3.1 Local map and transition map

Recall that $f\in \mathrm {Diff}^r(\mathcal {M})$ with $r=2,\ldots ,\infty ,\omega $ has a heterodimensional cycle involving two hyperbolic periodic orbits $L_1$ and $L_2$ , where their indices $d_1$ and $d_2$ satisfy $d_1+1=d_2$ . Let $\Gamma ^0$ be a heteroclinic orbit in $W^u(L_1)\cap W^s(L_2)$ and $\Gamma ^1$ be a heteroclinic orbit in $W^s(L_1)\cap W^u(L_2)$ . We further denote the heterodimensional cycle as the closed invariant set $ \Gamma =L_1\cup L_2\cup \Gamma ^0\cup \Gamma ^1. $

Take two points $O_1$ and $O_2$ from the two periodic orbits $L_1$ and $L_2$ . For $i=1,2$ , let $U_{0i}$ be some sufficiently small neighborhoods of $O_i$ . We define the local maps $F_i:=f^{per(O_i)}|_{U_{0i}}$ , where $per(O_i)$ are the periods of $O_i$ . Next, using the heteroclinic orbits $\Gamma ^0$ and $\Gamma ^1$ , we define transition maps connecting neighborhoods of $O_1$ and $O_2$ . Take in $\Gamma ^0$ a pair of points $M^-_1\in W^u_{\mathrm {loc}}(O_1)$ and $M^+_2\in W^s_{\mathrm {loc}}(O_2)$ so that $f^{n_{1}}(M^-_1)=M^+_2$ for some integer $n_1>0$ , and in $\Gamma ^1$ , a pair of points $M^-_2\in W^u_{\mathrm {loc}}(O_2)$ and $M^+_1\in W^s_{\mathrm {loc}}(O_1)$ so that $f^{n_{2}}(M^-_2)=M^+_1$ for some integer $n_2>0$ . The transition maps are then defined as $F_{12}:=f^{n_1}$ , taking a small neighborhood of $M^-_1$ to a small neighborhood of $M^+_2$ ; and $F_{21}:=f^{n_2}$ , taking a small neighborhood of $M^-_2$ to a small neighborhood of $M^+_1$ . The first return maps are defined as compositions of the local map and transition map.

As mentioned, we assume that the cycle of f is non-degenerate in the sense of [Reference Li and Turaev26], which can always be achieved by an arbitrarily $C^r$ -small perturbation. There are four non-degeneracy conditions, whose precise definitions are given in Appendix A. Roughly speaking, the first three require that:

• • $T_1(W^u_{\mathrm {loc}}(O_1))$ intersects transversely the local extended stable manifold $W^{sE}(O_2)$ at $M^+_2$ and $T^{-1}_1(W^s_{\mathrm {loc}}(O_2))$ intersects transversely the local extended unstable manifold $W^{uE}(O_1)$ at $M^-_1$ ;

• • $T_2^{-1}(W^s_{\mathrm {loc}}(O_1))$ is not tangent to the strong-unstable leaf $\ell ^{uu}\subset W^u(O_2)$ at $M^-_2$ and $T_2(W^u_{\mathrm {loc}}(O_2))$ is not tangent to the strong-stable leaf $\ell ^{ss}\subset W^s(O_1)$ at $M^+_1$ ; and

• • $M^+_1$ does not belong to the strong-stable manifold $W^{ss}_{\mathrm {loc}}(O_1)$ and $M^-_2$ does not belong to the strong-unstable manifold $W^{uu}_{\mathrm {loc}}(O_2)$ .

Here, $\ell ^{uu},\ell ^{ss},W^{ss}_{\mathrm {loc}}(O_1), W^{uu}_{\mathrm {loc}}(O_2)$ are defined in the usual way; $W^{sE}_{\mathrm {loc}}(O_2)$ is the invariant manifold corresponding to the stable and weak-unstable directions at $O_2$ , and similarly for $W^{uE}_{\mathrm {loc}}(O_1)$ . Figure 2 provides an illustration. The last non-degeneracy condition is more delicate, requiring that the projections of the transverse intersection points $M^+_1$ and $M^-_2$ to the central directions do not lie in certain lines. See Appendix A for details.

Figure 2

Here, $W^u_{\mathrm {loc}}(O_1)$ is the y-axis, $W^s_{\mathrm {loc}}(O_1)$ is the $(x,z)$ -plane, $W^s_{\mathrm {loc}}(O_2)$ is the v-axis, and $W^u_{\mathrm {loc}}(O_2)$ is the $(u,w)$ -plane. The two red curves are $\ell _1=W^s_{\mathrm {loc}}(O_1)\cap F_{21}(W^u_{\mathrm { loc}}(O_2))$ and $\ell _2=F^{-1}_{21}(\ell _1)$ (color online).

Figure 2 Long description

The diagram consists of two three-dimensional coordinate systems connected by curved arrows.

On the left, the first system is centered at origin O sub 1. The vertical axis is y, the horizontal axis is x, and the depth axis is z. A vertical curved surface is labeled F sub 1 2 super negative 1 open parenthesis W sub l o c super s open parenthesis O sub 2 close parenthesis close parenthesis. A point M sub 1 super minus lies on this surface along the y-axis. To the right of the origin, a smaller curved surface contains a red curve labeled ell sub 1 and a point M sub 1 super plus.

On the right, the second system is centered at origin O sub 2. The vertical axis is v, the horizontal axis is u, and the depth axis is w. A vertical curved surface is labeled F sub 1 2 open parenthesis W sub l o c super u open parenthesis O sub 1 close parenthesis close parenthesis. A point M sub 2 super plus lies on this surface along the v-axis. Below the u-axis, a smaller curved surface contains a red curve labeled ell sub 2 and a point M sub 2 super minus.

Two large curved arrows connect the systems. The top arrow points from left to right and is labeled F sub 1 2. The bottom arrow points from right to left and is labeled F sub 2 1. Labels for stable and unstable manifolds, such as W sub l o c super u open parenthesis O sub 1 close parenthesis and W sub l o c super s open parenthesis O sub 2 close parenthesis, are positioned near the vertical axes.

3.2 Saddle-focus case

The first return map of a saddle-focus heterodimensional cycle is defined as the composition $T_{k,m}:=F_{21}\circ F^m_2\circ F_{12}\circ F^k_1$ , which, for suitable choices of $(k,m)$ , takes points in a small neighborhood of $M^+_1$ back to the neighborhood. It is shown in [Reference Li and Turaev26, §§6.1.1–6.1.3] that there exist coordinates $(X,Y,Z)\in \mathbb {R}\times \mathbb {R}^{d_1}\times \mathbb {R}^{d-d_1-1}$ near $M^+_1$ such that the first return map can be expressed in a cross-form, which manifests the partial hyperbolicity, for a collection of pairs $(k,m)$ . In what follows, we introduce this formula.

Recall that we denote the pair of complex conjugate center-stable multipliers by $\unicode{x3bb} e^{\pm i\omega }\, (0<\omega <\pi )$ and the real center-unstable multiplier by $\gamma $ . For $\delta>0$ , define

(4) $$ \begin{align} \Pi=[-\delta,\delta]\times[-\delta,\delta]^{d_1}\times [-\delta,\delta]^{d-d_1-1}, \end{align} $$

where $d_1=\operatorname {ind}(O_1)$ .

It is proven that there exist coefficients $A,B,b,\eta _{1,2}\in \mathbb {R}$ (which are expressed in terms of the derivatives $\mathrm {D} F_{12}|_{M^-_1}$ and $\mathrm {D} F_{21}|_{M^+_2}$ ), $u^-\in \mathbb {R}$ (which is given by the position of $M^-_2$ , see (A.1)) and a positive number $C(\delta )$ depending on $\delta $ such that, for all sufficiently small $\delta $ and sufficiently large k and m satisfying

(5) $$ \begin{align} |\unicode{x3bb}^k\gamma^m B\sin (k\omega +\eta_2)-bu^-|<C(\delta), \end{align} $$

one has that $(\bar X,\bar Y,\bar Z):=T_{k,m}(X,Y,Z)\in \Pi $ for a point $(X,Y,Z)\in \Pi $ if and only if

(6) $$ \begin{align} \begin{aligned} \bar X &= A_{k,m} X + B_{k,m} +\phi_{1}(X,\bar Y,Z),\\ Y &=\phi_2(X,\bar Y,Z),\quad \bar Z = \phi_3(X, \bar Y,Z), \end{aligned} \end{align} $$

where the functions $\phi $ satisfy

$$ \begin{align*}\phi_1=o(\delta)+o(1)_{k,m\to\infty},\quad \dfrac{\partial \phi_1}{\partial (X,\bar Y,Z)}=o(1)_{\delta\to 0}+o(1)_{k,m\to\infty},\quad \|\phi_{2,3}\|_{C^1}\!=o(1)_{k,m\to\infty},\end{align*} $$

and

(7) $$ \begin{align} A_{k,m}=\unicode{x3bb}^k\gamma^m A \sin(k\omega + \eta_1),\quad B_{k,m}=\unicode{x3bb}^k\gamma^m B\sin(k\omega + \eta_2)- b u^-. \end{align} $$

(Equation (6) is obtained from [Reference Li and Turaev26, (6.18)] by dropping the subscript of $X_1$ and grouping the coordinates $(X_2,Z)=:Z$ .)

