}$ is a compact manifold of dimension $\dim G(u,ss+c)=u(ss+c-u)$. Then, the dimension of $\mathcal{S}^{{G}}$ is $ss+c+u(ss+c-u)={ss+}u(m-u)+c-u^{2}$. In fact, since $c=u^{2}$ we have that this dimension coincides with $d^{{G}}_{ss}=ss+u(m-u)$.

Lemma 3.5**.**

The set $\mathcal{H}^{{G}}=\mathcal{S}^{{G}}\cap\overline{\mathcal{B}^{{G}}}$ is a (${\alpha,}\nu,\delta)$-horizontal $d^{{G}}_{ss}$-dimensional $C^{{r-1}}$-disc in $\mathcal{B}^{{G}}$.

Proof.

First notice that $\mathcal{H}^{{G}}$ is a $d^{{G}}_{ss}$-dimensional $C^{{r-1}}$-disc in $\mathcal{B}^{{G}}$. This follows from Definition 3.1, since given any $x\in[-2,2]^{ss}$ and any $E\in\overline{\mathcal{C}^{{G}}}$ we have a unique $t=t(x,E)\in[{-\epsilon,\epsilon}]^{c}$ which varies $C^{r}$-continuously with $x$ and $E$ such that $E\subset T_{z}\mathcal{S}$ with $z=\mathcal{S}(x,t)$. Thus we can parametrize $\mathcal{H}^{G}$ as

[TABLE]

is a $C^{{r-1}}$-disc in $\mathcal{B}^{{G}}$. On the other hand, the unstable and central coordinates of this disc are given by

[TABLE]

where

[TABLE]

are the unstable and central coordinates of the folding manifold $\mathcal{S}$. Here $\mathscr{P}_{u}$ and $\mathscr{P}_{c}$ denote the canonical projections on $\mathbb{R}^{u}$ and $\mathbb{R}^{c}$ respectively. Hence, again, by the definition of folding manifold we have $y\in B$ such that $d(h^{{G}}(x,E),y)=d(h(x,t(x,E)),y)<\delta$,

[TABLE]

and

[TABLE]

This proves that $\mathcal{H}^{{G}}$ is $({\alpha,}\nu,\delta)$-horizontal disc concludes the proof. ∎

Remark 3.6**.**

The $({\alpha,}\nu,\delta)$-horizontal $C^{{r-1}}$-disc $\mathcal{H}^{{G}}$ obtained from the $C^{r}$-folding manifold in Example 3.2 is $C^{0}$-close to a horizontal disc but $C^{1}$-far from it.

3.4. Robust tangencies with a folding manifold

Recall that the $C^{r}$-diffeomorphism $\Phi$ of $\mathbb{R}^{m}$ we are considering in this section was introduced in §3.2. This map has a $cs$-blender $\Gamma$ of central dimension $c>0$, where a superposition region contains the family of $({\alpha,}\nu,\delta)$-horizonal discs in $\mathcal{B}$ with $\delta<\lambda L/2$. Now, we will prove the following key result:

Proposition 3.7**.**

There is a $C^{2}$-neighborhood $\mathscr{U}$ of $\Phi$ such that for any $({\alpha,}\nu,\delta)$-folding $C^{r}$-manifold $\mathcal{S}$ with respect to $\mathcal{B}^{G}=\mathcal{B}\times\mathcal{C}^{G}$ it holds that for any $g\in\mathscr{U}$ there are points $z\in\Gamma_{g}$ and $x\in W^{u}_{loc}(z)\cap\mathcal{S}$ such that

[TABLE]

In particular, since the codimension of $W^{u}_{loc}(z)$ coincides with the dimension of $\mathcal{S}$, these two manifolds intersect at $x$ in a tangency of codimension $u$.

Proof.

We recall that $\Gamma^{{G}}=\Gamma\times\{E^{u}\}$ is a $cs$-blender of central dimension $c>0$ for the induced $C^{1}$-diffeomorphism $\Phi^{{G}}$, whose superposition region contains the set $\mathscr{D}$ of $({\alpha,}\nu,\delta)$-horizontal $d^{{G}}_{ss}$-dimensional $C^{1}$-discs in $\mathcal{B}^{{G}}=\mathcal{B}\times\mathcal{C}^{{G}}$. Hence, by definition of a blender, there is a $C^{1}$-neighborhood $\mathscr{U}^{{G}}$ of $\Phi^{{G}}$ where for each map $\Psi\in\mathscr{U}^{G}$ we have an intersection between each disc in $\mathscr{D}$ and the local unstable manifold of the continuation of $\Gamma^{{G}}$ for $\Psi$. We take a $C^{2}$-neighborhood $\mathscr{U}$ of $\Phi$ so that for every $g\in\mathscr{U}$ its induced $C^{1}$-diffeomorphism $g^{{G}}$ on $G_{u}(\mathbb{R}^{m})$ belongs to $\mathscr{U}^{{G}}$. Hence, the continuation $\Gamma^{G}_{g}$ of $\Gamma^{G}$ for $g^{{G}}$ is a $cs$-blender. Moreover, $W^{u}_{loc}(\Gamma_{g}^{{G}})$ is laminated by plaques of dimension $u$ which project one-to-one onto $W^{u}_{loc}(\Gamma_{g})$. In particular, shrinking $\mathscr{U}$ if necessary,

[TABLE]

On the hand, if $\mathcal{S}$ is a $({\alpha,}\nu,\delta)$-folding manifold with respect to $\mathcal{B}^{G}$, then by Lemma 3.5, the manifold $\mathcal{S}^{{G}}$ contains a ${({\alpha,}\nu,\delta)}$-horizontal $d^{{G}}_{ss}$-dimensional $C^{1}$-disc $\mathcal{H}^{{G}}$ in $\mathcal{B}^{{G}}$. Hence $\mathcal{H}^{{G}}\in\mathscr{D}$. Thus, $W^{u}_{loc}(\Gamma_{g}^{{G}})\cap\mathcal{H}^{{G}}\not=\emptyset$. Consequently, there is $(x,E)$ belonging to $\mathcal{H}^{{G}}\subset\mathcal{S}^{{G}}$ and $W^{u}_{loc}(\Gamma_{g}^{{G}})$. In particular, from (6), we get that $x\in\mathcal{S}\cap W^{u}_{loc}(\Gamma_{g})$ and $T_{x}W^{u}_{loc}(z)=E\subset T_{x}\mathcal{S}$ for some $z\in\Gamma_{g}$. This completes the proof. ∎

3.5. Proof of Theorem A

Finally we prove Theorem A by assuming that the global stable manifold of a periodic point $P$ in the affine $cs$-blender $\Gamma$ contains the folding manifold with respect to $\mathcal{B}^{{G}}=\mathcal{B}\times\mathcal{C}^{{G}}$ given in Example 3.2. As was explained in Remark 3.4 this folding manifold is $C^{2}$-robust. Thus, the stable manifold $W^{s}(P_{g})$ of the continuation $P_{g}$ of $P$ contains a folding manifold with respect to $\mathcal{B}^{{G}}$ for all small enough $C^{2}$-perturbations $g$ of $\Phi$. Then, Proposition 3.7 implies that there is $z\in\Gamma_{g}$ such that $W^{u}_{loc}(z)$ and $W^{s}(P_{g})$ have a tangency of codimension $u>0$. Thus, we get that $\Phi$ has a $C^{2}$-robust homoclininc tangency of codimension $u$. Moreover, using  (5) we can conclude that the tangency cannot be inside a strong partially hyperbolic set. To see this, notice that $T_{x}\mathcal{S}=E^{ss}\oplus F$ where $E^{ss}=\mathbb{R}^{ss}\times\{0^{u}\}\times\{0^{c}\}$, $T_{x}W^{u}_{loc}(z)\subset F$ and $T_{x}W^{u}_{loc}(z)\in\mathcal{C}^{G}$. Oberve that $E^{ss}$ is in the strong stable cone-field on $\mathcal{B}$ of $\Phi$, while $\mathcal{C}^{G}$ can be identified with the unstable cone on $\mathcal{B}$ of $\Phi$. Since both cone-fields are disjoint, we obtain that $T_{x}W^{u}_{loc}(z)$ cannot be the strong stable direction. This proves that the tangency must be inside a weak partially hyperbolic set. Finally, recall that $c=u^{2}$ and then $m=ss+c+u>u^{2}+u$, completing the proof.

4. Degenerate unfoldings of tangencies

Recall that by a tangency we understand the opposite of a transverse intersection. We will introduce the notion of degenerate unfoldings of a tangency between two submanifolds $\mathcal{L}_{0}$ and $\mathcal{S}_{0}$. Let $\mathcal{L}=(\mathcal{L}_{a})_{a}$ and $\mathcal{S}=(\mathcal{S}_{a})_{a}$ be $k$-parameter families of submanifolds $\mathcal{L}_{a}$ and $\mathcal{S}_{a}$ of $\mathcal{M}$ diffeomorphic to $\mathcal{L}_{0}$ and $\mathcal{S}_{0}$ respectively by families of diffeomorphisms $C^{d,r}$-close to the identity with $0<d\leq r$.

Definition 4.1**.**

We say that $\mathcal{L}$ and $\mathcal{S}$ has a tangency at $a=0$ which unfolds $C^{d}$-degenerate if there exist $x=(x_{a})_{a},y=(y_{a})_{a}\in C^{d}(\mathbb{I}^{k},\mathcal{M})$ such that

[TABLE]

A useful formalism to define a $C^{d}$-degenerate unfolding of a tangency is to consider the space of jets $J^{d}_{0}(\mathbb{I}^{k},\mathcal{M})$ whose elements are the coefficients of the truncated Taylor series at $a=0$,

[TABLE]

For a more precise definition see §5.1. Then

[TABLE]

The set $J_{0}^{d}(\mathbb{I}^{k},\mathcal{M})$ can be endowed with a smooth manifold structure sometimes called the manifold of $(d,k)$-velocities over $\mathcal{M}$.

