Homoclinic tangencies leading to robust heterodimensional cycles

Abstract

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

Appendices

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

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

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

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

(7.2)

Applying [4,  Theorem B$'$] to the hyperbolic measure $\mu$, given any $\delta >0$ we get $C^1$ perturbation $h \in \mathrm {Diff}^r(M)$ of f with an affine horseshoe¹⁰$\Gamma _\delta$ whose hyperbolic splitting $E_\delta ^s\oplus E_\delta ^u$ satisfies

• $\Gamma _\delta$ has a constant linear part $A_\delta$ which is a diagonal matrix with distinct real positive eigenvalues,

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

  $$\begin{aligned} \mathrm {Jac}_{E^s_\delta }(A_\delta ) \rightarrow \mathrm {J}^s_\mu \quad \text{ and } \quad \chi ^{cs}(A_\delta ) \rightarrow \chi ^{cs}_\mu \quad \text{ as }\,\delta \rightarrow 0, \end{aligned}$$

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

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

  (7.3)

• $\Gamma _\delta$ is $\delta$-close to the support of $\mu$ in Hausdorff distance,

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

The last inequality implies that

$$\begin{aligned} \begin{aligned} h_{\mathrm {top}}({\Gamma _\delta },{h_\delta })&>h_{\mu }(f)-\delta \overset{(8.1)}{\geqslant } -\log \mathrm {J}^{s}_{\mu }+ \frac{1}{2 r} \chi ^{cs}_{\mu } + \varepsilon - \delta \\&\overset{(8.2)}{>} -\log \mathrm {Jac}_{E_\delta ^s}(A_\delta )+ \frac{1}{2 r} \chi ^{cs}(A_\delta ). \end{aligned} \end{aligned}$$

(7.4)

Observe that the inequality in (7.4) allows us to apply [4,  Theorem C] to the affine horseshoe $\Gamma _\delta$ of $h_\delta$, getting a perturbation $g_\delta$ of $h_\delta$ supported in a small neighbourhood of $\Gamma _\delta$ such that the continuation $\Gamma _{g_\delta }$ of $\Gamma _\delta$ is a cs-blender of central dimension $d_{cs}=m-1$ with a dominated splitting whose bundles are one-dimensional. Taking now a sequence $(\delta _k)\rightarrow 0$, for each large k we get diffeomorphisms $h_k=h_{\delta _k}$ and perturbations $g_k$ of $h_k$ with blenders $\Gamma _k$ as before. These blenders satisfy conditions (1), (2), and (3) in the theorem.

To conclude the proof of the theorem, it remains to see that if the saddle P is homoclinically related to $\mu$ then we can chose the perturbations $g_k$ such that the continuations $P_k$ of P and $\Gamma _k$ of $\Gamma$ are homoclinically related. To see this, we need to review the steps in the proof of [4,  Theorem B$'$], which is a combination of [4,  Theorem B] and Katok’s approximation theorem [33] and its extensions in [4,  Theorem. 3.3] and [22,  Theorem 2.12].

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

• (a) $\Lambda$ is homoclinically related to $\mu$,

• (b) $\Lambda$ is close to the support of $\mu$ in the Hausdorff distance,

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

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

• (e) the Lyapunov exponents of any ergodic measure of $\Lambda$ are close to the Lyapunov exponents of $\mu$.

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

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

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

Appendix: Dominated splittings

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

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

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

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

A Df invariant bundle $T_\Lambda M= E_1\oplus \cdots \oplus E_k$ with k bundles, $k \geqslant 3$, is dominated if the bundles $T_\Lambda M= (E_1\oplus \cdots \oplus E_i) \oplus ( E_{i+1} \oplus \cdots \oplus E_k)$ are dominated for every $i\in \{1,\dots , k-1\}$.

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

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

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

Given a splitting $E\oplus F$ and $\alpha >0$, the cone field of size $\alpha >0$ around the bundle F, denoted by $\mathcal {C}_{\alpha ,F}$, consist of all vectors $v=v_E + v_F$, $v_E \in E$ and $v_F$ in F, such that the norm of $v_F$ is larger or equal than the norm of $\alpha v_E$. Below, for simplicity, we omit the dependence on $\alpha$ of the cone fields.

The following proposition is a well-known result about dominated splittings. We refer to [20,  Chapter B.2] for details.

Proposition 7.3

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

• (1) the bundles of the dominated splitting depend continuously with the point x and the map g,

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

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

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