The non-degeneracy conditions ensure (see [Reference Li and Turaev26, §6.1.3]) that $A> 0,B> 0,b\neq 0, u^-\neq 0$ and

(8) $$ \begin{align} \tan\eta_1\neq\tan\eta_2. \end{align} $$

The partial hyperbolicity of $T_{k,m}$ follows immediately from a computation of the invariant cone fields using (6). Denote by $(\Delta X,\Delta Y,\Delta Z)$ the vectors in the tangent spaces.

Lemma 3.2. [Reference Li and Turaev26, Lemma 6.1]

Given any $K>0$ , one can choose $\delta $ sufficiently small such that, for all sufficiently large $k,m$ for which $\sin (k\omega +\eta _1)$ and $\sin (k\omega +\eta _2)$ are bounded away from zero, the cone fields

(9) $$ \begin{align} \begin{aligned} &\mathcal{C}^{cu} = \{(\Delta X,\Delta Y,\Delta Z): \|\Delta Z\| \leqslant K (|\Delta X|+\|\Delta Y\|)\},\\ &\mathcal{C}^{uu} = \{(\Delta X,\Delta Y,\Delta Z): \max\{|\Delta X|,\|\Delta Z\|\}\leqslant K \|\Delta Y\|\}, \end{aligned} \end{align} $$

which are defined on $\Pi $ , are forward-invariant in the sense that if a point $M\in \Pi $ has its image $\bar M=T_{k,m}(M)$ in $\Pi $ , then the cone at M is mapped strictly into the cone at $\bar M$ by $\mathrm {D} T_{k,m}$ ; the cone fields

(10) $$ \begin{align} \begin{aligned} &\mathcal{C}^{cs} = \{(\Delta X,\Delta Y,\Delta Z): \|\Delta Y\|\leqslant K (|\Delta X|+\|\Delta Z\|)\},\\ &\mathcal{C}^{ss}=\{(\Delta X,\Delta Y,\Delta Z):\max\{|\Delta X|,\|\Delta Y\|\}\leqslant K \|\Delta Z\|\}, \end{aligned} \end{align} $$

which are defined on $\Pi $ , are backward-invariant in the sense that if a point $\bar {M}\in \Pi $ has its preimage $M=T^{-1}_{k,m}(\bar M)$ in $\Pi $ , then the cone at $\bar {M}$ is mapped strictly into the cone at M by $\mathrm {D} T^{-1}_{k,m}$ . Moreover, vectors in $\mathcal {C}^{uu}$ and, if $|A_{k,m}|>1$ , also vectors in $\mathcal {C}^{cu}$ are expanded by $\mathrm {D} T_{k,m}$ ; vectors in $\mathcal {C}^{ss}$ and, if $|A_{k,m}|<1$ , also vectors in $\mathcal {C}^{cs}$ are contracted by $\mathrm {D} T_{k,m}$ .

3.3 Double-focus case

In this case, we consider the first return map $T_{k,m}:=F^m_2\circ F_{12}\circ F^k_1\circ F_{21}$ near $M^-_2$ . A cross-form formula for $T_{k,m}$ is obtained in [Reference Li and Turaev26, §§6.4.1–6.4.3] with coordinates $(U,V,W)\in \mathbb {R}\times \mathbb {R}^{d-d_2}\times \mathbb {R}^{d_2-1}$ in a small neighborhood $M^-_2$ , which we describe below. For simplicity, many coefficients here carry the same notation as those in the saddle-focus case, but they are different coefficients. This does not cause ambiguity as we never discuss the two cases at the same time in the proofs.

Recall that for a double-focus heterodimensional cycle, we denote the center-stable multipliers by $\unicode{x3bb} e^{\pm i\omega _1}\, (0<\omega _1<\pi )$ and the center-unstable multipliers by $\gamma e^{\pm i\omega _2}\, (0<\omega _2<\pi )$ . For $\delta>0$ , define

(11) $$ \begin{align} \Pi=[-\delta,\delta]\times[-\delta,\delta]^{d-d_2}\times [-\delta,\delta]^{d_2-1}, \end{align} $$

where $d_2=\operatorname {ind}(O_2)$ .

It is proven that there exist coefficients $A^*,B^*,b,\eta _{1,2,3}$ (which are expressed in terms of derivatives $\mathrm {D} F_{12}|_{M^-_1}$ and $\mathrm {D} F_{21}|_{M^+_2}$ ), $u^-_{1,2}$ (which is given by the position of $M^-_2$ , see (A.1)) and a positive number $c(\delta )=o(1)_{\delta \to 0}$ depending on $\delta $ such that, for all sufficiently small $\delta $ , and sufficiently large k and m satisfying

(12) $$ \begin{align} \!\!\!\min\{|\cos m\omega_2|,\, |\cos(m\omega_2+\eta_3)|,\, |\cos(m\omega_2+\eta_3) + b\sin(m\omega_2+\eta_3)|\}>c(\delta),\ \end{align} $$

one has that $(\bar U,\bar V,\bar W):=T_{k,m}(U,V,W)\in \Pi $ for a point $(U,V,W)\in \Pi $ if and only if

(13) $$ \begin{align} \begin{gathered} \bar U = A_{k,m}U+B_{k,m} + \phi_1(U,V,\bar W),\\ \bar{V}=\phi_2(U,V,\bar W),\quad W=\phi_3(U,V,\bar W), \end{gathered} \end{align} $$

where the functions $\phi $ satisfy

$$ \begin{align*} & \phi_1=o(\delta)+o(1)_{k,m\to\infty},\quad \dfrac{\partial \phi_1}{\partial (U, V,\bar W)} \\ &\quad =o(1)_{\delta\to 0}+o(1)_{k,m\to\infty},\quad \|\phi_{2,3}\|_{C^1}=o(\delta)+o(1)_{k,m\to\infty}, \end{align*} $$

and

(14) $$ \begin{align} A_{k,m}=\unicode{x3bb}^k\gamma^mA(m\omega_2)\sin(k\omega_1+\eta_1),\quad B_{k,m}=\unicode{x3bb}^k\gamma^m B(m\omega_2)\sin(k\omega_1+\eta_2) -\xi(m\omega_2) \end{align} $$

with

(15) $$ \begin{align} \begin{aligned} A(m\omega_2)&=\dfrac{ A^*}{\cos(m\omega_2+\eta_3) + b\sin(m\omega_2+\eta_3)}, \\ B(m\omega_2)&=\dfrac{ B^*}{\cos(m\omega_2+\eta_3) + b\sin(m\omega_2+\eta_3)}, \\ \xi(m\omega_2)&=\dfrac{\cos(m\omega_2+\eta_3)u^-_1+\sin(m\omega_2+\eta_3)u^-_2}{\cos(m\omega_2+\eta_3) + b\sin(m\omega_2+\eta_3)}. \end{aligned} \end{align} $$

(Equation (13) is obtained from [Reference Li and Turaev26, (6.44)] by dropping the subscript of $U_1$ , grouping the coordinates $(U_2,W)=:W$ , and dividing $(1+b_{41}\tan (m\omega +\eta _3))$ to both sides of (6.44).)

The non-degeneracy conditions ensure that $A^*> 0,B^*> 0,b\neq 0,u^-\neq 0$ and $\tan \eta _1\neq \tan \eta _2$ , see [Reference Li and Turaev26, §6.4.3].

One sees that (13) assumes the same form as (6), with replacing $U,V,W$ by $X,Z,Y$ , respectively. By a completely parallel computation, one finds the following counterpart to Lemma 3.2, which characterizes the partial hyperbolicity of $T_{k,m}$ .

Lemma 3.3. Given any $K>0$ , one can choose $\delta $ sufficiently small such that for all sufficiently large $k,m$ satisfying (12) and (30), the cone fields

(16) $$ \begin{align} \begin{aligned} &\mathcal{C}^{cu} = \{(\Delta U,\Delta V,\Delta W): \|\Delta V\| \leqslant K (|\Delta U|+\|\Delta W\|)\},\\ &\mathcal{C}^{uu} = \{(\Delta U,\Delta V,\Delta W): \max\{|\Delta U|,\|\Delta V\|\}\leqslant K \|\Delta W \|\}, \end{aligned} \end{align} $$

which are defined on $\Pi $ , are forward-invariant in the sense that if a point $M\in \Pi $ has its image $\bar M=T_{k,m}(M)$ in $\Pi $ , then the cone at M is mapped into the cone at $\bar M$ by $\mathrm {D} T_{k,m}$ ; the cone fields

(17) $$ \begin{align} \begin{aligned} &\mathcal{C}^{cs} = \{(\Delta U,\Delta V,\Delta W): \|\Delta W\|\leqslant K (|\Delta U|+\|\Delta V\|)\},\\ &\mathcal{C}^{ss}=\{(\Delta U,\Delta V,\Delta W):\max\{|\Delta U|,\|\Delta W\|\}\leqslant K \|\Delta V\|\}, \end{aligned} \end{align} $$

which are defined on $\Pi $ , are backward-invariant in the sense that if a point $\bar {M}\in \Pi $ has its preimage $M=T^{-1}_{k,m}(\bar M)$ in $\Pi $ , then the cone at $\bar {M}$ is mapped into the cone at M by $\mathrm {D} T^{-1}_{k,m}$ . Moreover, vectors in $\mathcal {C}^{uu}$ and, if $|A_{k,m}|>1$ , also vectors in $\mathcal {C}^{cu}$ are expanded by $\mathrm {D} T_{k,m}$ ; vectors in $\mathcal {C}^{ss}$ and, if $|A_{k,m}|<1$ , also vectors in $\mathcal {C}^{cs}$ are contracted by $\mathrm {D} T_{k,m}$ .