Next, we will be interested in unfoldings which control not only the separation of points on the manifold, but also the separation of the tangent spaces. Hence, to control this separation, we assume $d<r$ and introduce the following definition. Let $G_{{\ell}}(\mathcal{M})$ be the ${\ell}$-th Grassmannian bundle of $\mathcal{M}$. That is, the fiber bundle over $\mathcal{M}$ whose fibers are the ${\ell}$-th Grassmannian manifold of the tangent space $T_{p}\mathcal{M}$, i.e.,

[TABLE]

where $G({{\ell}},T_{p}\mathcal{M})$ is the set of all ${{\ell}}$-dimensional linear subspaces of $T_{p}M$.

Definition 4.2**.**

We say that $\mathcal{L}$ and $\mathcal{S}$ has a tangency of dimension $\ell>0$ at $a=0$ which unfolds $C^{d}$-degenerate if there exist $x=(x_{a})_{a},y=(y_{a})_{a}\in C^{d}(\mathbb{I}^{k},G_{{\ell}}(\mathcal{M}))$ such that

[TABLE]

*Using the formalism of jets, the unfolding is $C^{d}$-degenerate if and only if $J^{d}_{0}(x)=J_{0}^{d}(y)$. *

Remark 4.3**.**

In the terminology of [Ber16], $C^{d}$-degenerate unfoldings of a tangency of dimension one (between curves in dimension two) are called $C^{d}$-paratangencies.

5. Parablenders

The concept of parablender was initially introduced by Berger [Ber16] for endomorphisms (see also [BCP16, Ber17a, Ber17b]). The following generalizes both, the blender (Definition 2.1) and the definition of parablender for diffeomorphisms given also by Berger in [Ber17b, Example 1.21,Def. 1.23].

Definition 5.1**.**

Let $\Gamma_{0}$ be a $cs$-blender of central dimension $c\geq 1$ and strong stable dimension $d_{ss}=\mathrm{Ind}^{s}(\Gamma_{0})-c$ of a $C^{r}$-diffeomorphism $f_{0}$ of $\mathcal{M}$. Consider a $k$-parameter $C^{d}$-family $f=(f_{a})_{a}$ of $C^{r}$-diffeomorphisms of $\mathcal{M}$ unfolding $f_{0}$ at $a=0$. A family $\Gamma=(\Gamma_{a})_{a}$ of compact sets $\Gamma_{a}$ of $\mathcal{M}$ is said to be * $C^{d}$-$cs$-parablender at $a=0$ of central dimension $c$ for $f$ if*

1. i)

$\Gamma_{a}$* is the continuation for $f_{a}$ of the $cs$-blender $\Gamma_{0}$ for all $a\in\mathbb{I}^{k}$,* 2. ii)

there exists an open set $\mathscr{D}$ of $k$-parameter $C^{d}$-families $\mathcal{D}=(\mathcal{D}_{a})_{a}$, where each $\mathcal{D}_{a}$ is a $C^{r}$-embedded $d_{ss}$-dimensional disc into $\mathcal{M}$, 3. iii)

there exists a $C^{d,r}$-neighborhood $\mathscr{U}$ of $\Phi$,

such that for every $g=(g_{a})_{a}\in\mathscr{U}$ and $\mathcal{D}=(\mathcal{D}_{a})_{a}\in\mathscr{D}^{ss}$ it holds that

[TABLE]

That is, there are $x=(x_{a})_{a}$, $y=(y_{a})_{a}$, $z=(z_{a})_{a}\in C^{d}(\mathbb{I}^{k},\mathcal{M})$ with

[TABLE]

where $\Gamma_{a,g}$ is the continuation for $g_{a}$ of the $cs$-blender $\Gamma_{a}$ for all $a\in\mathbb{I}^{k}$.

*A $C^{d}$-$cu$-parablender at $a=0$ of central dimension $c$ is $C^{d}$-$cs$-parablender for $f^{-1}=(f^{-1}_{a})_{a}$. *

Remark 5.2**.**

For $k=0$ and $d=r=1$, i.e., when there are no parameters and the class is $C^{1}$, the above definition of a parablender coincides with the definition of a blender. As was mentioned after Definition 2.1, the tangency between $W^{u}_{loc}(z_{0})$ and $\mathcal{D}_{0}$ has codimension at least $c$. In general, it is a quasi-transverse intersection of codimension exactly $c$.

Remark 5.3**.**

For simplicity, to introduce parablenders, we have chosen the parameter $a=0$. However, we can also define a parablender at any other parameter $a=a_{0}$ with $a_{0}\in\mathbb{I}^{k}$. Moreover, we will say that a $k$-parameter family $f=(f_{a})_{a}$ has a parablender $\Gamma=(\Gamma_{a})_{a}$ at any parameter when $\Gamma$ is a parablender for $f$ at $a=a_{0}$ for all $a_{0}\in\mathbb{I}^{k}$ with $\mathscr{D}$ and $\mathscr{U}$ independent of the value $a_{0}$.

Parablenders are a mechanism to provide $C^{d,r}$-open sets of families of diffeomorphisms which are $C^{d}$-degenerate unfoldings of tangencies (of dimension zero in general). The following theorem proves the existence of such open sets.

Theorem 5.4**.**

Any manifold of dimension $m>c+1$ admits a $k$ - parameter $C^{d}$-family of $C^{r}$-diffeomorphisms with $0<d<r$ having a $C^{d}$-parablender of central dimension $c\geq 1$ at any parameter.

We split the proof of this theorem into several parts. The basic idea to obtain a parablender is by constructing a blender for the induced dynamics $\widehat{f}$ in the space of jets using a parametric family $f=(f_{a})_{a}$ of diffeomorphisms. To do this, first in §5.1, we introduce the jet space and the induced dynamics. After that, we consider in §5.2 a class of parametric families of diffeomorphisms with a family $\Gamma=(\Gamma_{a})_{a}$, where each $\Gamma_{a}$ is an affine $cs$-blender as constructed in §2.2. In order to see that this family of blenders is, indeed, a parablender we show in §5.3.1 that the induced dynamics on the jet space has a $cs$-blender $\widehat{\Gamma}$. To conclude that $\Gamma$ is a parablender we also need to provide an open set $\mathscr{D}$ of $k$-parametric families of discs. This is done in §5.3.2 where additionally we show that each family of discs $\mathcal{D}=(\mathcal{D}_{a})_{a}$ in $\mathscr{D}$ induces a disc $\widehat{D}$ of jets in a superposition region of the blender $\widehat{\Gamma}$. Finally, we show in §5.3.3 that the robust intersection between each of these discs $\widehat{\mathcal{D}}$ of jets and the local unstable manifolds of $\widehat{\Gamma}$ implies a degenerate unfolding of a tangency between the family $\mathcal{D}$ and the local unstable manifold of $\Gamma$ in the sense of Definition 5.1.

First of all, notice that we will provide the existence of parablenders by constructing these objets in local coordinates. Thus, again we will work in an open set of $\mathbb{R}^{m}=\mathbb{R}^{n}\times\mathbb{R}^{c}$ with $n\geq 2$ and $c\geq 1$. We also ask that $n=ss+u$.

5.1. Jet space

Let $f=(f_{a})_{a}$ be a $k$-parameter $C^{d}$-family of $C^{r}$-diffeomorphisms of $\mathbb{R}^{m}$ with $0<d\leq r$. To analyze the unfolding of $f_{a}$ for $a\in\mathbb{I}^{k}$, we will consider on $J^{d}_{0}(\mathbb{I}^{k},\mathbb{R}^{m})$ the map $\widehat{f}$  induced by the family $f=(f_{a})_{a}$ and given by

[TABLE]

Here $J^{d}_{0}(\mathbb{I}^{k},\mathbb{R}^{m})$ denotes the $d$-th order jet space at $a=0$, i.e., the set of equivalence classes $J^{d}_{0}(z)$ where $z=(z_{a})_{a}\in C^{d}(\mathbb{I}^{k},\mathbb{R}^{m})$. The equivalent relation is defined by declaring that $J^{d}_{0}(u)=J^{d}_{0}(v)$ if the functions $u$ and $v$ have all of their partial derivatives equal at $a=0$ up to $d$-th order. A useful choice of a representative for $J_{0}^{d}(z)$ is the $d$-th order Taylor approximation of $z$ at $a=0$. This polynomial is completely determined by the derivatives of $z$ at $a=0$, a finite list of numbers. Therefore, it make sense to identify $J_{0}^{d}(\mathbb{I}^{k},\mathbb{R}^{m})$ with $\mathbb{R}^{m}\times\mathcal{J}^{d}(k,m)$ where

[TABLE]

and $\mathscr{L}^{i}_{sym}(\mathbb{R}^{k},\mathbb{R}^{m})$ denotes the space of symmetric $i$-linear maps from $\mathbb{R}^{k}$ to $\mathbb{R}^{m}$. Hence, clearly $J_{0}^{d}(\mathbb{I}^{k},\mathbb{R}^{m})$ is a Euclidian vector space of dimension

[TABLE]

Remark 5.5**.**

Notice that the map $\widehat{f}$ is of class $C^{r-d}$.

