Proof of Lemma 3.2.

Notice that $f_p := (\gamma _p)^{-1}\circ F^{\pi (p)}\circ \gamma _p \in \mathrm {Diff}_\partial ^{1+}([0, 1])$ . Since $\unicode{x3bb} ^c(p) < 0$ , we have $0 < f^{\prime }_p(0) < 1$ and, therefore, $f_p$ has a smallest positive fixed point $0 < s_p \leq 1$ . It follows that ${\ell _p}^{-1}\circ f_p\circ {\ell _p}\in \mathrm {Diff}_\partial ^{1+}([0, 1])$ is a diffeomorphism with exactly two fixed points $0,1$ , where $0$ is a hyperbolic sink.

By the above argument, $I_p^c := W^s(p)\cap \mathcal {F}^c(p) = \gamma _p([0, s_p))$ . Therefore, we have

$$ \begin{align*}W^s(p) \supseteq \bigcup_{x\in I_p^c}\mathcal{F}^s(x).\end{align*} $$

Now, take $y\in W^s({p})$ . Then, we have $\pi _0(y)\in \mathcal {F}^s(\pi _0({p}))$ . Recall that $\pi _0: M \to M_0$ is the natural projection along central leaves. It follows that $y\in \mathcal {F}^{cs}({p})$ . Assume that $y\in \mathcal {F}^s(z)$ for some $z\in \mathcal {F}^c({p})$ . Then, $z\in W^s(p)$ and, hence, $z\in I_p^c$ since $W^s(p)$ is s-saturated. This completes the proof of Lemma 3.2.

Proof of Theorem 3.1.

It suffices to show that for any two non-empty open subsets ${U, V\subseteq M}$ , there exists $N\in \mathbb {N}$ such that $F^N(U)\cap V\neq \varnothing $ . Equivalently, we only need to show that $U^*\cap V^*\neq \varnothing $ , where

$$ \begin{align*}U^* := \bigcup_{n\in\mathbb{N}}F^n(U)\quad \text{and}\quad V^* := \bigcup_{n\in\mathbb{N}}F^{-n}(V).\end{align*} $$

Note that $F(U^*)\subseteq U^*$ , hence, $U^*$ is u-saturated; $F^{-1}(V^*)\subseteq V^*$ ; and $V^*$ is s-saturated.

Since F is boundary interconnected, there exist two pairs of periodic points $p_i,q_i\in M_i$ such that

$$ \begin{align*} \unicode{x3bb}^c(p_0) < 0 < \unicode{x3bb}^c(p_1) &\quad\text{and}\quad W^s(p_0)\pitchfork W^u(p_1)\neq\varnothing;\\ \unicode{x3bb}^c(q_1) < 0 < \unicode{x3bb}^c(q_0) &\quad\text{and}\quad W^s(q_1)\pitchfork W^u(q_0)\neq\varnothing. \end{align*} $$

Without loss of generality, we assume that the periodic points $p_i$ and $q_i$ are fixed points. Define $\varphi : M \to \mathbb {R}$ by

$$ \begin{align*}\varphi(x) := \ln\|DF|_{E^c(x)}\|.\end{align*} $$

Then, $\varphi $ is Hölder continuous, and there exist constants $C_\varphi \geq 1$ and $0 < \theta < 1$ such that

$$ \begin{align*}|\varphi(x) - \varphi(y)| \leq C_\varphi d(x, y)^\theta\quad \text{for all } x, y\in M.\end{align*} $$

Recall that we denote by

$$ \begin{align*}S_\varphi F_0(p) := \sum_{i = 0}^{\pi(p) - 1}\varphi(F_0^i(p)) = \pi(p)\unicode{x3bb}^c(p)\end{align*} $$

the Birkhoff sum of the function $\varphi $ for $p\in \mathrm {Per}(F_0)$ with period $\pi (p)$ . As a result,

$$ \begin{align*}S_\varphi F_0(p_0) = \varphi(p_0) = \unicode{x3bb}^c(p_0) < 0 < \unicode{x3bb}^c(q_0) = \varphi(q_0) = S_\varphi F_0(q_0).\end{align*} $$

Generate new periodic points for asymptotically rational independence. Now, compared with the results in [Reference XiaXia24], we do not know whether $S_\varphi F_0(p_0)$ and $S_\varphi F_0(q_0)$ are rationally independent. Therefore, we need the following asymptotic version.