4.1 A criterion for the existence of blenders

Let us first introduce the notion of activating pair to further characterize the embedding set $\mathcal {D}$ in Definition 1.1. Take a cube

$$ \begin{align*}Q=\{(x_1,\ldots,x_d)\mid x_i\in I_i\}\subset \mathbb{R}^d,\end{align*} $$

where $I_i$ are closed intervals in $\mathbb {R}$ . A k-dimensional disc S is said to cross Q if the intersection $S\cap Q$ is given by $(x_{i_{k+1}},\ldots ,x_{i_d})=s(x_{i_1},\ldots ,x_{i_k})$ , where $(i_1,\ldots ,i_d)$ is some permutation of $(1,\ldots ,d)$ and s is a smooth function defined on $I_{i_1}\times \cdots \times I_{i_k}$ . It crosses Q properly with respect to a cone field $\mathcal {C}$ defined on Q if the tangent spaces of $S\cap Q$ lie in $\mathcal {C}$ , that is, the tangent spaces of S lie in $\mathcal {C}$ .

For an invariant cone field $\mathcal {C}$ , we use $\dim (\mathcal {C})$ to denote the largest dimension of linear subspaces of the tangent spaces contained in $\mathcal {C}$ .

For example, if $(\Pi ,\mathcal {C}^{ss})$ , with $\Pi $ in (4) and $\mathcal {C}^{ss}$ in (10), is an activating pair of some cs-blender, then all curves crossing $\Pi $ properly with respect to $\mathcal {C}^{ss}$ belong to the embedding set $\mathcal {D}$ in Definition 1.1. Each such curve is the graph of some smooth function $(X,Y)=s(Z)$ , which is defined on $[-\delta ,\delta ]^{d-d_1-1}$ and whose tangent spaces lie in $\mathcal {C}^{ss}$ .

Let $\delta>0$ and take a cube

$$ \begin{align*}Q=[-\delta,\delta]\times [-\delta,\delta]^{d_Y}\times [-\delta,\delta]^{d_Z}\end{align*} $$

with $1+d_Y+d_Z=d$ . Let $\{T_{n}\}_{n\in \mathbb N}$ be a sequence of maps defined on pairwise disjoint subsets $\{\sigma _n\}$ of Q such that $T_i(\sigma _i)\cap T_j(\sigma _j)\cap Q =\emptyset $ for $i\neq j$ . Define $T:\bigcup _n \sigma _n \to \mathbb {R}^d$ by $T(X,Y,Z)=T_n(X,Y,Z)$ if $(X,Y,Z)\in \sigma _n$ . Suppose for every $n\in \mathbb N$ and $(X,Y,Z)\in Q$ , one has that $(\bar X,\bar Y,\bar Z)=T_n (X,Y,Z)$ if and only if

(18) $$ \begin{align} \begin{aligned} \bar X &= A_{n} X + B_{n} + \phi_1(X,\bar{Y},Z;n), \\ Y &=\phi_2(X,\bar{Y},Z;n), \\ \bar Z &= \phi_3(X,\bar{Y},Z;n), \end{aligned} \end{align} $$

where the constants $A_n$ and $B_n$ , and the $C^1$ functions $\phi $ defined on Q are allowed to depend on $\delta $ , and we have the estimates

$$ \begin{align*} \begin{aligned} \phi_1&=o(\delta)+o(1)_{n\to\infty},\\ \dfrac{\partial \phi_1}{\partial (X,\bar Y,Z)} & =o(1)_{\delta\to 0}+o(1)_{n\to\infty},\\ \|\phi_{2,3}\|_{C^1}&=o(1)_{\delta\to 0}+o(1)_{n\to\infty}. \end{aligned} \end{align*} $$

Proposition 4.2. [Reference Li and Turaev26, Proposition 4.12]

Suppose there exists $\delta _0>0$ such that the constants $A_n$ and $B_n$ satisfy the following.

1. (1) For each $\delta \in (0,\delta _0)$ , the set $\{B_n\}$ is dense in an interval I that contains zero and is independent of $\delta $ .

2. (2) For all $\delta \in (0,\delta _0)$ and all $n\in \mathbb N$ either:

   1. (2.1) $0<C_1<|A_n|<C_2<1$ with $C_{1,2}$ independent of n; or

   2. (2.2) $1<C_3<|A_n|<C_4<\infty $ with $C_{3,4}$ independent of n.

Then, there exist $c\in (0,1)$ and

(19) $$ \begin{align} Q'=[-c\delta,c\delta]\times [-\delta,\delta]^{d_Y}\times [-\delta,\delta]^{d_Z}, \end{align} $$

such that for every $\delta \in (0,\delta _0)$ , one can find a finite set $\mathcal {N}\subset \mathbb {N}$ with arbitrarily large elements such that the restriction $T_{\mathcal {N}}$ of T to $\bigcup _{n\in \mathcal {N}} \sigma _n\cap Q$ has a blender, which consists of points whose orbits never leave Q under $T_{\mathcal {N}}$ , and has an activating pair $(Q',\mathcal {C})$ with the cone field $\mathcal {C}$ independent of $\delta $ and n. Moreover:

• • if condition (2.1) holds, $\Lambda $ is a cs-blender of index $d_Y$ and cones in $\mathcal {C}$ are around Z-coordinates;

• • if condition (2.2) holds, $\Lambda $ is a cu-blender of index $(d_Y+1)$ and cones in $\mathcal {C}$ are around Y-coordinates.

In our case, the maps $T_n$ are the return maps $T_{k_n,m_n}$ for some sequence of pairs satisfying conditions (1) and (2). For completeness, we sketch the proof of this proposition in Appendix B.1.

4.2 Existence of blenders when at least one of $\theta $ and $\omega /2\pi $ is irrational

Recall that for a saddle-focus heterodimensional cycle, we denote the complex center-stable multipliers by $\unicode{x3bb} e^{\pm i\omega }$ , where $\unicode{x3bb}>0$ and $0<\omega <\pi $ , and the real center-unstable multiplier by $\gamma $ . Let $A>0,B>0,b,u^-,\eta_{1,2}$ be the coefficients in (7). We introduce a condition on the modulus $\omega $ .

1. A1: There exist infinitely many $k>0$ such that

   (20) $$ \begin{align} bu^-\sin(k\omega +\eta_2)>0,\quad \sin(k\omega +\eta_1)\neq 0,\quad \alpha_k:=\dfrac{bu^-A\sin(k\omega+\eta_1)}{\sin(k\omega+\eta_2)}\neq\pm 1. \end{align} $$

Observe that condition A1 is automatically satisfied if $\omega /2\pi $ is irrational or if $\omega /2\pi = p/q$ with $q\geqslant 9$ . To see the latter, note first that the last inequality in (20) can be rewritten as

$$ \begin{align*}|bu^- \sin(k\omega +\eta_1)|\neq|\sin(k\omega +\eta_1)\cos(\eta_2-\eta_1) +\sin(k\omega +\eta_1)\sin(\eta_2-\eta_1)|\end{align*} $$

or

$$ \begin{align*}|(bu^--\cos(\eta_2-\eta_1))\tan(k\omega +\eta_1)|\neq|\sin(\eta_2-\eta_1)|.\end{align*} $$

It follows that to achieve simultaneously this inequality and the second one in (20), it is enough to have at least four different values of $k\omega \bmod 2\pi $ in $[0,1/2)$ or in $[1/2,1)$ . It is easy to check that when $\omega /2\pi =p/q$ , there are at least $ \left \lfloor {(q-1)/2} \right \rfloor $ different values of $k\omega \bmod 2\pi $ both in $[0,1/2)$ and in $[1/2,0)$ , where $ \left \lfloor {a} \right \rfloor $ means the largest integer that is smaller than or equal to a. Choosing the values from the suitable interval also gives $bu^-\sin (k\omega +\eta _2)>0$ . Thus, when $ \left \lfloor {(q-1)/2} \right \rfloor \geqslant 4$ or $q\geqslant 9$ , there exists some $r\in \{0,\ldots ,q-1\}$ such that all $k=r+iq$ with $i=0,1,2,\ldots $ satisfy the inequalities in (20).

Proposition 4.3. Let $\delta $ be the size of $\Pi $ defined in (4), and $\mathcal {C}^{ss}$ and $\mathcal {C}^{uu}$ be given by Lemma 3.2 with a sufficiently small K. Let a non-degenerate saddle-focus heterodimensional cycle satisfy condition A1 and U be any neighborhood of the cycle.

1. (1) If $\theta =\ln \unicode{x3bb} /\ln |\gamma |$ is irrational and $\omega /2\pi =p/q$ , then, for every sufficiently large k satisfying condition A1, there exists a cs-blender in U when $|\alpha _k|<1$ (defined in (20)) or a cu-blender in U when $|\alpha _k|>1$ .

2. (2) If $\omega /2\pi $ is irrational, and $\theta ,\omega /2\pi $ and 1 are not rationally independent, then there exists a cu-blender in U.

3. (3) If $\theta ,\omega /2\pi $ and 1 are rationally independent, then there exist simultaneously a cs-blender and a cu-blender in U.

In each of the above cases, the cs-blender has index $d_1$ and an activating pair $(\Pi ',\mathcal {C}^{ss})$ , while the cu-blender has index $(d_2=d_1+1)$ and an activating pair $(\Pi ',\mathcal {C}^{uu})$ , where the cube $\Pi '$ is defined as

(21) $$ \begin{align} \Pi'=[-c\delta,c\delta]\times[-\delta,\delta]^{d_1}\times [-\delta,\delta]^{d-d_1-1}\subset \Pi, \end{align} $$

for some $c\in (0,1)$ .