Notation**.**

In order to simplify notation write

[TABLE]

Sometimes, by considering $z_{a}=(x_{a},y_{a})\in\mathbb{R}^{n}\times\mathbb{R}^{c}$, we will split the manifold of $(d,k)$-velocities over $\mathbb{R}^{m}$ (i.e., the space of $d$-jets from $\mathbb{I}^{k}$ to $\mathbb{R}^{m}$ at $a=0$) in the form of $J(\mathbb{R}^{m})=J(\mathbb{R}^{n})\times J(\mathbb{R}^{c})$ and

[TABLE]

Moreover, denote by $J({\Lambda})$ the subset of $J(\mathbb{R}^{*})$ of $d$-jets $J(z)$ at $a=0$ of families of points $z=(z_{a})_{a}\in C^{d}(\mathbb{I}^{k},\mathbb{R}^{*})$ such that $z_{0}\in\Lambda$, where $\Lambda\subset\mathbb{R}^{*}$ and $*\in\{m,n,c\}$. Also, denote by $\mathscr{P}_{*}:J(\mathbb{R}^{m})\to\mathbb{R}^{*}$ the canonical projection onto $\mathbb{R}^{*}$ with $*\in\{ss,u,n,c,m\}$.

5.2. A family of affine blenders

We will take a $C^{r}$-diffeomorphism $\Phi_{0}$ of $\mathbb{R}^{m}$ locally defined as the skew-product given in §2.2. In particular, we have an affine $cs$-blender $\Gamma_{0}$ for $\Phi_{0}$ in the cube $[-2,2]^{m}$ having the family of almost-horizonal $C^{1}$-discs in $\mathcal{B}=\mathsf{R}\times B$ as a superposition region. Here $B$ is an open neighborhood of [math] in $D=(-2,2)^{c}$ satisfying the covering property (3) and $\mathsf{R}=\mathsf{R}_{1}\cup\dots\cup\mathsf{R}_{\kappa}$. Now, we will take a particular family $\Phi=(\Phi_{a})_{a}$, unfolding $\Phi_{0}$ at $a=0$. Namely, we consider $C^{r}$-diffeomorphisms $\Phi_{a}$ locally defined in a similar way by means of skew-products of the form

[TABLE]

where $\phi_{\ell,a}$ are $k$-parameter $C^{d}$-families of affine $(\lambda,\beta)$-contractions on $D$ for $\nu<\lambda<\beta<1$. That is, $\phi_{\ell}=(\phi_{\ell,a})_{a}$ is a $k$-parameter $C^{d}$-family of $C^{r}$-diffeomorphisms $\phi_{\ell,a}$ of $\mathbb{R}^{c}$ such that $\phi_{\ell,a}(\overline{D})\subset D$ and there are linear maps $T_{\ell,a}:\mathbb{R}^{c}\to\mathbb{R}^{c}$ so that

[TABLE]

Moreover, we ask that a bounded open neighborhood $\widehat{B}$ of the $d$-jet [math] in $J(\mathbb{R}^{c})$ such that

[TABLE]

where $\widehat{\phi}_{\ell}$ is the induced map on $J(D)$ by the family $\phi_{\ell}=(\phi_{\ell,a})_{a}$, i.e.,

[TABLE]

with $y=(y_{a})_{a}\in C^{d}(\mathbb{I}^{k},\mathbb{R}^{c})$ such that $y_{0}\in D$. Without restriction of generality we can assume that $B\times\{0\}\subset\widehat{B}$ where $B$ is the open set given in (3).

On the other hand, let $\Gamma_{a}$ be the affine $cs$-blender continuation of $\Gamma_{0}$ for $\Phi_{a}$. To conclude the proof we need to prove that $\Gamma=(\Gamma_{a})_{a}$ is a $cs$-parablender of $\Phi=(\Phi_{a})_{a}$ at $a=0$.

Remark 5.6**.**

The family $\Phi=(\Phi_{a})_{a}$ can be seen as an unfolding of $\Phi_{a_{0}}$ for any $a_{0}\in\mathbb{I}^{k}$. Since $\phi_{\ell,a}$ varies $C^{d}$-continuously with $a\in\mathbb{I}^{k}$, a similar covering property as in (7) holds for the maps $\widehat{\phi}_{\ell}=J_{a_{0}}^{d}[\phi_{\ell}]$. These are induced by the families of fiber maps $\phi_{\ell}=(\phi_{\ell,a})_{a}$ on the $d$-jet space $J_{a_{0}}^{d}(\mathbb{I}^{k},\mathbb{R}^{c})$ at $a=a_{0}$ for all sufficiently small parameter $a_{0}$. That is,

[TABLE]

with $y=(y_{a})_{a}\in C^{d}(\mathbb{I}^{k},\mathbb{R}^{c})$ such that $y_{a_{0}}\in D$. In what follows, we will show that $\Gamma$ is a $C^{d}$-$cs$-parablender of $\Phi$ at $a=0$. However, the choice of $a=0$ is only for convenience to fix an unfolding parameter (and thus a jet space). The same argument works to prove that $\Gamma$ is a $C^{d}$-$cs$-parablender of $\Phi$ at $a=a_{0}$ for any $a_{0}$ close enough to [math]. In fact, by continuity with respect to the parameter, we can take an uniform open set $\mathscr{D}$ of families of discs and an uniform neighborhood $\mathscr{U}$ of the family $\Phi$ for all $a_{0}$ close to [math]. Therefore $\Gamma=(\Gamma_{a})_{a}$ will be, up to scaling the parametrization, a $C^{d}$-$cs$-parablender of the $k$-parametric family $\Phi=(\Phi_{a})_{a}$ at any value of the parameter $a\in\mathbb{I}^{k}$.

Example 5.7**.**

Let $\phi(t)=\lambda t$ for $t\in[-2,2]$ with $1/2<\lambda<1$. Set

[TABLE]

Each $\delta\in\Delta$ is seen as a function which maps $\iota\in\Upsilon$ to $\delta(\iota)\in\{-(1-\lambda),+(1-\lambda)\}$. Take

[TABLE]

where

[TABLE]

Finally, consider

[TABLE]

Here, $\kappa=\varrho^{c}$ where $\varrho$ is the cardinal of $\Delta$. When there are no parameters, i.e., for $k=0$, we recover the Example 2.4. Moreover, $D\phi_{\ell,a}(y)$ is the diagonal matrix $\lambda I$ where $I$ is the identity matrix and thus it does not depend on $a$ for all $y\in\overline{D}$. Hence, we can rewrite (8) as

[TABLE]

where using multilinear algebra

[TABLE]

with $\partial^{i}\ell\in\mathscr{L}^{i}_{\mathrm{sym}}(\mathbb{R}^{k},\mathbb{R}^{c})$ determined by

[TABLE]

Now, we can easily compute the induced map $\widehat{\phi}_{\ell}$ on $J(\mathbb{R}^{c})$. To do this, consider $y=(y_{a})_{a}\in C^{d}(\mathbb{I}^{k},\mathbb{R}^{c})$ such that $y_{0}\in D$. Denoting by

[TABLE]

from (9) we get that

[TABLE]

Thus,

[TABLE]

Consequently, $\widehat{\phi}_{\ell}$ is the composition of a contracting hyperbolic linear map on $J(\mathbb{R}^{c})$ with a translation by the jet $(\partial^{0}\ell,\partial^{1}\ell,\dots,\partial^{i}\ell)$. Since $\ell$ runs over $\Delta^{c}$, we find that the open neighborhood $\widehat{B}=(-1,1)^{\widehat{d}_{c}}$ of [math] in $J(\mathbb{R}^{c})$ satisfies (7), where $\widehat{d}_{c}=\dim J(\mathbb{R}^{c})$.

5.3. Parablenders in the $C^{d,r}$-topology for $0<d<r$

According to Remark 5.5, in order to construct blenders for the induced map $\widehat{\Phi}$ by the $C^{d,r}$-family $\Phi=(\Phi_{a})_{a}$ we will restrict our analysis to $0<d<r$ to obtain that $\widehat{\Phi}$ is at least $C^{1}$.

5.3.1. Blender induced in the jet space

Consider $z=(z_{a})_{a}\in C^{d}(\mathbb{I}^{k},\mathbb{R}^{m})$ and write $z_{a}=(x_{a},y_{a})$ with $x=(x_{a})_{a}\in C^{d}(\mathbb{I}^{k},\mathbb{R}^{n})$ and $y=(y_{a})_{a}\in C^{d}(\mathbb{I}^{k},\mathbb{R}^{c})$. For each $i=1,\dots,d$, since $F$ is an affine map which does not depend on $a$, the partial derivative is

[TABLE]

Hence, the map $\widehat{\Phi}$ on $J(\mathbb{R}^{m})$ induced by the family $\Phi=(\Phi_{a})_{a}$ restricted to $J(\mathcal{U})=J(\mathsf{R})\times J(D)$ is given by the skew-product

[TABLE]

where $\widehat{F}$ acts on $J({\mathsf{R}})$ given by

[TABLE]

and $\widehat{\phi}_{\ell}$ is the induced map on $J(D)$ by the family $\phi_{\ell}=(\phi_{\ell,a})_{a}$ for $\ell=1,\dots,\kappa$. Moreover, $\widehat{F}$ has a horseshoe $\widehat{\mathsf{\Lambda}}=\mathsf{\Lambda}\times\{0\}$ with, except for multiplicity, the same eigenvalues of $F$ and stable index

[TABLE]

Also the width of the stable cone-field on $J(\mathsf{R})$ of $\hat{F}$ is the same width $\alpha$ that the stable cone-field of $F$ has with respect to $\mathsf{R}$.