Claim 3.3. There exists a sequence of periodic points $\{p_{0,m}\}\subseteq \mathrm {Per}(F_0)$ such that

$$ \begin{align*} (S_\varphi F_0(p_{0,m}),\ S_\varphi F_0(q_0)) \to 0\quad \text{and}\quad \unicode{x3bb}^c(p_{0,m}) \to \unicode{x3bb}^c(p_0)\quad \text{as } m \to +\infty. \end{align*} $$

Proof of Claim 3.3.

If $S_\varphi F_0(p_0)$ and $S_\varphi F_0(q_0)$ are indeed rationally independent, then we just take $p_{0,m} = p_0$ for any $m\in \mathbb {N}$ . Now, we assume that $S_\varphi F_0(p_0)$ and $S_\varphi F_0(q_0)$ are rationally dependent. By Lemma 2.8, the set $\{S_\varphi F_0(p): p\in \mathrm {Per}(F_0)\}$ is dense in $\mathbb {R}$ . Therefore, there exists a sequence of periodic points $\{\bar {p}_{0,m}\} \subseteq \mathrm {Per}(F_0)$ such that

(3.1) $$ \begin{align} 2^{-m} < |S_\varphi F_0(\bar{p}_{0,m}) - S_\varphi F_0(p_0)| < 2^{-m+1}. \end{align} $$

For fixed $m\in \mathbb {N}$ , we construct $p_{0,m}$ from $\bar {p}_{0,m}$ by the shadowing property.

An Anosov diffeomorphism on a nilmanifold $M_0$ , $F_0$ is topologically transitive. Hence, $\bar {p}_{0,m}$ and $p_0$ are homoclinically related, and there are points $x_0, y_0\in M_0$ with

$$ \begin{align*} x_0 \in \mathcal{F}^u(p_0) \pitchfork\mathcal{F}^s(\bar{p}_{0,m}) \quad \text{and}\quad y_0 \in \mathcal{F}^u(\bar{p}_{0,m}) \pitchfork \mathcal{F}^s(p_0), \end{align*} $$

see Figure 3. For simplicity, we omit the picture of the other points in $\mathrm {Orb}(\bar {p}_{0,m})$ .

Figure 3

Periodic pseudo-orbit (color online).

We then consider the following pseudo-orbit segment:

$$ \begin{align*} \mathcal{P}^{\prime}_m := (F_0^{-K_m}(y_0),\ldots, y_0, \ldots, F_0^{L_m - 1}(y_0), F_0^{-L_m}(x_0), \ldots, x_0, \ldots, F_0^{K_m - 1}(x_0)). \end{align*} $$

Here, $K_m,L_m\in \mathbb {N}$ are large multiples of $\pi (\bar {p}_{0,m})$ to be decided. The sum of $\varphi $ over $\mathcal {P}^{\prime }_m$ is

$$ \begin{align*} S_\varphi(\mathcal{P}^{\prime}_m):= \sum_{x\in\mathcal{P}^{\prime}_m}\varphi(x) = \sum_{i = -K_m}^{L_m - 1}\varphi(F_0^i(y_0)) + \sum_{i = -L_m}^{K_m - 1}\varphi(F_0^i(x_0)). \end{align*} $$

Since $F_0$ is uniformly contracting (respectively expanding) in stable (respectively unstable) manifolds and $\varphi $ is Hölder continuous, we have the following convergent series:

$$ \begin{align*} P&= \sum_{i = -\infty}^{-1} [\varphi(F_0^i(y_0)) - \varphi(F_0^i(\bar{p}_{0,m}))] + \sum_{i = 0}^{+\infty} [\varphi(F_0^i(y_0)) - \varphi(p_0)]\\ &\quad +\sum_{i = -\infty}^{-1} [\varphi(F_0^i(x_0)) - \varphi(p_0)] + \sum_{i = 0}^{+\infty} [\varphi(F_0^i(x_0)) - \varphi(F_0^i(\bar{p}_{0,m}))]. \end{align*} $$