Proof. We prove the proposition by finding, in each case, a sequence of pairs $\{(k_n,m_n)\}$ such that the sequence of return maps $\{T_{k_n,m_n}\}$ satisfies the conditions of Proposition 4.2. Let $A>0,B>0,b,u^-,\eta_{1,2}$ be the coefficients in (6).

Case (1): ${\theta }$ is irrational and $ {\omega /2\pi }$ is rational. Take $\omega /2\pi =p/q$ . Since condition A1 is satisfied, there exists $k=r\in \{0,\ldots ,q-1\}$ satisfying (20). We particularly have

(22) $$ \begin{align} \sin \bigg( \dfrac{2\pi rp}{q} +\eta_2\bigg)Bbu^-=: R Bbu^->0. \end{align} $$

Since $\theta $ is irrational, the set

$$ \begin{align*} \{m-(kq+r)\theta \}_{k\in \mathbb{N},m\in 2\mathbb N} \end{align*} $$

is dense in $\mathbb R$ and, hence, $\{\gamma ^{m-(kq+r)\theta }\}_{k\in \mathbb {N},m\in 2\mathbb N}$ is dense in $[0,\infty ]$ . This along with (22) implies that one can find a sequence $\{(k^{\prime }_nq+r,m^{\prime }_n)\}$ such that $\{\gamma ^{m^{\prime }_n-(k^{\prime }_nq+r)\theta }\}$ is dense in a small neighborhood of $bu^-/BR$ . Thus, the sequence

(23) $$ \begin{align} \{|\gamma|^{m^{\prime}_n-(k^{\prime}_nq+r)\theta }B\sin((k^{\prime}_nq+r)\omega+\eta_2)-bu^-\} \end{align} $$

is dense in a small neighborhood $\Delta $ of zero.

Recall $\theta =-\ln \unicode{x3bb} /\ln |\gamma |$ (note that $\unicode{x3bb}>0$ ). When the integer m is even, we can write

(24) $$ \begin{align} \unicode{x3bb}^k\gamma^m=|\gamma|^{m-\theta k}. \end{align} $$

Now, denote $k_n=k^{\prime }_nq+r$ and $m_n=m^{\prime }_n$ . The preceding paragraph shows that $\sin (k_n\omega +\eta _2)$ is bounded away from zero. In addition, since $\unicode{x3bb}>0$ and $m_n$ are even, (24) implies that $\unicode{x3bb} ^{k_n}\gamma ^{m_n}=\gamma ^{m_n-k_n\theta }$ . It follows that by taking $\Delta $ sufficiently small, condition (5) is satisfied with $(k,m)=(k_n,m_n)$ . Hence, the first return maps $T_n:=T_{k_n,m_n}$ take the form (6). By comparing (7) with (23), we find that the sequence (23) is exactly the constant term $B_{k_n,m_n}$ in (6). Thus, setting $X=X_1$ and $Z^{new}=(X_2,Z)$ , we find that the sequence $\{T_n\}$ satisfies condition (1) of Proposition 4.2.

Next, let us check condition (2) of Proposition 4.2. To find the values of $|A_{k_n,m_n}|$ , we use (7) to write

(25) $$ \begin{align} A_{k_n,m_n}=\dfrac{(B_{k_n,m_n}+bu^-)A\sin(k_n\omega+\eta_1)}{\sin(k_n\omega+\eta_2)}. \end{align} $$

On the one hand, by taking $\Delta $ sufficiently small, the contribution of $B_{k_n,m_n}$ to $A_{k_n,m_n}$ can be ignored. On the other hand, by the choice of $k_n$ , one has $\sin (k_n\omega +\eta _1)=\sin (r\omega +\eta _1)$ and $\sin (k_n\omega +\eta _2)=\sin (r\omega +\eta _2)$ . Thus, we have

(26) $$ \begin{align} A_{k_n,m_n}\sim \dfrac{bu^-A\sin(r\omega+\eta_1)}{\sin(r\omega+\eta_2)}=:\alpha_r. \end{align} $$

Since assumption (20) ensures $|\alpha _r|\neq 1$ , this gives condition (2.1) of Proposition 4.2 if $|\alpha _r|<1$ or condition (2.2) if $|\alpha _r|>1$ . Therefore, using Proposition 4.2, we obtain, in the former case, a cs-blender with an activating pair $(\Pi ',\mathcal {C}_1)$ for some cone field $\mathcal {C}_1$ and, in the latter case, a cu-blender with an activating pair $(\Pi ',\mathcal {C}_2)$ for some cone field $\mathcal {C}_2$ .

Since these two cone fields are independent of $\delta $ and n, one can take $\delta $ small and n large so that $\mathcal {C}^{ss}$ and $\mathcal {C}^{uu}$ in Lemma 3.2 are contained in $\mathcal {C}_1$ and $\mathcal {C}_2$ , respectively. As a result, $(\Pi ',\mathcal {C}^{ss})$ and $(\Pi ',\mathcal {C}^{uu})$ are also activating pairs.

Case (2): $ {\omega /2\pi }$ is irrational, and $ {\theta ,\omega /2\pi }$ and 1 are rationally dependent. We can write

$$ \begin{align*} \theta =\dfrac{p}{q} \dfrac{\omega}{2\pi} + \dfrac{p'}{q'}. \end{align*} $$

Consider the function

$$ \begin{align*}h(s)=|\gamma|^{ -({p}/{q})s}B\sin (2\pi s +\eta_2)-bu^-,\end{align*} $$

which is defined on $\mathbb {R}$ . Let $s=s_0$ be a zero of h, that is, $h(s_0)=0$ . It is obvious that $s_0$ can be taken with arbitrarily large absolute value. Since the set

$$ \begin{align*} \bigg\{qq'k\dfrac{\omega}{2\pi}-qq'l\bigg\}_{k,l\in 2\mathbb{N}} \end{align*} $$

is dense in $\mathbb R$ , one can find a sequence $\{qq'k^{\prime }_n\omega /2\pi -qq'l_n\}$ dense in a small neighborhood of $s_0$ such that, by continuity of $h(s)$ , the sequence

(27) $$ \begin{align} \{|\gamma|^{m_0 -({p}/{q})(qq'k^{\prime}_n({\omega}/{2\pi})-qq'l_n)}B\sin (qq'k^{\prime}_n \omega +\eta_2)-bu^-\} \end{align} $$

is dense in a small neighborhood $\Delta $ of zero.

By (24) and (A1), for even m, we have

$$ \begin{align*}\unicode{x3bb}^k\gamma^m=|\gamma|^{m-k({p}/{q})({\omega}/{2\pi})-k({p'}/{q'})}.\end{align*} $$

Taking $k_n=qq'k^{\prime }_n$ and $m_n=p'qk^{\prime }_n+pq'l_n$ , one readily finds that $B_{k_n,m_n}$ in (6) equals (27). Arguing as in case (1), we obtain condition (1) of Proposition 4.2 with $\{T_n:=T_{k_n,m_n}\}$ .

It is immediate from (27) that $\sin (k_n\omega +\eta _2)$ can be arbitrarily close to zero by taking $|s_0|$ sufficiently large. Since $B_{k_n,m_n}$ is negligible by taking $\Delta $ small, it follows from (8) and (25) that $|A_{k_n,m_n}|>1$ for all n, and, particularly, they are bounded away from $1$ and infinity. So, applying Proposition 4.2 with condition (2.2) gives a cu-blender. The activating pair is obtained as in case (1).

Case (3): $ {\theta ,\omega /2\pi }$ and 1 are rationally independent. The result is an easy consequence of the denseness of the set

$$ \begin{align*} \bigg\{\bigg(k\theta -m,k\dfrac{\omega}{2\pi} - l\bigg)\bigg\}_{k,\in\mathbb N,m\in2\mathbb N,l\in\mathbb Z} \end{align*} $$

in $\mathbb R^2$ . Indeed, one just needs to choose a sequence $\{(k_n,m_n)\}$ satisfying that $\sin (k_n\omega +\eta _2)$ is dense in a small interval $\Delta '$ and $\unicode{x3bb} ^{k_n}\gamma ^{m_n}=|\gamma |^{m_n-k_n\theta } \to t$ for some constant t such that

$$ \begin{align*} \{B_{k_n,m_n}=|\gamma|^{m_n-k_n\theta }B\sin(k_n\omega+\eta_2)-bu^-\} \end{align*} $$

is dense in a small neighborhood $\Delta $ of zero. The latter fact gives condition (1) of Proposition 4.2. The former one implies that, with sufficiently small $\Delta $ , the values $|A_{k_n,m_n}|$ in (25) can lie between zero and 1 by taking $\Delta '$ close to zero, which leads to a cs-blender, or lie between 1 and infinity by taking $\Delta '$ such that $\sin (k_n\omega +\eta _2)$ is close to zero, which leads to a cu-blender. The activating pair is obtained as in case (1).