Similarly, $\widehat{\phi}_{\ell}$ on $J(D)$ has also, except multiplicity, the same eigenvalues of $\phi_{\ell,0}$ on $D$ for all $\ell=1,\dots,\kappa$. Thus, $\widehat{F}|_{\widehat{\Lambda}}$ dominates the fiber dynamics $\widehat{\phi}_{1},\dots,\widehat{\phi}_{\kappa}$ and also by assumption the covering property (7) holds. Hence, according to Theorem 2.2, we have a $cs$-blender $\widehat{\Gamma}$ of central dimension $\widehat{d}_{c}$ for $\widehat{\Phi}$, where

[TABLE]

Additionally, the family of $({\alpha},\nu,\delta)$-horizonal discs in $\widehat{\mathcal{B}}=\widehat{\mathsf{R}}\times\widehat{B}$ is a superposition region of $\widehat{\Gamma}$, where now $\delta<\lambda\hat{L}/2$ and $0<\hat{L}\leq L$ is the Lebesgue number of the cover (7). Here $\widehat{\mathsf{R}}$ is a bounded open neighborhood on $J(\mathsf{R})$ of $\widehat{\Lambda}=\Lambda\times\{0\}$. Moreover, by construction, $\mathscr{P}_{m}(\widehat{\Gamma})=\Gamma_{0}$ where $\mathscr{P}_{m}:J(\mathbb{R}^{m})\to\mathbb{R}^{m}$ is the canonical projection.

5.3.2. An open set of families of discs for the

family of affine blenders

Recall that $\widehat{\mathsf{R}}$ was taken as a bounded neighborhood on $J(\mathsf{R})$ of $\widehat{\Lambda}=\Lambda\times\{0\}$. Since $J(\mathbb{R}^{n})=J(\mathbb{R}^{ss})\times J(\mathbb{R}^{u})$, there is no loss of generality in assuming that

[TABLE]

In fact, we can assume that $\widehat{\mathsf{R}}_{ss}=(-2,2)^{ss}\times B_{{\rho}}(0)$, where $B_{\rho}(0)$ denotes the subset of $\mathcal{J}^{d}(k,ss)$ so that the symmetric $i$-linear maps have all norm less than $\rho$. Notice that the closure of $\widehat{\mathsf{R}}_{ss}$ can be identified with $[-2,2]^{d_{ss}}\times[-\rho,\rho]^{\widehat{d}_{ss}-d_{ss}}$. Hence, without loss of generality, this set can be used to parameterize the ({\color[rgb]{0,0,1}\alpha,}\nu,\delta)-horizontal $\widehat{d}_{ss}$-dimensional discs in $\widehat{\mathcal{B}}=\widehat{\mathsf{R}}\times\widehat{B}$.

Consider a $({\alpha,}\nu,\delta)$-horizontal $ss$-dimensional $C^{r}$-disc $\mathcal{H}_{0}$ in $\mathcal{B}=\mathsf{R}\times B$. Take the $k$-parametric constant family associated with $\mathcal{H}_{0}$ given by

[TABLE]

Lemma 5.8**.**

The set $\widehat{\mathcal{H}}$ in $J(\mathbb{R}^{m})$ parameterized by

[TABLE]

*is a $({\alpha,}\nu,\delta)$-horizontal $\widehat{d}_{ss}$-dimensional $C^{1}$-disc in $\widehat{\mathcal{B}}=\widehat{\mathsf{R}}\times\widehat{B}$ for any $\alpha>0$ large enough and $\nu>0$ small enough. *

Proof.

According to Definition 2.3, we need to show that $\widehat{h}=\mathscr{P}\circ\widehat{\mathcal{H}}$ is $\delta$-close in the $C^{0}$-topology to a constant function on $\widehat{B}$ where $\mathscr{P}$ the canonical projection on the central coordinate, i.e., onto $J(\mathbb{R}^{c})$. Moreover, we also need to show that $\widehat{h}$ is $C^{1}$-dominated by a constant $C>0$ so that $C\nu<\delta$ and $\|D\widehat{g}\|_{\infty}\leq\alpha$ where $\widehat{g}$ is the $J(\mathbb{R}^{u})$-coordinate of $\widehat{\mathcal{H}}$. Since $\widehat{h}$ and $\widehat{g}$ are of class $C^{r-d}$ with $r>d$, then $C=\|D\widehat{h}\|_{\infty}<\infty$ and $D=\|D\widehat{g}\|_{\infty}<\infty$ over the closure of $\widehat{\mathsf{R}}_{ss}$. Thus taking $\nu>0$ small enough and $\alpha>0$ large enough, we can guarantee that $C\nu<\delta$ and $D\leq\alpha$. So, we only need to prove that there is a point $\widehat{y}\in\widehat{B}$ such that

[TABLE]

Since $\mathcal{H}=(\mathcal{H}_{a})_{a}$ is a constant family of discs, then $\widehat{h}=\widehat{h}_{0}$ where $h_{0}=\mathscr{P}\circ\mathcal{H}_{0}$ and

[TABLE]

Moreover, as $\mathcal{H}_{0}$ is a $({\alpha,}\nu,\delta)$-horizontal disc in $\mathcal{B}$, there is $y\in B$ such that $d(h_{0}(x),y)<\delta$ for all $x\in[-2,2]^{ss}$. Set $\widehat{y}=(y,0)\in\widehat{B}$. For any $x\in[-2,2]^{ss}$, we consider $\xi=(\xi_{a})_{a}$ given by $\xi_{a}=x$ for all $a\in\mathbb{I}^{k}$. Then $J(\xi)=(x,0)$ and since $h_{0}(\xi_{a})=h_{0}(x)$ for all $a\in\mathbb{I}^{k}$ we have $\widehat{h}_{0}(J(\xi))=(h_{0}(x),0)$. Therefore $d(\widehat{h}(J(\xi)),\widehat{y})<\delta$ for all $J(\xi)\in[-2,2]^{ss}\times\{0\}$. By continuity, and since ${\rho}>0$ can be taken arbitrarily small, it follows that $d(\widehat{h}(J(\xi)),\widehat{y})<\delta$ for all $J(\xi)$ in the closure of $\widehat{\mathsf{R}}_{ss}=(-2,2)^{ss}\times B_{{\rho}}(0)$. This completes the proof of the lemma. ∎

Remark 5.9**.**

If $\mathcal{H}_{0}$ is a horizontal $ss$-dimensional $C^{r}$-disc in $\mathcal{B}$ then $\widehat{\mathcal{H}}$ is also a horizontal $\widehat{d}_{ss}$-dimensional $C^{1}$-disc in $\widehat{\mathcal{B}}$. Thus, in this case, we do not need a strong contraction for the dynamics on the base. It is only required the domination assumption $\nu<\lambda$.

Since being an almost-horizontal $C^{1}$-disc is an open property, any small enough $C^{d,r}$-perturbation $\mathcal{D}=(\mathcal{D}_{a})_{a}$ of $\mathcal{H}=(\mathcal{H}_{a})_{a}$ still provides an almost-horizontal $\widehat{d}_{ss}$-dimensional $C^{1}$-disc $\widehat{\mathcal{D}}$ in $\widehat{\mathcal{B}}$ close to $\widehat{\mathcal{H}}$ given by

[TABLE]

In fact, taking $\xi_{a}\in[-2,2]^{ss}$ and $z_{a}=\mathcal{D}_{a}(\xi_{a})$ for all $a\in\mathbb{I}^{k}$, it is not difficult to see that the image of this embedding is given by

[TABLE]

In this way, we take $\mathscr{D}=\mathscr{D}({\mathcal{H}})$, a small enough $C^{d,r}$-neighborhood of $\mathcal{H}$.

Remark 5.10**.**

The superposition region $\mathscr{D}$ of an affine $cs$-parablender contains the open set of almost-constant $k$-parameter $C^{d}$-families of almost-horizontal $ss$-dimensional $C^{r}$-discs in $\mathcal{B}$.

5.3.3. Parablenders from blenders in the jet space

We will get that $\Gamma=(\Gamma_{a})_{a}$ is a $cs$-parablender of $\Phi=(\Phi_{a})_{a}$ as a consequence of the following general result.

Proposition 5.11**.**

Let $\Gamma_{0}$ be a $cs$-blender of central dimension $c$ and strong stable dimension $d_{ss}=\mathrm{ind}^{s}(\Gamma_{0})-c$ of a $C^{r}$-diffeomorphism $f_{0}$ of a manifold $\mathcal{M}$. Consider a $k$-parameter $C^{d}$-family $f=(f_{a})_{a}$ unfolding $f_{0}$ at $a=0$ such that the induced map $\widehat{f}$ on the manifold of $(d,k)$-velocities $J(\mathcal{M})=J^{d}_{0}(\mathbb{I}^{k},\mathcal{M})$ over $\mathcal{M}$, which is given by

[TABLE]

has a $cs$-blender $\widehat{\Gamma}$ satisfying the following assumptions:

1. i)

$\widehat{\Gamma}$* projects on $\mathcal{M}$ onto $\Gamma_{0}$;* 2. ii)

there is a $k$-parameter $C^{d}$-family $\mathcal{H}=(\mathcal{H}_{a})_{a}$ of $d_{ss}$-dimensional $C^{r}$-embedded discs $\mathcal{H}_{a}$ into $\mathcal{M}$ so that $\widehat{\mathcal{H}}_{0}\in\widehat{\mathscr{D}}$, where $\widehat{\mathcal{H}}_{0}$ is contained in

[TABLE]

and $\widehat{\mathscr{D}}$ is a superposition region of the blender $\widehat{\Gamma}$.

Then $\Gamma=(\Gamma_{a})_{a}$ is a $C^{d}$-$cs$-parablender at $a=0$ of central dimension $c$ for $f$, where $\Gamma_{a}$ is the continuation of $\Gamma_{0}$ for $f_{a}$.

Proof.

First of all, we will provide the open set of embedded discs. To do this, similarly as in §5.3.2, we take a small $C^{d,r}$-neighborhood $\mathscr{D}=\mathscr{D}(\mathcal{H})$ of the family $\mathcal{H}$, so that any family $\mathcal{D}$ in $\mathscr{D}$ still gives a disc $\widehat{\mathcal{D}}_{0}\in\widehat{\mathscr{D}}$ contained in $\widehat{\mathcal{D}}$.