As a result, given a sequence of positive numbers $\bar {\varepsilon }_m \to 0$ , there exist $K_m^0, L_m^0\in \mathbb {N}$ being multiples of $\pi (\bar {p}_{0,m})$ such that when $K_m \geq K_m^0$ and $L_m \geq L_m^0$ , we have

(3.2) $$ \begin{align} |S_\varphi(\mathcal{P}^{\prime}_m) - (P + 2K_m\unicode{x3bb}^c(\bar{p}_{0,m}) + 2L_m\unicode{x3bb}^c(p_0))| < \bar{\varepsilon}_m. \end{align} $$

However, by the classical exponential Anosov shadowing lemma, there are constants ${0 < \delta _0 < 1}$ , $C_0 \geq 1$ , and $0 < \mu _0 < 1$ such that for every $0 < \delta < \delta _0$ , every $\delta $ -pseudo-orbit $\{x_i: i_A\leq i \leq i_B\}\ (i_A \leq 0 \leq i_B)$ is exponentially shadowed by a periodic point p in the sense that

$$ \begin{align*} d(F_0^i(p),\, x_i) < C_0\mu_0^{|i|}\delta\quad \text{for all } i_A \leq i \leq i_B. \end{align*} $$

As a result, there exist $K_m^1, L_m^1\in \mathbb {N}$ such that when $K_m \geq K_m^1$ and $L_m \geq L_m^1$ , $\mathcal {P}^{\prime }_m$ is a $\delta _0\bar {\varepsilon }_m$ -pseudo-orbit, exponentially shadowed by a periodic point $p_{0,m}'\in \mathrm {Per}(F_0)$ . In fact, it suffices to take $K_m^1$ and $L_m^1$ to be multiples of $\pi (\bar {p}_{0,m})$ and so large that

$$ \begin{align*} &\max\{d(F_0^{L_m^1}(y_0),\, p_0), \ d(F_0^{-L_m^1}(x_0),\, p_0)\} < \tfrac{1}{2}\delta_0\bar{\varepsilon}_m;\\&\max\{d(F_0^{K_m^1}(x_0),\, \bar{p}_{0,m}), \ d(F_0^{-K_m^1}(y_0),\, \bar{p}_{0,m})\} < \tfrac{1}{2}\delta_0\bar{\varepsilon}_m. \end{align*} $$

Now, we consider $S_\varphi F_0(p_{0,m}')$ , which is the sum of $\varphi $ over the orbit of $p_{0,m}'$ . Since $\varphi $ is Hölder continuous, the difference between $S_\varphi F_0(p_{0,m}')$ and $S_\varphi (\mathcal {P}^{\prime }_m)$ is controlled by

(3.3) $$ \begin{align} |S_\varphi F_0(p_{0,m}') - S_\varphi(\mathcal{P}^{\prime}_m)| < C_\varphi C_0^\theta\frac{2\delta_0^\theta}{1 - \mu_0^\theta}\cdot \bar{\varepsilon}_m^\theta. \end{align} $$

Combining (3.2) and (3.3), there exists a constant $C \geq 1$ such that

(3.4) $$ \begin{align} |S_\varphi F_0(p_{0,m}') - (P + 2K_m\unicode{x3bb}^c(\bar{p}_{0,m}) + 2L_m\unicode{x3bb}^c(p_0))| < C\bar{\varepsilon}_m^\theta. \end{align} $$

Here, we choose $K_m {\kern-1pt}={\kern-1pt} \max \{K_m^0, K_m^1, m\pi (\bar {p}_{0,m})\}$ and ${L_m {\kern-1pt}={\kern-1pt} \max \{L_m^0, L_m^1, m(K_m+|P|)\}}$ so that the central Lyapunov exponent is also controlled:

$$ \begin{align*} |\unicode{x3bb}^c(p_{0,m}') - \unicode{x3bb}^c(p_0)| &= \bigg|\frac{S_\varphi F_0(p_{0,m}')}{\pi(p_{0,m}')} - \unicode{x3bb}^c(p_0)\bigg|\\ &\leq \frac{|P| + 2K_m|\unicode{x3bb}^c(\bar{p}_{0,m}) - \unicode{x3bb}^c(p_0)| + C\bar{\varepsilon}_m^\theta}{2K_m + 2L_m}\\ &\to 0\quad (\mathrm{as}\ m \to +\infty). \end{align*} $$