4.3 Homoclinic relations in the saddle-focus case

We first make some observations on the blenders obtained in §4.2. The cs-blender in Proposition 4.3, denoted by $\Lambda _1$ , is given by Proposition 4.2 and, hence, corresponds to a finite set $\mathcal {P}_1$ of pairs $(k,m)$ such that $0<C_1<|A_{k,m}|<C_2<1$ for $(k,m)\in \mathcal {P}_1$ , where $A_{k,m}$ is as in (6) and $C_{1,2}$ are some constants. Similarly, a cu-blender in Proposition 4.3, denoted by $\Lambda _2$ , corresponds to a finite set $\mathcal {P}_2$ of pairs $(k,m)$ such that $1<C_3<|A_{k,m}|<C_4<\infty $ for $(k,m)\in \mathcal {P}_2$ , where $C_{3,4}$ are some constants. The sets $\Lambda _i\, (i=1,2)$ consist of points whose orbits under $T_{k,m}$ with $(k,m)\in \mathcal {P}_i$ never leave the cube $\Pi $ defined in (4) and, hence, are hyperbolic basic sets equipped with the cone fields in Lemma 3.2.

For each $i=1,2$ and a point $M_i\in \Lambda _i\cap \Pi '$ , define its local invariant manifolds $W^s_{\mathrm {loc}}(M_i)$ and $W^u_{\mathrm { loc}}(M_i)$ as the connected components through $M_i$ of $W^s_{\mathrm {loc}}(\Lambda _i)\cap \Pi $ and of $W^u_{\mathrm {loc}}(\Lambda _i)\cap \Pi $ , respectively. One readily finds from Lemma 3.2 that those manifolds are given by smooth functions of the following forms:

• • $W^s_{\mathrm {loc}}(M_1):Y=\psi ^s_{M_1}(X,Z)$ defined on $[-\delta ,\delta ]\times [-\delta ,\delta ]^{d-d_1-1}$ ;

• • $W^u_{\mathrm {loc}}(M_1):(X,Z)=\psi ^u_{M_1}(Y)$ defined on $[-\delta ,\delta ]^{d_1}$ and, in particular, $W^u_{\mathrm {loc}}(M_1)$ crosses $\Pi '$ properly with respect to $\mathcal {C}^{uu}$ defined in (9);

• • $W^s_{\mathrm {loc}}(M_2):(X,Y)=\psi ^s_{M_2}(Z)$ defined on $[-\delta ,\delta ]^{d-d_1-1}$ and, in particular, $W^s_{\mathrm {loc}}(M_2)$ crosses $\Pi '$ properly with respect to $\mathcal {C}^{ss}$ defined in (10);

• • $W^u_{\mathrm {loc}}(M_2):Z=\psi ^u_{M_2}(X,Y)$ defined on $[-\delta ,\delta ]\times [-\delta ,\delta ]^{d_1}$ .

An immediate observation is that when a cs-blender and a cu-blender found in Proposition 4.3 coexist (that is, case (3)), they activate each other in the sense of Definition 2.1.

For the other cases in Proposition 4.3, we need the fact (see (3.23) of [Reference Li and Turaev26]) that

(28) $$ \begin{align} W^s_{\mathrm{loc}}(O_1)\cap\Pi=\{Y=0\}\quad\text{and}\quad F_{21}(W^u_{\mathrm{loc}}(O_2))\cap\Pi=\{Z=0\}, \end{align} $$

and the following lemma.

Lemma 4.4. [Reference Li and Turaev26, Lemma 6.5]

If $\omega /2\pi $ is irrational, then there exist infinitely many k such that $F^{-k}_1\circ F^{-1}_{12}(W^s_{\mathrm {loc}}(O_2))$ is a disc of the form

$$ \begin{align*} (X,Y)=s(Z), \end{align*} $$

which crosses $\Pi '$ defined by (21) properly with respect to $\mathcal {C}^{ss}$ defined by (10).

Comparing the equations in (28) with those for $W^u_{\mathrm {loc}}(M_1)$ and $W^s_{\mathrm {loc}}(M_2)$ , we obtain

(29) $$ \begin{align} W^u(O_2)\pitchfork W^s(\Lambda_2)\neq \emptyset \quad\text{and}\quad W^s(O_1)\pitchfork W^u(\Lambda_1)\neq \emptyset. \end{align} $$

Here, the second intersection is used later in the proof of the perturbative results. By the first intersection, the formulas for $W^s_{\mathrm {loc}}(M_2)$ and $W^u_{\mathrm {loc}}(M_2)$ , and Lemma 4.4, one finds that $O_2$ is homoclinically related to the cu-blender found in Proposition 4.3 when $\omega /2\pi $ is irrational (that is, cases (1) and (2)). In this case, it follows from the lambda-lemma that $O_2$ also activates the cs-blender if the cu-blender does. In summary, we obtain the following lemma.

4.4 Proofs of Theorem 1 and the first case in Theorem 3

Theorem 1 follows immediately from Proposition 4.3 and Lemma 4.5.

5.1 Existence of blenders when at least one of $\theta ,{\omega _1}/2\pi ,{\omega _2}/2\pi $ is irrational

Recall that for a double-focus heterodimensional cycle, we denote the center-stable multipliers by $\unicode{x3bb} e^{\pm i\omega _1}$ and the center-unstable multipliers by $\gamma e^{\pm i\omega _2}$ , where $\unicode{x3bb}>0,\gamma >0$ and $0<\omega _{1,2}< \pi $ . Let $A^*>0,B^*>0,b,u^-,\eta_{1,2,3}$ be the coefficients in (14) and (15). We first introduce a condition on the moduli $\omega _1$ and $\omega _2$ .

1. A2: There exist infinitely many pairs $(k,m)$ satisfying (12) and the inequalities

   (30) $$ \begin{align} &(\cos(m\omega_2+\eta_3)u^-_1+\sin(m\omega_2+\eta_3)u^-_2)\sin(k\omega_1+\eta_2)>0, \hspace{9pt} \sin(k\omega_1 +\eta_1)\neq 0, \nonumber\\[5pt] &\alpha_{k,m}:=\dfrac{A^*(\cos(m\omega_2+\eta_3)u^-_1+\sin(m\omega_2+\eta_3)u^-_2)}{(\cos(m\omega_2+\eta_3) + b\sin(m\omega_2+\eta_3))^2} \cdot \dfrac{\sin(k\omega_1+\eta_1)}{\sin(k\omega_1+\eta_2)}\neq\pm 1. \end{align} $$

Obviously, condition A2 is satisfied when both $\omega _1/2\pi $ and $\omega _2/2\pi $ are irrational. This condition also holds if $\omega _1/2\pi $ is irrational and $\omega _2/2\pi =p/q$ with $q\geqslant 8$ , and if $\omega _1/2\pi =p_1/q_1$ with $q_1\geqslant 9$ , and if $\omega _2/2\pi =p_2/q_2$ with $q_2\geqslant 8$ . Indeed, when $q_2\geqslant 8$ , there are at least six different values of $m\omega _2\bmod 2\pi $ in $[0,2\pi )$ , so at least one value will satisfy (12) and $\cos (m\omega _2+\eta _3)u^-_1+\sin (m\omega _2+\eta _3)u^-_2\neq 0$ . Taking this value fixes the factors involving m in (30). Then, arguing as in the saddle-focus case for condition A1, we find that, when $\omega _1$ is irrational or $q_1\geqslant 9$ , there is at least one value of $k\omega _1 \bmod 2\pi $ validating all inequalities in (30).

For the sake of clarity in proving our results, we classify the various possibilities of the arithmetic relations among $\theta ,\omega _1/2\pi ,\omega _2/2\pi $ into seven cases.

Proposition 5.1. Let $\delta $ be the size of $\Pi $ defined in (11), and $\mathcal {C}^{ss}$ and $\mathcal {C}^{uu}$ be given by Lemma 3.3 with a sufficiently small K. Let a non-degenerate double-focus heterodimensional cycle satisfy condition A2 and U be any neighborhood of the cycle.

1. (1) If $\theta $ is irrational and $\omega _1/2\pi ,\omega _2/2\pi $ are rational, then, for every sufficiently large pair $(k,m)$ satisfying condition A2, there exists a cs-blender in U if $|\alpha _{k,m}|<1$ (defined in (30)), or a cu-blender in U if $|\alpha _{k,m}|>1$ .

2. (2.1) If $\omega _1/2\pi $ is irrational, $\omega _2/2\pi $ is rational, and $\theta ,\omega _1/2\pi ,1$ are rationally independent, then there exist simultaneously a cs-blender and a cu-blender in U.

3. (2.2) If $\omega _1/2\pi $ is irrational, $\omega _2/2\pi $ is rational, and $\theta ,\omega _1/2\pi ,1$ are rationally dependent, then there exists a cu-blender in U.

4. (3.1) If $\omega _1/2\pi ,\omega _2/2\pi $ are irrational, and $\theta ,\omega _1/2\pi ,\theta \omega _2/2\pi ,1$ are rationally independent, then there exist simultaneously a cs-blender and a cu-blender in U.

5. (3.2) If $\omega _1/2\pi ,\omega _2/2\pi $ are irrational, $\theta ,\omega _1/2\pi ,\theta \omega _2/2\pi ,1$ are rationally dependent, but $\theta ,\omega _1/2\pi ,1$ are rationally independent, then there exist simultaneously a cs-blender and a cu-blender in U.

6. (3.3.1) If $\omega _1/2\pi ,\omega _2/2\pi $ are irrational, $\theta ,\omega _1/2\pi ,1$ are rationally dependent, and $\omega _1/2\pi ,\theta \omega _2/2\pi ,1$ are rationally dependent, then there exists a cu-blender in U.