Next, we will construct the open set of families of diffeomorphisms. Consider the neighborhood $\widehat{\mathscr{U}}$ of the induced map $\widehat{f}$ coming from the definition of the blender. Take the $C^{d,r}$-neighborhood $\mathscr{U}$ of the family $f=(f_{a})_{a}$, so that for every $g=(g_{a})_{a}\in\mathscr{U}$ its induced map $\widehat{g}$ on $J(\mathcal{M})$ belongs to $\widehat{\mathscr{U}}$.

Now, we will prove the existence of a degenerate unfolding at $a=0$ of a tangency between any family of $ss$-dimensional discs $\mathcal{D}=(\mathcal{D}_{a})_{a}\in\mathscr{D}$ and the unstable manifold of $\Gamma_{g}=(\Gamma_{a,g})_{a}$ for any $g=(g_{a})_{a}\in\mathscr{U}$, where $\Gamma_{a,g}$ is the continuation of $\Gamma_{a}$ for $g_{a}$. Since $\widehat{\mathcal{D}}$ contains a disc $\widehat{\mathcal{D}}_{0}$ in the superposition region $\widehat{\mathscr{D}}$ of the $cs$-blender $\widehat{\Gamma}$ of $\widehat{f}$, then

[TABLE]

where $\widehat{\Gamma}_{g}$ is the continuation of $\widehat{\Gamma}$ for the induced map $\widehat{g}$. It is clear that $\mathscr{P}_{m}(\widehat{\Gamma}_{g})=\Gamma_{0,g}$ and that $\widehat{\Gamma}_{g}$ is a hyperbolic set of $\widehat{g}$. If $J(z)\in\widehat{\Gamma}_{g}$, where $z=(z_{a})_{a}\in C^{d}(\mathbb{I}^{k},\mathcal{M})$, then $z_{0}\in\Gamma_{0,g}$ and the point $z_{a}$ must be the continuation in $\Gamma_{a,g}$ of $z_{0}$ for $g_{a}$. Similarly, $\mathscr{P}_{m}(W^{u}_{loc}(J(z)))=W^{u}_{loc}(z_{0})$ and if $\widehat{x}\in W^{u}_{loc}(J(z))$ then

[TABLE]

In summary, we can find a point $\widehat{q}\in W^{u}_{loc}(\widehat{\Gamma}_{g})\cap\widehat{\mathcal{D}}_{0}$. Since $\widehat{q}$ belongs to the local unstable manifold of $\widehat{\Gamma}_{g}$ there are functions $x=(x_{a})_{a},z=(z_{a})_{a}\in C^{d}(\mathbb{I}^{k},\mathcal{M})$ such that

[TABLE]

On the other hand, since $\widehat{q}\in\widehat{\mathcal{D}}$,

[TABLE]

Thus $J(x)=J(y)$. This concludes that $\mathcal{W}=(W^{u}_{loc}(z_{a}))_{a}$ and $\mathcal{D}=(\mathcal{D}_{a})_{a}$ has a tangency at $a=0$ which unfolds $C^{d}$-degenerately. Therefore $\Gamma=(\Gamma_{a})_{a}$ is a $C^{d}$-$cs$-parablender at $a=0$ of central dimension $c\geq 1$ for $f=(f_{a})_{a}$ and we complete the proof of the proposition. ∎

5.3.4. Proof of Theorem 5.4

Take the family of $cs$-blenders $\Gamma=(\Gamma_{a})_{a}$ of central dimension $c>0$ of the particular family of locally defined affine skew-products $\Phi=(\Phi_{a})_{a}$ constructed in §5.2. From §5.3.1, we get a $cs$-blender $\widehat{\Gamma}$ for the induced map $\widehat{\Phi}$ on $J(\mathbb{R}^{m})$ which projects in $\mathbb{R}^{m}$ onto $\Gamma_{0}$. In §5.3.2 it was obtained that any $k$-parameter constant family of horizontal discs induced a $\widehat{d}_{ss}$-dimensional $C^{1}$-disc into the superposition domain $\widehat{\mathcal{B}}$ of $\widehat{\Gamma}$. Thus, this disc belongs to the superposition region of the induced blender. Hence, according to Proposition 5.11, $\Gamma$ is a $C^{d}$-$cs$-parablender of central dimension $c>0$ at $a=0$. Finally, by Remark 5.6 and reparameterizing if necessary, $\Phi=(\Phi_{a})_{a}$ is $k$-parametric $C^{d}$-family of $C^{r}$-diffeomorphisms having a $C^{d}$-$cs$-parablender $\Gamma=(\Gamma_{a})_{a}$ of central dimension $c\geq 1$ at any parameter. This completes the proof.

6. Robust degenerate unfolding of heterodimensional cycles

Now we will prove Theorem C. We will consider a $C^{d}$-family $f=(f_{a})_{a}$ of $C^{r}$-diffeomorphisms of a manifold $\mathcal{M}$ parameterized by $a\in\mathbb{I}^{k}$ with a $C^{d}$-$cs$-parablender $\Gamma=(\Gamma_{a})_{a}$ of codimension $c\geq 1$ at any parameter. For simplicity, we will assume that $\Gamma$ is the family of affine blenders constructed to prove Theorem 5.4. We will assume that $f_{0}$ has a heterodimensional cycle of co-index $c\geq 1$ associated with $\Gamma_{0}$ and another hyperbolic periodic point $P_{0}$. We suppose that $W^{s}(P_{0})$ contains a $ss$-dimensional horizontal disc $\mathcal{H}_{0}$ in the superposition domain $\mathcal{B}$ of the $cs$-blender $\Gamma_{0}$ of $f_{0}$. Moreover, as the construction is local, we ask that $W^{s}(P_{a})$ contains the same disc $\mathcal{H}_{0}$ for all $a\in\mathbb{I}^{k}$ where $P_{a}$ denotes the continuation of $P_{0}$ for $f_{a}$. Hence the constant family of discs $\mathcal{H}=(\mathcal{H}_{a})$ where $\mathcal{H}_{a}=\mathcal{H}_{0}$ for all $a\in\mathbb{I}^{k}$ belongs to the open set $\mathscr{D}$ of families of embedded discs associated with the $C^{d}$-$cs$-parablender $\Gamma$. Thus, for every $C^{d,r}$-close enough family $g=(g_{a})_{a}$ of $f=(f_{a})_{a}$ the family of stable manifolds $W^{s}(P_{g})=(W^{s}(P_{a,g}))_{a}$ of the continuation $P_{a,g}$ of $P_{a}$ contains a family of discs $\mathcal{D}=(D_{a})_{a}\in\mathscr{D}$. Therefore, $W^{u}_{loc}(\Gamma_{g})$ and $W^{s}(P_{g})$ has a tangency at $a=0$ which unfolds $C^{d}$-degenerately. In fact, since $\Gamma$ is a $C^{d}$-$cs$-parablender at any parameter, the same argument also works for any parameter $a=a_{0}$. This concludes the proof of the theorem.

7. Robust degenerate unfoldings of homoclinic tangencies

In this section we prove Theorem B. We begin by mentioning a few words about the strategy of the proof. Recall that to prove Theorem C we first show that any manifold $\mathcal{M}$ of dimension at least $3$ admits a family $f=(f_{a})_{a}$ of diffeomorphisms $f_{a}$ of $\mathcal{M}$ having a parablender at any parameter (see Theorem 5.4). Now, to prove Theorem B, we will proceed similarly by showing first the following result:

Theorem 7.1**.**

Any manifold $\mathcal{M}$ of dimension $m>c+u$ admits a $k$-parameter $C^{d}$-family $f=(f_{a})_{a}$ of $C^{r}$-diffeomorphisms with $0<d<r-1$ such that the $k$-parameter induced $C^{d}$-family $f^{{G}}=(f^{{G}}_{a})_{a}$ of $C^{r-1}$-diffeomorphisms on the $u$-th Grassmannian bundle of $\mathcal{M}$,

[TABLE]

has a $C^{d}$-parablender $\Gamma^{{G}}=(\Gamma^{{G}}_{a})_{a}$ of central dimension $c\geq 1$ at any parameter.

As in the proof of Theorem 5.4, we will obtain a parablender for the $k$-parameter family $f^{{G}}=(f_{a}^{{G}})_{a}$ for the induced dynamics by constructing a blender with respect to the induced dynamics $\widehat{f}^{{G}}$ on the manifold of $(c,k)$-velocities over $G_{u}(\mathcal{M})$, i.e., on the jet space $J^{d}_{0}(\mathbb{I}^{k},G_{u}(\mathcal{M}))$.

Remark 7.2**.**

Notice that the map $\widehat{f}^{{G}}$ is of class $C^{r-1-d}$.

In what follows, we fix $0<d<r-1$. As in the previous section, we will provide the proof of Theorem 7.1 using the local coordinates in $\mathcal{M}$. Thus, as usual, we will work in $\mathbb{R}^{m}=\mathbb{R}^{ss}\times\mathbb{R}^{u}\times\mathbb{R}^{c}$ with $u,c\geq 1$ and $n=ss+u$.