In the same manner as above, we can make a slight modification on the choice of the pseudo-orbit segment. Precisely, we add the periodic orbit of $\bar {p}_{0,m}$ and consider the following pseudo-orbit segment:

$$ \begin{align*} \mathcal{P}^{\prime\prime}_m := (\bar{p}_{0,m},\, F_0(\bar{p}_{0,m}),\, \ldots,\, F_0^{\pi(\bar{p}_{0,m}) - 1}(\bar{p}_{0,m}))\cup \mathcal{P}_m'. \end{align*} $$

Then, we have a periodic point $p_{0,m}"\in \mathrm {Per}(F_0)$ shadowing $\mathcal {P}^{\prime \prime }_m$ and satisfying a similar estimate as (3.4):

(3.5) $$ \begin{align} |S_\varphi F_0(p_{0,m}") - (P + 2K_m\unicode{x3bb}^c(\bar{p}_{0,m}) + 2L_m\unicode{x3bb}^c(p_0) + S_\varphi F_0(\bar{p}_{0,m}))| < C\bar{\varepsilon}_m^\theta. \end{align} $$

Moreover, we also have that

$$ \begin{align*}|\unicode{x3bb}^c(p_{0,m}") - \unicode{x3bb}^c(p_0)| \to 0\quad \text{as } m \to +\infty.\end{align*} $$

Finally, we choose $p_{0,m} \in \{p_{0,m}',\, p_{0,m}"\}$ so that $(S_\varphi F_0(p_{0,m}),\ S_\varphi F_0(p_0)) \to 0$ as ${m \to +\infty }$ . In fact, combining (3.4) and (3.5), we have

(3.6) $$ \begin{align} |S_\varphi F_0(p_{0,m}") - S_\varphi F_0(p_{0,m}') - S_\varphi F_0(\bar{p}_{0,m})| < 2C\bar{\varepsilon}_m^\theta. \end{align} $$

Now, taking $\bar {\varepsilon }_m$ so small that $2C\bar {\varepsilon }_m^\theta < {2^{-m}}$ , then we have the following estimate by combining (3.1) and (3.6):

(3.7) $$ \begin{align} {0} < |S_\varphi F_0(p_{0,m}") - S_\varphi F_0(p_{0,m}') - S_\varphi F_0(p_0)| < {3\cdot2^{-m}}. \end{align} $$

Taking further $3\cdot 2^{-m}< |S_\varphi F_0(p_0)|$ , then by Lemma 2.5, there exists ${p_{0,m} \in \{p_{0,m}', p_{0,m}"\}}$ such that

$$ \begin{align*} (S_\varphi F_0(p_{0,m}),\ S_\varphi F_0(p_0)) < {\sqrt{3|S_\varphi F_0(p_0)|}\cdot 2^{ - m/2}}. \end{align*} $$

Since $S_\varphi F_0(p_0)$ and $S_\varphi F_0(q_0)$ are rationally dependent, there exists a sequence of positive numbers $\varepsilon _m \to 0^+$ such that $(S_\varphi F_0(p_{0,m}),\ S_\varphi F_0(q_0)) < \varepsilon _m$ .

This completes the proof of Claim 3.3.

In fact, we can modify the choice of $\{p_{0,m}\}$ to obtain more information about the sequence.

Proof of Claim 3.4.

Since $\unicode{x3bb} ^c(p_{0,m}) \to \unicode{x3bb} ^c(p_0) < 0$ , by taking a subsequence, we assume that

$$ \begin{align*} \unicode{x3bb}^c(p_{0,m}) < \frac{\unicode{x3bb}^c(p_0)}{2}\quad \text{for all } m \in\mathbb{N}. \end{align*} $$

By the classical Pliss lemma, for each $m\in \mathbb {N}$ , there exists a hyperbolic time $n_m$ such that

$$ \begin{align*} \frac{1}{k}\sum_{i = 0}^{k - 1}\ln\|DF|_{E^c(F_0^{n_m + i}(p_{0,m}))}\| \leq \frac{\unicode{x3bb}^c(p_0)}{3}\quad \text{for all } k\in\mathbb{N}. \end{align*} $$

Hence, we can replace $p_{0,m}$ by $F_0^{n_m}(p_{0,m})\in \mathrm {Orb}(p_{0,m})$ . Since $p_{0,m}$ is a periodic point, this replacement does not influence the properties that we need in Claim 3.3.