7. (3.3.2) If $\omega _1/2\pi ,\omega _2/2\pi $ are irrational, $\theta ,\omega _1/2\pi ,1$ are rationally dependent, and $\omega _1/2\pi ,\theta \omega _2/2\pi ,1$ are rationally independent, then there exists a cu-blender in U.

In each of the above cases, the cs-blender has index $d_1$ and an activating pair $(\Pi ',\mathcal {C}^{ss})$ , while the cu-blender has index $(d_2=d_1+1)$ and an activating pair $(\Pi ',\mathcal {C}^{uu})$ , where the cube $\Pi '$ is defined as

(31) $$ \begin{align} \Pi'=[-c\delta,c\delta]\times[-\delta,\delta]^{d-d_2}\times [-\delta,\delta]^{d_2-1}\subset \Pi \end{align} $$

for some $c\in (0,1)$ .

We postpone the proof of this proposition to §5.3, after we prove the theorems.

5.2 Homoclinic relations in the double-focus case

Let us find out the homoclinic relations among the found blenders in Proposition 5.1, and the periodic points $O_1$ and $O_2$ . Denote by $\Lambda _1$ and $\Lambda _2$ the cs- and cu-blenders. Since the proof of Proposition 5.1 (see §5.4) is also based on the use of Proposition 4.2, the structures of $\Lambda _i\, (i=1,2)$ are completely the same as those in the saddle-focus case, that is, they consist of points whose orbits under the corresponding sets of first return maps never leave the cube $\Pi $ defined in (11), and their activating domain is $\Pi '$ defined in (31). (See the discussion at the beginning of §4.3). One concludes from Lemma 3.3 that for any points $M_1\in \Lambda _1\cap \Pi '$ and $M_2\in \Lambda _2\cap \Pi '$ , their local invariant manifolds are given by smooth functions of the following forms:

• • $W^s_{\mathrm {loc}}(M_1):W=\psi ^s_{M_1}(U,V)$ defined on $[-\delta ,\delta ]\times [-\delta ,\delta ]^{d-d_2}$ with c in (31);

• • $W^u_{\mathrm {loc}}(M_1):(U,V)=\psi ^u_{M_1}(W)$ defined on $[-\delta ,\delta ]^{d_2-1}$ and, in particular, $W^u_{\mathrm {loc}}(M_1)$ crosses $\Pi $ properly with respect to $\mathcal {C}^{uu}$ defined in (16);

• • $W^s_{\mathrm {loc}}(M_2):(U,W)=\psi ^s_{M_2}(V)$ defined on $[-\delta ,\delta ]^{d-d_2}$ and, in particular, $W^s_{\mathrm {loc}}(M_2)$ crosses $\Pi $ properly with respect to $\mathcal {C}^{ss}$ defined in (17);

• • $W^u_{\mathrm {loc}}(M_2):V=\psi ^u_{M_2}(U,W)$ defined on $[-\delta ,\delta ]\times [-\delta ,\delta ]^{d_2-1}$ .

Obviously, whenever two blenders in Proposition 5.1 coexist (that is, cases (2.1), (3.1) and (3.2)), they activate each other in the sense of Definition 2.1.

For other cases in Proposition 5.1, we need the fact, given at the end of [Reference Li and Turaev26, §6.4], that

$$ \begin{align*} W^u_{\mathrm{loc}}(O_2)=\{V=0\} \quad\text{and}\quad F^{-1}_{21}(W^s_{\mathrm{loc}}(O_1))\cap \Pi'=\{W=\psi(U,V)\}, \end{align*} $$

where $\psi =O(U^2+\delta V^2)$ is some smooth function defined on $[-c\delta ,c\delta ]\times [-\delta ,\delta ]^{d-d_2}$ . Comparing these equations with those for $W^s_{\mathrm {loc}}(M_2)$ and $W^u_{\mathrm {loc}}(M_1)$ , we obtain that

(32) $$ \begin{align} W^u(O_2)\pitchfork W^s(\Lambda_2)\neq \emptyset \quad\text{and}\quad W^s(O_1)\pitchfork W^u(\Lambda_1)\neq \emptyset. \end{align} $$

For other heteroclinic intersections, we use the following lemma.

Thus, if $\omega _2/2\pi $ is irrational, then $W^u(O_1)\pitchfork W^s(\Lambda _1)\neq \emptyset $ , which, along with (32), implies that $O_1$ is homoclinically related to the cs-blender $\Lambda _1$ ; if $\omega _1/2\pi $ is irrational, then $W^s(O_2)\pitchfork W^u(\Lambda _2)\neq \emptyset $ , which, along with (32), implies that hence $O_2$ is homoclinically related to the cu-blender $\Lambda _2$ . In summary, we obtain the following lemma.

5.3 Proofs of Theorem 2 and the second case in Theorem 3

5.4 Proof of Proposition 5.1

We first prove an auxiliary result (Lemma 5.5). Define two functions

(33) $$ \begin{align} &g:\mathbb{R}\times S^1\times S^1\to \mathbb R,\quad g(t,s,w)=\gamma^tA(2\pi w)\sin(2\pi s+\eta_1), \end{align} $$

(34) $$ \begin{align} &h:\mathbb{R}\times S^1\times S^1\to \mathbb R,\quad h(t,s,w)=\gamma^tB(2\pi w)\sin (2\pi s+\eta_2) - \xi(2\pi w),\end{align} $$

where $S^1=[0,1]$ with the endpoints identified, and $A,B,\xi $ are given in (15). We consider two classes of lines and one class of planes in $\mathbb {R}^3$ :

(35) $$ \begin{align} \text{(proper horizontal line)}\qquad\qquad\qquad\qquad\{s=s^*,w=w^*\},\qquad\quad \end{align} $$

where $s^*$ and $w^*$ are constants satisfying

(36) $$ \begin{align} \begin{aligned} & 0<\xi(2\pi w^*)B(2\pi w^*)\sin(2\pi s^*+\eta_2)<\infty,\\ &\dfrac{\xi(2\pi w^*)A(2\pi w^*)\sin(2\pi s^*+\eta_1)}{\sin(2\pi s^*+\eta_2)}\not\in\{0,\pm 1,\infty\}; \end{aligned} \end{align} $$

(37) $$ \begin{align} \text{(proper tilted line)}\qquad\{t=a_1s+b,w=a_2s\}\quad\text{or}\quad \{t=a_1s+b,w=w^*\},\quad\end{align} $$

where $a_1^2+b^2\neq 0,a_2\neq 0$ , and $w^*$ satisfies $A(2\pi w^*)B(2\pi w^*)\xi (2\pi w^*)\not \in \{0,\infty \}$ ;

(38) $$ \begin{align} \text{(proper plane)} \qquad\qquad\quad\qquad \{w=at+bs\}\quad\text{or}\quad \{w=w^*\},\end{align} $$

where $a^2+b^2\neq 0$ and $w^*$ satisfies $A(2\pi w^*)B(2\pi w^*)\xi (2\pi w^*)\not \in \{0,\infty \}$ .

Proof. To simplify the proof, we consider g and h as functions $\mathbb {R}^3\to \mathbb {R}$ , which makes no difference by the formulas of g and h. Also note that if a sequence approximates a proper line or a proper plane in $\mathbb {R}\times S^1\times S^1$ , then its lift in $\mathbb {R}^3$ also approximates the lift of the proper line or proper plane.

Case (1). Let the proper horizontal line be given by (35). Its image by h is parameterized as

$$ \begin{align*}h_1(t):=h(t,s^*,w^*)=\gamma^tB(2\pi w^*)\sin (2\pi s^*+\eta_2) - \xi(2\pi w^*).\end{align*} $$

It follows from the first inequality in (36) that this image covers a neighborhood of zero. Since $cl(E)$ contains the proper horizontal line, by continuity of $h_1$ , we obtain a subsequence for which $\{h(t_n,s_n,w_n)\}$ is dense near zero. Let us denote the subsequence by the same notation. By (33) and (34), we have

(39) $$ \begin{align} |g(t_n,s_n,w_n)| &=\bigg|\dfrac{(h(t_n,s_n,w_n)+\xi(2\pi w_n))A(2\pi w_n)\sin(2\pi s_n+\eta_1)}{B(2\pi w_n)\sin( 2\pi s_n+\eta_2)}\bigg|\nonumber\\ &\to \bigg|\dfrac{\xi(2\pi w^*)A(2\pi w^*)\sin( 2\pi s^*+\eta_1)}{B(2\pi w^*)\sin( 2\pi s^*+\eta_2)}\bigg|. \end{align} $$

This gives the desired result on $|g(t_n,s_n,w_n)|$ due to the second inequality in (36).