We also recall some notation from §3.2:

[TABLE]

Sometimes, when no confusion arises, we write $\mathcal{U}^{{G}}$ by a change of coordinates as

[TABLE]

7.1. A parablender on the manifold of velocities over the Grassmanian manifold

We will start considering the $k$-parameter $C^{d}$-family of locally defined affine $C^{r}$ skew-product maps

[TABLE]

introduced in §5.2. For simplicity, we assume that there are $0<\nu<\lambda<\beta<1$ and diagonal linear maps $S:\mathbb{R}^{ss}\to\mathbb{R}^{ss}$, $U:\mathbb{R}^{u}\to\mathbb{R}^{u}$ and $T:\mathbb{R}^{c}\to\mathbb{R}^{c}$ such that

[TABLE]

and

[TABLE]

Under theses assumptions, we get that $D\Phi_{a}(x,y)$ is the same linear map $D\Phi$ for all $(x,y)\in\mathsf{R}\times D$ which has $E^{u}_{a}=E^{u}$ with $E^{u}=\{0^{ss}\}\times\mathbb{R}^{u}\times\{0^{c}\}$ as a fixed point and $\mathcal{C}^{{G}}$ as a neighborhood of attraction for all $a\in\mathbb{I}^{k}$. Thus, following §3.2, the induced $C^{r}$-diffeomorphism $\Phi^{{G}}_{a}$ of $\Phi_{a}$ on the Grassmannian manifold $G_{u}(\mathbb{R}^{m})$ is given by

[TABLE]

where $F^{{G}}=F\times D\Phi$ on $\mathsf{R}^{{G}}=\mathsf{R}\times\mathcal{C}^{{G}}$. Moreover, for each $a\in\mathbb{I}^{k}$ we have a $cs$-blender $\Gamma^{{G}}_{a}=\Gamma_{a}\times\{E^{u}\}$ of central dimension $c\geq 1$ where $\Gamma_{a}$ is the $cs$-blender of $\Phi_{a}$. Now, we will show that the family $\Gamma^{{G}}=(\Gamma^{{G}}_{a})_{a}$ is a $cs$-parablender of central dimension $c$ at $a=0$ for ${\Phi}^{{G}}=({\Phi}^{{G}}_{a})_{a}$.

To prove this, we will work with the induced $C^{1}$-map on $J(G_{u}(\mathbb{R}^{m}))\stackrel{{\scriptstyle\scriptscriptstyle\rm def}}{{=}}J^{d}_{0}(\mathbb{I}^{k},G_{u}(\mathbb{R}^{m}))$ by the family $\Phi^{{G}}=({\Phi}^{{G}}_{a})_{a}$ given by

[TABLE]

According to Proposition 5.11 to prove that $\Gamma^{{G}}=(\Gamma^{{G}}_{a})_{a}$ is a $C^{d}$-$cs$-parablender for $\Phi^{{G}}=(\Phi^{{G}}_{a})_{a}$ at $a=0$ we need to show the following. First, we must prove that $\widehat{\Phi}^{{G}}$ has a $cs$-blender $\widehat{\Gamma}^{{G}}$ which projects onto $\Gamma^{{G}}_{0}$ and after provide a particular $C^{d}$-family $\mathcal{H}^{{G}}=(\mathcal{H}^{{G}}_{a})_{a}$ of $C^{r-1}$-discs which induce a disc $\widehat{\mathcal{H}}^{{G}}$ in the open set of $C^{1}$-discs. This will be done in the two next sections.

7.1.1. Blender

Using local coordinates (c.f. [Mic80, DK00]) in the manifold of $(d,k)$-velocities over $G_{u}(\mathbb{R}^{m})=\mathbb{R}^{m}\times G(u,m)$ we can identify $J(\mathcal{U}^{{G}})=J(\mathsf{R}^{{G}})\times J(D)$. Thus, it is not difficult to see that $\widehat{\Phi}^{{G}}$ restricted to $J(\mathcal{U}^{{G}})$ can be written as a skew-product map

[TABLE]

where $\widehat{F}^{{G}}$ is the induced map on $J(\mathsf{R}^{{G}})$ by the map $F^{{G}}$ and $\widehat{\phi}_{\ell}$ are the induced maps on $J(D)$ by the family $\phi_{\ell}=(\phi_{\ell,a})_{a}$ for $\ell=1,\dots,\kappa$. Then, according to (7) and Theorem 2.2 we only need to prove that the base dynamics of $\widehat{\Phi}^{{G}}$ has a horseshoe which dominates the fiber dynamics. To do this, first we identify $J(\mathsf{R}^{{G}})=J(\mathsf{R})\times J(\mathcal{C}^{{G}})$. In this way, we write the base dynamics of $\widehat{\Phi}^{{G}}$ as a direct product map

[TABLE]

where $\widehat{F}$ acts on $J(\mathsf{R})$ by means of

[TABLE]

and $\widehat{D\Phi}$ acts on $J(\mathcal{C}^{{G}})$ defined as

[TABLE]

with $E=(E_{a})_{a}\in C^{d}(\mathbb{I}^{k},G(u,m))$ and $E_{0}\in\mathcal{C}^{{G}}$. As in §5.3.1, using that $F$ is an affine map and is independent of $a$, we have that

[TABLE]

with $x=(x_{a})_{a}\in C^{d}(\mathbb{I}^{k},\mathbb{R}^{n})$ such that $x_{0}\in\mathsf{R}$. From here we get that $\widehat{F}$ has a horseshoe $\widehat{\Lambda}=\Lambda\times\{0\}$ as an invariant set. Moreover, the eigenvalues of the linear part of $\widehat{F}$ are the same as of $DF$ at $\mathsf{R}$ and thus, as in §5.3.1, they dominate the fiber dynamics.

On the other hand, it is not difficult to see that $\widehat{D\Phi}$ has the fixed point $J(E^{u})$ where $E^{u}=(E^{u}_{a})_{a}\in C^{d}(\mathbb{I}^{k},G(u,m))$ is given by $E^{u}_{a}=\{0^{ss}\}\times\mathbb{R}^{u}\times\{0^{c}\}$ for all $a\in\mathbb{I}^{k}$. Hence

[TABLE]

is a horseshoe for $\widehat{F}^{{G}}$ with stable index

[TABLE]

Also, analogously with §5.3.1 and §3.2, the width of the stable cone-field of $\widehat{F}^{G}$ on $J(\mathsf{R}^{G})$ coincides with the width $\alpha$ of the stable cone-field of $F$ on $\mathsf{R}$.

Now we need to prove that $\widehat{F}^{{G}}$ restricted to $\widehat{\Lambda}^{{G}}$ dominates $\widehat{\phi}_{1},\dots,\widehat{\phi}_{\kappa}$. Since, $\widehat{F}$ dominates the fiber dynamics, it suffices to show that $\widehat{D\Phi}$ at $J(E^{u})$ also dominates $\widehat{\phi}_{1},\dots,\widehat{\phi}_{\kappa}$. In local coordinates around $E^{u}$ we can write

[TABLE]

being $P$ a diagonal $(d_{u}\times d_{u})$-matrix whose eigenvalues are dominated by $\beta\nu<1$ and $d_{u}=\dim G(u,m)=u(m-u)$. Similarly, we can identify $J(E)\equiv(e,{\partial^{1}e,\dots,\partial^{d}e})\in\mathbb{R}^{d_{u}}\times\mathcal{J}^{d}(c,d_{u})$ and take as a representative of $J(E)$ the function $E=(E_{a})_{a}$ given by

[TABLE]

Substituting (12) into (11), in local coordinates we have that

[TABLE]

where $O(2)$ is a function that envolves the products of $\partial^{s}e\cdot\partial^{t}e$ with $s+t=2$. In local coordinates $J(E^{u})\equiv(e,\partial^{1}e,\dots,\partial^{d}e)=0$ is a fixed point of $\widehat{D\Phi}$. Moreover, from (13) the linear part at this point is given by a triangular matrix whose diagonal elements are the eigenvalues of $P$. Since these eigenvalues are dominated by $\beta\nu<\nu<\lambda$, then $\widehat{D\Phi}$ dominates the fiber dynamics. This concludes the proof of the existence of a $cs$-blender $\widehat{\Gamma}^{{G}}$ of $\widehat{\Phi}^{{G}}$ projecting on $\Gamma^{{G}}_{0}=\Gamma_{0}\times\{E^{u}\}$. Moreover, the family of $(\alpha,\nu,\delta)$-horizontal $C^{1}$-discs in $\widehat{\mathcal{B}}^{{G}}=\widehat{\mathsf{R}}^{{G}}\times\widehat{B}$ is a superposition region of $\widehat{\Gamma}^{G}$, where $0<\delta<\hat{L}\lambda/2$.

7.1.2. Discs on the manifold of velocities induced by

folding manifolds

Let $\widehat{\mathcal{B}}^{{G}}=\widehat{\mathsf{R}}^{{G}}\times\widehat{B}$ be the superposition domain of the blender $\widehat{\Gamma}^{{G}}$ where $\widehat{\mathsf{R}}^{{G}}$ is a neighborhood on $J(\mathsf{R}^{{G}})=J(\mathsf{R})\times J(\mathcal{C}^{{G}})$ of $\widehat{\Lambda}^{{G}}$. Similar as in §5.3.2, since $\mathsf{R}\subset\mathbb{R}^{n}=\mathbb{R}^{ss}\times\mathbb{R}^{u}$ and $\mathcal{C}^{{G}}\subset G(u,m)$ we can take

[TABLE]

In fact, we have that ${\widehat{\mathsf{R}}_{G}}$ can be taken as an arbitrarily small neighborhood in $J(G(u,m))$ of $J(E^{u})\equiv(e,{\partial^{1}e,\dots,\partial^{d}e})=0$ and $\widehat{\mathsf{R}}^{ss}=(-2,2)^{ss}\times B_{{\rho}}(0)$. Again here denotes $B_{\rho}(0)$ denotes the subset of $\mathcal{J}^{d}(k,ss)$ so that the symmetric $i$-linear maps have all norm less than $\rho$.