Case (2). Let the proper tilted line be given by (37). By (15), one can find a small interval $\Delta _i$ arbitrarily close to $s=(-\eta _2/2\pi )+i$ with $i\in \mathbb {Z}$ such that

(40) $$ \begin{align} 0<\xi(2\pi w)B(2\pi w)\sin(2\pi s+\eta_2)<\infty,\quad \dfrac{\xi(2\pi w)A(2\pi w)\sin(2\pi s+\eta_1)}{\sin(2\pi s+\eta_2)}\not\in\{0,\pm 1,\infty\} \end{align} $$

for all $s\in \Delta _i$ with $w=a_2s$ or $w=w^*$ . Since $\xi (2\pi w)$ is bounded, taking n sufficiently large and $\Delta _i$ sufficiently close to $s=(-\eta _2/2\pi )+i$ ensures that the images of the functions

$$ \begin{align*}h_{21}(s):=h(a_1s+b,s,a_2 s)=\gamma^{a_1s+b}B(2\pi a_2s)\sin (2\pi s+\eta_2) - \xi(2\pi a_2s),\quad s\in\Delta_i\end{align*} $$

obtained with $w=a_2 s$ in (37), and

$$ \begin{align*}h_{22}(s):=h(a_1s+b,s,w^*)=\gamma^{a_1s+b}B(2\pi w^*)\sin (2\pi s+\eta_2) - \xi(2\pi w^*),\quad s\in\Delta_i\end{align*} $$

obtained with $w=w^*$ in (37) both cover a small neighborhood of zero. Now, take a subsequence of E approximating the line segment obtained by intersecting the tilted line with $\{(t,w,w)\mid s\in \Delta _i\}$ . Obviously, for this subsequence, $\{h(t_n,s_n,w_n)\}$ is dense near zero. Moreover, when $\Delta _i$ is taken sufficiently close, the value of $\sin (2\pi s+\eta _2)$ is arbitrarily close to zero, which together with (39) implies that $\{|g(t_n,s_n,w_n)|\}$ lie in some closed interval in $(1,\infty )$ .

Case (3). We now take a small interval $\Delta _i$ arbitrarily close to $s=(-\eta _2/2\pi ) +i$ and a small interval $\Delta ^{\prime }_i$ arbitrarily close to $s=-\eta _1/2\pi +i$ . Define two functions

$$ \begin{align*}h_{31}(t,s):=h(t,s,at+bs)=\gamma^{t}B(2\pi (at+bs))\sin (2\pi s+\eta_2) - \xi(2\pi(at+bs)),\end{align*} $$

in the first case of (38), and

$$ \begin{align*}h_{32}(t,s):=h(t,s,w^*)=\gamma^{t}B(2\pi w^*)\sin (2\pi s+\eta_2) - \xi(2\pi\omega^*)\end{align*} $$

in the second case of (38). Obviously, one can find $t^*$ such that, for all $s\in \Delta _i$ , and either $w=at^*+bs$ or $w=w^*$ , the inequalities in (40) are satisfied and $h_{31}(\Delta _i)$ or $h_{32}(\Delta _i)$ , depending on whether $w=at^*+bs$ or $w=w^*$ , covers a small neighborhood of zero. Then, consider the subsequence approximating the intersection of the proper plane with $\{t=t^*, s\in \Delta _i\}$ . Proceeding as in case (2) shows that $\{h(t_n,s_n,w_n)\}$ is dense near zero and $\{|g(t_n,s_n,w_n)|\}$ lie in some closed interval in $(1,\infty )$ . Considering a similar number $t^*$ for $s\in \Delta _i'$ gives another required subsequence.

Proof of Proposition 5.1

Denote

(41) $$ \begin{align} t_n=m_n-k_n\theta,\quad s_n=k_n\dfrac{\omega_1}{2\pi} \bmod 1,\quad w_n=m_n\dfrac{\omega_2}{2\pi} \bmod 1. \end{align} $$

Recall that in the double-focus case, we have $\unicode{x3bb}>0,\gamma >0$ , so $\unicode{x3bb} ^k\gamma ^m=\gamma ^{m-k\theta }.$ Comparing $A_{k,m}$ and $B_{k,m}$ in (14) with functions g and h given by (33) and (34), yields

$$ \begin{align*}A_{k_n,m_n}=g(t_n,s_n,w_n) \quad\text{and}\quad B_{k_n,m_n}=h(t_n,s_n,w_n). \end{align*} $$

It follows that, for any sequence $\{(t_n,s_n,w_n)\}$ falling in one of the cases of Lemma 5.5, Proposition 4.2 is applicable to the system $\{T_n:=T_{k_n,m_n}\}$ (passing to the subsequence in the lemma) given by (13), with the correspondence $U\to X, W\to Y, V\to Z.$ More specifically, we have that:

• • if $\{(t_n,s_n,w_n)\}$ approximates a proper horizontal line, then there exists a blender whose type depends on the coefficients in (13);

• • if $\{(t_n,s_n,w_n)\}$ approximates a proper tilted line, then there exists a cu-blender; and

• • if $\{(t_n,s_n,w_n)\}$ approximates a proper plane, then there exist simultaneously a cs-blender and a cu-blender.

Moreover, since the cone field in Proposition 4.2 does not depend on $\delta $ and k, one concludes (as in the proof of Proposition 4.3) that the obtained blender has an activating pair $(\Pi ',\mathcal {C}^{ss})$ if it is center-stable or $(\Pi ',\mathcal {C}^{uu})$ if it is center-unstable, where $\Pi $ is defined in (31), and $\mathcal {C}^{ss}$ and $\mathcal {C}^{uu}$ are given by Lemma 3.3.

We thus prove Proposition 5.1 by showing that, for each of the cases in the statement of the proposition, one can find a sequence $\{(k_n,m_n)\}$ such that $\{(t_n,s_n,w_n)\}$ given by (41) approximates a proper line or a proper plane. Recall that $\omega _{1,2}\in (0,\pi )$ and by $p/q$ , we always mean that p and q are positive coprime integers.

Case (1): ${\theta }$ is irrational, ${\omega _1/2\pi ,\omega _2/2\pi }$ are rational. Denote

$$ \begin{align*}\dfrac{\omega_1}{2\pi}=\dfrac{p_1}{q_1}\quad\text{and}\quad \dfrac{\omega_2}{2\pi}=\dfrac{p_2}{q_2}. \end{align*} $$

Since condition A2 is satisfied, there exist integers $r_1\in \{0,\ldots ,q_1-1\}$ and $r_2\in \{0,\ldots ,q_2-1\}$ such that inequalities in (36) are satisfied with $s^*=r_1{\omega _1}/{2\pi }$ and $w^*=r_2\omega _2/2\pi $ . Then,

$$ \begin{align*}\bigg\{s=r_1\dfrac{\omega_1}{2\pi}\bmod 1,w=r_2\dfrac{\omega_2}{2\pi}\bmod 1\bigg\}\end{align*} $$

is a proper horizontal line of the form (35). The irrationality of $\theta $ implies that we can take a sequence $\{(k^{\prime }_n,m^{\prime }_n)\}$ of pairs of positive integers such that the set

$$ \begin{align*}\{q_2m^{\prime}_n+r_2-(q_1k^{\prime}_n+r_1)\theta\}\end{align*} $$

is dense in $\mathbb {R}$ . Setting $k_n=q_1k^{\prime }_n+r_1$ and $ m_n=q_2m^{\prime }_n+r_2$ in (41) gives a sequence whose closure contains the above proper horizontal line. So, we get a cs-blender if $|\alpha _{r_1,r_2}|<1$ or a cu-blender if $|\alpha _{r_1,r_2}|>1$ .

Case (2.1): $ {\omega _1/2\pi }$ is irrational, $ {\omega _2/2\pi }$ is rational, and $ {\theta ,\omega _1/2\pi ,1}$ are rationally independent. Let $p_2,q_2,r_2$ be as in the preceding case. By the rational independence among $\theta ,\omega _1/2\pi ,1$ , one can find a sequence $\{(k^{\prime }_n,m^{\prime }_n,i^{\prime }_n)\in \mathbb {N}^3\}$ such that the set

$$ \begin{align*}\bigg\{\bigg(q_2m^{\prime}_n+r_2-k^{\prime}_n\theta,k^{\prime}_n\dfrac{\omega_1}{2\pi} - i^{\prime}_n\bigg)\bigg\}\end{align*} $$

is dense in $\mathbb R^2$ . Setting $k_n=k^{\prime }_n$ and $m_n=q_2m^{\prime }_n+r_2$ in (41) gives a sequence whose closure contains the proper plane $\{w=r_2\omega _2/2\pi \}.$ So, we get simultaneously a cs-blender and a cu-blender.

Before proceeding to other cases, we make the following claim.

Claim. To prove that there exists $\{(t_n,s_n,w_n)\}$ given by (41) approximating a proper line or a proper plane, it suffices to show that each point of this proper line or proper plane is accumulated by points of the set

(42) $$ \begin{align} \bigg\{\bigg(m_n-k_n\theta,k_n\dfrac{\omega_1}{2\pi}\bmod 1,m_n\dfrac{\omega_2}{2\pi}\bmod 1\bigg)\bigg\} \end{align} $$

for some sequence $\{(k_n,m_n)\}$ .

Indeed, the assumption particularly implies that, for each rational point in the proper line or proper plane, we have such a sequence $\{(k_n,m_n)\}$ and, hence, the union of these countably many sequences gives a new sequence which defines the desired $\{(t_n,s_n,w_n)\}$ by (41).

Case (2.2): $ {\omega _1/2\pi }$ is irrational, $ {\omega _2/2\pi }$ is rational, and $ {\theta ,\omega _1/2\pi ,1}$ are rationally dependent. Denote

(43) $$ \begin{align} \theta =\dfrac{p_1}{q_1} \dfrac{\omega_1}{2\pi} + \dfrac{p_2}{q_2}\quad\text{and}\quad \dfrac{\omega_2}{2\pi}=\dfrac{p_3}{q_3}. \end{align} $$

With $r_2$ as in case (1), the set $\{t=-sp_1/q_1+r_2,w=r_2p_3/q_3\bmod 1\}$ is a proper tilted line. Let us show that each point of this line can be approximated by the set (42).

By the irrationality of $\omega _1/2\pi $ , the set

$$ \begin{align*} \bigg\{q_2q_3k\dfrac{\omega_1}{2\pi}-q_1q_3i\bigg\}_{k,i\in\mathbb N^2} \end{align*} $$

is dense in $\mathbb {R}$ . So, for any $s\in [0,1]$ , one can find a sequence $\{(k_n',i_n')\}$ such that

$$ \begin{align*}q_2q_3k^{\prime}_n\dfrac{\omega_1}{2\pi}-q_1q_3i^{\prime}_n\to s.\end{align*} $$

Denote $k_n=q_2q_3k^{\prime }_n $ and $ m_n=r_2+p_2q_3k^{\prime }_n+p_1q_3i^{\prime }_n.$ The relation (43) then implies

$$ \begin{align*} m_n-k_n\theta &=(r_2+p_2q_3k^{\prime}_n+p_1q_3i^{\prime}_n)-\dfrac{p_1}{q_1} k_n \dfrac{\omega_1}{2\pi} - p_2q_3k_n'\\ &=r_2 -\dfrac{p_1}{q_1}\bigg(k_n \dfrac{\omega_1}{2\pi}-q_1q_3i^{\prime}_n\bigg)\\ &\to r_2-s\dfrac{p_1}{q_1},\\ m_n\dfrac{\omega_2}{2\pi}\mod {\!1}&=(r_2+p_2q_3k^{\prime}_n+p_1q_3 i^{\prime}_n)\dfrac{p_3}{q_3} \mod {\!1}\\ &=\dfrac{r_2p_3}{q_3} \mod {\!1}, \end{align*} $$

which gives the desired approximation. By the claim, we have a cu-blender.

Case (3.1): $ {\omega _1/2\pi ,\omega _2/2\pi }$ are irrational and ${\theta ,\omega _1/2\pi ,\theta \omega _2/2\pi ,1}$ are rationally independent. The rational independence implies that, for any point $(t,s,w)$ , there exists $\{(k_n,m_n,i_n,\ell _n)\}$ such that

(44) $$ \begin{align} m_n-k_n\theta\to t,\quad k_n\dfrac{\omega_1}{2\pi}-i_n\to s,\quad k_n\theta\dfrac{\omega_2}{2\pi}-\ell_n\to w- t\dfrac{\omega_2}{2\pi}. \end{align} $$

Combining the first and third relations yields

$$ \begin{align*}(m_n-t)\dfrac{\omega_2}{2\pi}-\ell_n \to w-t\dfrac{\omega_2}{2\pi},\end{align*} $$

which implies

(45) $$ \begin{align} m_n-k_n\theta\to t,\quad k_n\dfrac{\omega_1}{2\pi}-i_n\to s,\quad m_n\dfrac{\omega_2}{2\pi}-\ell_n\to w. \end{align} $$

Therefore, any point of any proper plane can be approximated by a sequence of the form (42). By the claim, we get simultaneously a cs-blender and a cu-blender.

Case (3.2): ${\omega _1/2\pi ,\omega _2/2\pi }$ are irrational, ${\theta ,\omega _1/2\pi ,\theta \omega _2/2\pi ,1}$ are rationally dependent, but ${\theta ,\omega _1/2\pi ,1}$ are rationally independent. Denote

(46) $$ \begin{align} \theta\dfrac{\omega_2}{2\pi}= \dfrac{p_1}{q_1}\theta + \dfrac{p_2}{q_2}\dfrac{\omega_1}{2\pi}+\dfrac{p_3}{q_3}. \end{align} $$

Since $\theta ,\omega _1/2\pi ,1$ are rationally independent, the set

$$ \begin{align*}\bigg\{\bigg(q_1m-q_3k\theta,q_3k\dfrac{\omega_1}{2\pi}-q_2 i\bigg)\bigg\}_{k,m,i\in\mathbb{N}}\end{align*} $$

is dense in $\mathbb {R}^2$ . Hence, for any point $(t,s)$ , there exists a sequence $(k^{\prime }_n,m^{\prime }_n,i^{\prime }_n)$ such that

$$ \begin{align*} q_1m^{\prime}_n-q_3k^{\prime}_n\theta\to t \quad\text{and}\quad q_3k^{\prime}_n\dfrac{\omega_1}{2\pi}-q_2i^{\prime}_n \mod1\to s. \end{align*} $$

Denote $k_n=q_3k^{\prime }_n $ and $ m_n=q_1m^{\prime }_n.$ The relation (46) shows that

$$ \begin{align*} m_n\dfrac{\omega_2}{2\pi}\mod1 &= k_n\bigg(\dfrac{p_1}{q_1}\theta + \dfrac{p_2}{q_2}\dfrac{\omega_1}{2\pi}+\dfrac{p_3}{q_3}\bigg)-\dfrac{\omega_2}{2\pi}t\mod1\\ &\to \bigg(\dfrac{p_1}{q_1}-\dfrac{\omega_2}{2\pi}\bigg)t+\dfrac{p_2}{q_2}s\mod1\\ &=:a t+bs\mod1. \end{align*} $$

Thus, any point of the proper plane $\{w=at+bs\bmod 1\}$ can be approximated by a sequence of the form (42). By the claim, we get simultaneously a cs-blender and a cu-blender.

Case (3.3.1): $ {\omega _1/2\pi ,\omega _2/2\pi }$ are irrational, $ {\theta ,\omega _1/2\pi ,1}$ are rationally dependent, and $ {\omega _1/2\pi ,\theta \omega _2/2\pi ,1}$ are rationally dependent. Denote

(47) $$ \begin{align} \theta=\dfrac{p_1}{q_1}\dfrac{\omega_1}{2\pi}+\dfrac{p_2}{q_2} \quad\text{and}\quad \theta\dfrac{\omega_2}{2\pi}= \dfrac{p_3}{q_3}\dfrac{\omega_1}{2\pi}+\dfrac{p_4}{q_4}. \end{align} $$

The irrationality of $\omega _1/2\pi $ implies that the set

$$ \begin{align*}\bigg\{q_2q_4k\dfrac{\omega_1}{2\pi}-q_1q_3i\bigg\}_{k,i\in\mathbb N}\end{align*} $$

is dense in $\mathbb {R}$ . For any $s\in [0,1]$ and any sequence $(k^{\prime }_n,i^{\prime }_n)$ such that

$$ \begin{align*} q_2q_4k^{\prime}_n\dfrac{\omega_1}{2\pi}-q_1q_3i^{\prime}_n \to s, \end{align*} $$

denote $k_n=q_2q_4k^{\prime }_n $ and $ m_n=p_2q_4k^{\prime }_n+p_1q_3i^{\prime }_n.$ The relations in (47) show that

$$ \begin{align*} m_n-k_n\theta &=p_2q_4k^{\prime}_n+p_1q_3i^{\prime}_n-q_2q_4k^{\prime}_n\dfrac{p_1}{q_1}\bigg(\dfrac{\omega_1}{2\pi}+\dfrac{p_2}{q_2}\bigg)\\ &=-\dfrac{p_1}{q_1}\bigg(q_2q_4k^{\prime}_n\dfrac{\omega_1}{2\pi}-q_1q_3i^{\prime}_n\bigg)\\ &\to -\dfrac{p_1}{q_1} s,\\[10pt] k_n\theta\dfrac{\omega_2}{2\pi}\mod {\!1} &=\dfrac{p_3}{q_3}\bigg(q_2q_4k^{\prime}_n\dfrac{\omega_1}{2\pi}-q_1q_3i^{\prime}_n\bigg)+p_3q_1i^{\prime}_n+p_4q_2k^{\prime}_n\mod {\!1}\\ &\to \dfrac{p_3}{q_3}s \mod1. \end{align*} $$

Observing the relation between (44) and (45), one finds

$$ \begin{align*}m_n\dfrac{\omega_2}{2\pi}\to \bigg(\dfrac{p_3}{q_3}-\dfrac{\omega_2}{2\pi}\dfrac{p_1}{q_1}\bigg)s=:as.\end{align*} $$

Thus, any point of the proper tilted line $\{t=-p_1s/q_1,w=as\}$ can be approximated by a sequence of the form (42). The claim then gives a cu-blender.

Case (3.3.2): $ {\omega _1/2\pi ,\omega _2/2\pi }$ are irrational, $ {\theta ,\omega _1/2\pi ,1}$ are rationally dependent, and $ {\omega _1/2\pi ,\theta \omega _2/2\pi ,1}$ are rationally independent. We still denote $\theta $ as in (47). For any $s\in [0,1]$ and any sequence $(k^{\prime }_n,i^{\prime }_n)$ such that

$$ \begin{align*}q_2k_n'\dfrac{\omega_1}{2\pi}-q_1i^{\prime}_n\to s,\end{align*} $$

denote $k_n=q_2k_n' $ and $ m_n=p_2k^{\prime }_n+p_1i^{\prime }_n.$ The first relation in (47) gives

$$ \begin{align*}m_n-k_n\theta =p_2k^{\prime}_n+p_1i^{\prime}_n- q_2k_n'\bigg( \dfrac{p_1}{q_1}\dfrac{\omega_1}{2\pi}+\dfrac{p_2}{q_2}\bigg) \to - \dfrac{p_1}{q_1}s.\end{align*} $$

It then follows from the rational independence among $\omega _1/2\pi ,\theta \omega _2/2\pi ,1$ , and the correspondence relation between (44) and (45) that any point of any proper tilted line in the plane $\{t=-p_1s/q_1\}$ can be approximated by a sequence of the form (42). The claim then gives a cu-blender.