Now, fix a $({\alpha,}\nu,\delta)$-folding $C^{r}$-manifold $\mathcal{S}_{0}$ with respect to $\mathcal{B}^{{G}}=\mathcal{B}\times\mathcal{C}^{{G}}$, where $\mathcal{B}=\mathsf{R}\times B$ is the superposition domain of the blender $\Gamma_{0}$ of $\Phi_{0}$. Consider the $k$-parametric constant family of $({\alpha,}\nu,\delta)$-folding $C^{r}$-manifolds associated with $\mathcal{S}_{0}$ given by

[TABLE]

According to Lemma 3.5, the set

[TABLE]

is an almost-horizonal $d^{{G}}_{ss}$-dimensional $C^{r-1}$-disc in $\mathcal{B}^{{G}}$ where $d^{{G}}_{ss}=ss+u(m-u)$. Hence, the constant family of folding $C^{r}$-manifolds $\mathcal{S}=(\mathcal{S}_{a})_{a}$ induces a constant family $\mathcal{H}^{{G}}=(\mathcal{H}^{{G}}_{a})_{a}$ of $C^{r}$-discs in $\mathcal{B}^{{G}}$ given by $\mathcal{H}^{{G}}_{a}=\mathcal{H}^{{G}}_{0}$ for all $a\in\mathbb{I}^{k}$. By means of a similar argument as in Lemma 5.8 we obtain the following:

Lemma 7.3**.**

The set $\widehat{\mathcal{H}}^{{G}}$ in $J(G_{u}(\mathbb{R}^{m}))$ parameterized by

[TABLE]

for $\xi=(\xi_{a})_{a}\in C^{d}(\mathbb{R}^{k},\mathbb{R}^{ss}\times G(u,m))$ with $J(\xi)$ belongs to the closure of $\widehat{\mathsf{R}}_{ss}\times\widehat{\mathsf{R}}_{G}$, is a $({\alpha,}\nu,\delta)$-horizontal $\widehat{d}_{ss}^{{G}}$-dimensional $C^{1}$-disc in $\widehat{\mathcal{B}}^{{G}}=\widehat{\mathsf{R}}^{{G}}\times\widehat{B}$ for any $\alpha>0$ large enough and $\nu>0$ small enough where

[TABLE]

Proof.

Since $d<r-1$, it is straight forward that $\widehat{\mathcal{H}}^{{G}}$ is a $\widehat{d}_{ss}^{{G}}$-dimensional $C^{1}$-disc in $\widehat{\mathcal{B}}^{{G}}$. Let $\widehat{h}^{{G}}$ and $\widehat{g}\ ^{G}$ be, respectively, the $J(\mathbb{R}^{c})$-coordinate and $J(\mathbb{R}^{u})$-coordinate of $\widehat{\mathcal{H}}^{{G}}$ which correspond with the central and unstable coordinates of the disc. In order to prove that $\widehat{\mathcal{H}}^{{G}}$ is a $({\alpha,}\nu,\delta)$-horizontal disc, notice that

[TABLE]

Hence, by taking $\alpha>0$ large enough and $\nu>0$ small enough we can always guarantee that $C\nu<\delta$ and $D\leq\alpha$. Thus, we only need to show that there is a point $\widehat{y}\in\widehat{B}$ such that

[TABLE]

Since $\mathcal{H}^{{G}}=(\mathcal{H}^{{G}}_{a})_{a}$ is a constant family of discs then $\widehat{h}^{{G}}=\widehat{h}^{{G}}_{0}$ where $h^{{G}}_{0}=\mathscr{P}\circ\mathcal{H}^{{G}}_{0}$ and

[TABLE]

with $J(\xi^{{G}})$ belonging to the closure of $\widehat{\mathsf{R}}_{ss}\times\widehat{\mathsf{R}}_{G}$. The same computation as in Lemma 5.8 proves that

[TABLE]

Hence $d(\widehat{h}^{{G}}(J(\xi^{{G}})),\widehat{y})<\delta$ for all $J(\xi^{{G}})\in([-2,2]^{ss}\times\{E^{u}\})\times\{0\}$, where $\widehat{y}=(y,0)\in\widehat{B}$ and $y\in B$ comes from the definition of the folding $C^{r}$-manifold. By continuity and since the neighborhood $\widehat{\mathsf{R}}_{G}$ of $J(E^{u})=(E^{u},0)$ can be taken arbitrarily small, it follows

[TABLE]

This completes the proof of the lemma. ∎

Remark 7.4**.**

Let $\mathcal{S}_{0}$ be the $({\alpha,}\nu,\delta)$-folding $C^{r}$-manifold with respect to $\mathcal{B}^{{G}}=\mathcal{B}\times\mathcal{C}^{{G}}$ introduced in Example 3.2. Proposition A.2 in Appendix A proves that for any $0<\nu<\delta$ the constant family $\mathcal{S}=(\mathcal{S}_{a})_{a}$ of $({\alpha,}\nu,\delta)$-folding $C^{r}$-manifolds induces a $({\alpha,}\nu,\delta)$-horizontal $C^{1}$-disc $\widehat{\mathcal{H}}^{{G}}$ in $\widehat{\mathcal{B}}^{{G}}$ from the constant family of $C^{r-1}$-discs $\mathcal{H}^{{G}}=(\mathcal{H}^{{G}}_{a})_{a}$ given by $\mathcal{H}^{{G}}_{a}=\mathcal{H}_{0}^{{G}}=\mathcal{S}^{{G}}_{0}\cap\overline{\mathcal{B}^{{G}}}$.

The previous lemma implies that $\widehat{\mathcal{H}}^{{G}}$ belongs to a superposition region of the blender $\widehat{\Gamma}^{{G}}$. This completes the proof of the particular $C^{d}$-family $\mathcal{H}^{{G}}=(\mathcal{H}^{{G}}_{a})_{a}$ of $C^{r-1}$-discs.

7.1.3. Proof of Theorem 7.1

The proof will follow from Proposition 5.11 and using a similar construction as in the proof of Theorem A. Indeed, we take a $k$-parameter family of $cs$-blenders $\Gamma=(\Gamma_{a})_{a}$ of central dimension $c\geq 1$ for a $C^{d}$-family $\Phi=(\Phi_{a})_{a}$ of $C^{r}$-diffeomorphisms of $\mathcal{M}$ locally defined as affine skew-product maps given at the beginning of §7.1. As we showed in §7.1.1, these maps provide a family $\Gamma^{{G}}=(\Gamma^{{G}}_{a})_{a}$ of $cs$-blenders of central dimension $c\geq 1$ for the induced dynamics $\Phi^{{G}}_{a}$ on $G_{u}(\mathcal{M})$ and as well as a $cs$-blender $\widehat{\Gamma}^{{G}}$ for the map $\widehat{\Phi}^{{G}}$ on $J(G_{u}(\mathcal{M}))$. Similarly, as in the proof of Theorem A, we take a $C^{d}$-family $\mathcal{S}=(\mathcal{S}_{a})_{a}$ of $C^{2}$-robust folding $C^{r}$-manifolds $\mathcal{S}_{a}$ with respect to the superposition domain $\mathcal{B}^{{G}}$ of $\Phi^{{G}}$ in the sense of Remark 3.4. Moreover, we assume that the $C^{d}$-family $\mathcal{S}$ induces a $C^{1}$-disc in the superposition region of $\widehat{\Gamma}^{{G}}$. This was done in §7.1.2 by taking a constant family of folding manifolds. Then, according to Proposition 5.11 we have that $\Gamma^{{G}}=(\Gamma^{{G}}_{a})_{a}$ is a $C^{d}$-parablender at $a=0$ of central dimension $c\geq 1$ of $\Phi^{{G}}=(\Phi^{{G}}_{a})_{a}$ at $a=0$. As in Remark 5.6, we can extend the result for any parameter $a_{0}$ close to $a=0$. This completes the proof of Theorem 7.1.

7.2. Proof of Theorem B

The following result is, basically, a consequence of Theorem 7.1.

Theorem 7.5**.**

For any $0<d<r-1$ and $k\geq 1$, there exists a $C^{d}$-family $f=(f_{a})_{a}$ of locally defined $C^{r}$-diffeomorphisms of $\mathcal{M}$ having a family of $cs$-blenders $\Gamma=(\Gamma_{a})_{a}$ with unstable dimension $u\geq 1$ and a family of folding manifolds $\mathcal{S}=(\mathcal{S}_{a})_{a}$ satisfying the following:

For any $a_{0}\in\mathbb{I}^{k}$, any family $g=(g_{a})_{a}$ close enough to $f$ in the $C^{d,r}$-topology and any $C^{d,r}$-perturbation $\mathcal{L}=(\mathcal{L}_{a})_{a}$ of $\mathcal{S}$ there exists $z=(z_{a})_{a}\in C^{d}(\mathbb{I}^{k},\mathcal{M})$ such that

1. i)

$z_{a}\in\Gamma_{g,a}$, where $\Gamma_{g,a}$ denotes the continuation for $g_{a}$ of the blender $\Gamma_{a}$, 2. ii)

the family of local unstable manifolds $\mathcal{W}=(W^{u}_{loc}(z_{a};g_{a}))_{a}$ and $\mathcal{L}$ have a tangency of dimension $u$ at $a=a_{0}$ which unfolds $C^{d}$-degenerately.

Remark 7.6**.**

The central dimension of the blenders $\Gamma_{a}$ is $c=u^{2}$ and the folding manifold $\mathcal{S}_{a}$ has dimension $ss+c$ where $ss\geq 1$ is the strong stable dimension of the blenders $\Gamma_{a}$. Thus the dimension of the manifold $\mathcal{M}$ is $m>u+u^{2}$. Theorem B follows immediately from the above result assuming that each folding manifold $\mathcal{S}_{a}$ is part of the stable manifold of a point of $\Gamma_{a}$.

Proof of Theorem 7.5.

Let $\Gamma=(\Gamma_{a})_{a}$ be the family of $cs$-blenders of the $C^{d}$-family $f=(f_{a})_{a}$ of $C^{r}$-diffeomorphisms of $\mathcal{M}$ given in Theorem 7.1. Consider the $C^{d}$-$cs$-parablender $\Gamma^{{G}}=(\Gamma^{{G}}_{a})_{a}$ at any parameter of central dimension $c\geq 1$ for the induced dynamics $f^{{G}}=(f^{{G}}_{a})_{a}$ in ${G}_{u}(\mathcal{M})$. From the proof of Theorem 7.1, we have a $C^{d}$-family of $C^{2}$-robust folding $C^{r}$-manifolds $\mathcal{S}=(\mathcal{S}_{a})_{a}$ with respect to the superposition domain $\mathcal{B}^{{G}}$ of $f^{{G}}$ such $\mathcal{S}$ induces a family of $C^{r}$-discs in $\mathcal{B}^{{G}}$ contained in $\mathcal{S}^{{G}}=(\mathcal{S}^{{G}}_{a})_{a}$ where

[TABLE]

According to the proof of Proposition 5.11, the open set of $k$-parameter $C^{d}$-families of $C^{r}$-discs contains a $C^{d,r}$-neighborhood $\mathscr{D}(\mathcal{S}^{{G}})$ of $\mathcal{S}^{{G}}=(\mathcal{S}^{{G}}_{a})_{a}$. Denote by $\mathscr{D}(\mathcal{S})$ a $C^{d,r}$-neighborhood of $\mathcal{S}=(\mathcal{S}_{a})_{a}$, so that if $\mathcal{L}=(\mathcal{L}_{a})_{a}\in\mathscr{D}(\mathcal{S})$ then the induced family of discs $\mathcal{L}^{{G}}=(\mathcal{L}^{{G}}_{a})_{a}\in\mathscr{D}(\mathcal{S}^{{G}})$. Similarly, let $\mathscr{U}(f)$ be a $C^{d,r}$-neighborhood of $f=(f_{a})_{a}$ so that if $g=(g_{a})_{a}\in\mathscr{U}(f)$ then the induced family of maps $g^{{G}}=(g^{{G}}_{a})_{a}\in\mathscr{U}(f^{{G}})$. Here, $\mathscr{U}(f^{{G}})$ comes from the definition of a parablender as the neighborhood of the $k$-parameter family $f^{{G}}=(f^{{G}}_{a})_{a}$. From Definition 5.1 applied at $a=a_{0}$, we have $x=(x_{a})_{a},y=(y_{a})_{a},z=(z_{a})_{a}\in C^{d}(\mathbb{I}^{k},G_{u}(\mathcal{M}))$ such that

[TABLE]

where $\Gamma^{{G}}_{a,g}$ is the continuation for $g^{{G}}_{a}$ of the $cs$-blender $\Gamma^{{G}}_{a}$ for all $a\in\mathbb{I}^{k}$. Set $\mathcal{W}_{a}=W^{u}_{loc}(\tilde{z}_{a})$ where $z_{a}=(\tilde{z}_{a},W_{a})\in\Gamma_{a}\times G(u,m)$ and denote $\mathcal{W}=(\mathcal{W}_{a})_{a}$. Hence,

[TABLE]

Similarly,

[TABLE]

Therefore, by Definition 4.2, $\mathcal{L}$ and $\mathcal{W}$ has a tangency of dimension $u>0$ at $a=a_{0}$ which unfolds $C^{d}$-degenerately. This completes the proof. ∎

Appendix A Estimates for the folding manifold of Example 3.2

We consider the $(ss+c)$-dimensional $C^{\infty}$-manifold of Example 3.2 given by

[TABLE]

where $t=(t_{1},\dots,t_{u},\dots,t_{c})\in[-\epsilon,\epsilon]^{c}$ and $h(t)=(h_{1}(t),\dots,h_{c}(t))$ with

[TABLE]

Let $t=t(E)$ be the $C^{r-1}$-function on $\mathcal{C}^{{G}}$ computes in Example 3.2. We have that:

Proposition A.1**.**

For every $\varepsilon>0$ suffices small there is a neighborhood $\mathcal{C}^{{G}}$ of $E^{u}$ such that

[TABLE]

Thus, $\mathcal{S}$ is a ({\color[rgb]{0,0,1}\alpha,}\nu,\delta)-folding manifold for any $\nu>0$ such that $(1+\varepsilon)\nu<\delta$.

Proof.

First of all, notice that for all $t\in[-\epsilon,\epsilon]^{c}$, it holds that

[TABLE]

Hence, taking $\epsilon>0$ small enough we have that $\|Dh\|_{\infty}=1$. On the other hand, by Cramer’s rule we get that the solution of the linear system $At=\vec{c}$  is given by

[TABLE]

where $A^{*}_{i}$ is the matrix formed by replacing the $i$-th column of $A$ by the column vector $\vec{c}$. In order to compute $Dt$ we write the variable of $t$ by $E=\langle v_{1},\dots,v_{u}\rangle$ with $v_{k}=(a_{k},b_{k},c_{k})\in\mathbb{R}^{ss}\times\mathbb{R}^{u}\times\mathbb{R}^{c}$ and then

[TABLE]

Moreover, since the matrix $A$ does not depend on the variables $a_{k}$ we get that $\partial_{a_{k}}t=0$. In the sequel we will use the symbol $D_{k}$ to denote any partial derivative of the form $\partial_{b_{k\iota}}$ or $\partial_{c_{k\iota}}$. For each $i=1,\dots,c$, using Jacobi’s formula it follows that

[TABLE]

In particular, since $\vec{c}(E^{u})=0$ then $\det A^{*}_{i}(E^{u})=0$, and $\det A(E^{u})=2$, we obtain that

[TABLE]

where $\mathrm{Adj}(A^{*}_{i})$ is the adjugate matrix of $A^{*}_{i}$. Notice that

[TABLE]

where $\sigma_{i}=2$ if $i\not=1$ and $\sigma_{i}=1$ otherwise ($i=1$). From this follows that

[TABLE]

If $D_{k}$ is either, $\partial_{b_{k\iota}}$ for $\iota=1,\dots,u$ or $\partial_{c_{k\iota}}$ for $\iota=u+1,\dots,c$ then $D_{k}\vec{c}=0$ and thus $D_{k}t_{i}(E^{u})=0$. Otherwise,

[TABLE]

Therefore

[TABLE]

Since $Dt$ varies continuously with respect to $E$ we have that $\|Dt(E)\|_{\infty}$ is close to $\|Dt(E^{u})\|_{\infty}$ for any $E$ close enough to $E^{u}$. Thus, shrinking $\mathcal{C}^{{G}}$ if necessary, this implies that $\|Dt\|_{\infty}\leq 1+\varepsilon$ over $\mathcal{C}^{{G}}_{\alpha}$. Hence, $C=\|Dh\|_{\infty}\cdot\max\{1,\|Dt\|_{\infty}\}\leq 1+\varepsilon$ for a fixed but arbitrarily small $\varepsilon>0$. ∎

This ({\color[rgb]{0,0,1}\alpha,}\nu,\delta)-folding $C^{r}$-manifold $\mathcal{S}$ induces a $C^{r-1}$-disc

[TABLE]

Set $h^{{G}}=\mathscr{P}\circ\mathcal{H}^{{G}}$. Here $\mathscr{P}$ denotes the standard projection onto $\mathbb{R}^{c}$. We consider

[TABLE]

with

[TABLE]

where $\overline{B}_{\rho}(0)$ denotes a closed ball of radius $\rho>0$ at [math] velocity of the jets over $\mathbb{R}^{ss}\times G(u,m)$.

Proposition A.2**.**

For every $\varepsilon>0$ suffices small there are $\rho>0$ and a neighborhood $\mathcal{C}^{{G}}$ of $E^{u}$ so that

[TABLE]

Thus, $\widehat{\mathcal{H}}^{{G}}$ is a ({\color[rgb]{0,0,1}\alpha,}\nu,\delta)-horizontal disc for any $\nu>0$ such that $(1+\varepsilon)\nu<\delta$.

Proof.

Observe that $h^{{G}}(E)=h(t(E))$. In this way,

[TABLE]

for $E=(E_{a})_{a}\in C^{d}(\mathbb{I}^{k},G(u,m))$ with $E_{0}\in\mathcal{C}^{{G}}_{\alpha}$. Denoting $t_{a}=t(E_{a})$ for all $a\in\mathbb{I}^{k}$, we can rewrite (A.1) as

[TABLE]

with $t_{0}$ small enough in norm. Therefore,

[TABLE]

where $\widehat{t}(J(E))=J(t\circ E)$. We want to compute $\|D\widehat{h}^{{G}}(J(E^{u}))\|_{\infty}$ where $E^{u}=(E^{u}_{a})_{a}$ with $E^{u}_{a}=E^{u}_{0}=\{0^{ss}\}\times\mathbb{R}^{u}\times\{0^{c}\}$ for all $a\in\mathbb{I}^{k}$. Hence $J(E^{u})=(E^{u}_{0},0)$. By a straightforward calculation using Faà di Bruno’s formula, we have

[TABLE]

By means of a similar computation we can show that

[TABLE]

where $t^{u}=(t^{u}_{a})_{a}$ with $t^{u}_{a}=t(E^{u}_{a})=t(E^{u}_{0})=0$ for all $a\in\mathbb{I}^{k}$ and hence $J(t^{u})=0$. Finally putting together (A.2)-(A.4) we get that

[TABLE]

By continuity with respect to $J(E)$, shrinking $\mathcal{C}^{{G}}$ if necessary and taking $\rho>0$ small enough, we have $\|D\widehat{h}^{{G}}\|_{\infty}\leq 1+\varepsilon$ over \big{(}[-2,2]^{ss}\times\mathcal{C}^{{G}}_{\alpha}\big{)}\times\overline{B}_{\rho}(0) for a fixed but arbitrarily small $\varepsilon>0$. This completes the proof. ∎

Acknowledgements

We thank P. Berger who told us the ideais behind his proof in [Ber17a]. We are grateful to J. Rojas for helping us to understand P. Berger’s articles and for the initial discussions on this project. We also thank the anonymous referee for his comments and pointing out some problems.