Now, we show that the length of the contracting center $I_{p_{0,m}}^c$ has a positive lower bound with respect to $m\in \mathbb {N}$ . Since $p_{0,m}$ now admits a better Birkhoff average, there exists a maximal closed interval $I_m$ containing $p_{0,m}$ as an endpoint, lying in $I_{p_{0,m}}^c$ , satisfying

$$ \begin{align*} \bar\varphi_k(x) \leq \frac{\unicode{x3bb}^c(p_0)}{6}\quad \text{for all } 1\leq k \leq \pi(p_{0,m}),\ \text{for all } x\in I_m, \end{align*} $$

where

$$ \begin{align*} \bar\varphi_k(x) := \frac{1}{k}\ln\|DF^k|_{E^c(x)}\| = \frac{1}{k}\sum_{i = 0}^{k - 1}\varphi(F^i(x)). \end{align*} $$

Then, for $x,y\in I_m$ , we have

$$ \begin{align*} |\bar\varphi_k(x) - \bar\varphi_k(y)| &\leq \frac{1}{k}\sum_{i = 0}^{k - 1}|\varphi(F^i(x)) - \varphi(F^i(y))| \leq \frac{1}{k}\sum_{i = 0}^{k - 1}C_\varphi d(F^i(x), F^i(y))^\theta\\ &\leq \frac{1}{k}\sum_{i = 0}^{k - 1}C_\varphi d_c(F^i(x), F^i(y))^\theta \leq \frac{1}{k}\sum_{i = 0}^{k - 1}C_\varphi \max_{z\in I_m}\|DF^i|_{E^c(z)}\|^\theta d_c(x, y)^\theta\\ &\leq \frac{1}{k}\sum_{i = 0}^{k - 1}C_\varphi e^{({1}/{6})\unicode{x3bb}^c(p_0)i\theta} d_c(x, y)^\theta \leq \frac{1}{k}\cdot\frac{C_\varphi}{1 - e^{({1}/{6})\unicode{x3bb}^c(p_0)\theta}} d_c(x, y)^\theta. \end{align*} $$

Let $\widetilde {p}_{0,m}$ be the other endpoint of $I_m$ . Since $I_m$ is maximal, there exists $1\leq k_m \leq \pi (p_{0,m})$ such that

$$ \begin{align*} \frac{\unicode{x3bb}^c(p_0)}{6} = \bar\varphi_{k_m}(\widetilde{p}_{0,m}) &\leq \bar\varphi_{k_m}(p_{0,m}) + |\bar\varphi_{k_m}(\widetilde{p}_{0,m}) - \bar\varphi_{k_m}(p_{0,m})|\\ &\leq \frac{\unicode{x3bb}^c(p_0)}{3} + \frac{1}{k_m}\frac{C_\varphi}{1 - e^{({1}/{6})\unicode{x3bb}^c(p_0)\theta}}|I_m|^\theta\\ &\leq \frac{\unicode{x3bb}^c(p_0)}{3} + \frac{C_\varphi}{1 - e^{({1}/{6})\unicode{x3bb}^c(p_0)\theta}}|I_m|^\theta. \end{align*} $$

As a result,

$$ \begin{align*} |I_m| \geq \bigg[-\frac{1}{6}\unicode{x3bb}^c(p_0)\cdot\frac{1}{C_\varphi}(1 - e^{({1}/{6})\unicode{x3bb}^c(p_0)\theta})\bigg]^{{1}/{\theta}}> 0. \end{align*} $$

It follows that $p_{0,m}$ has a uniform size of contracting center.

Finally, since $M_0$ is compact, we can take a subsequence to ensure that $p_{0,m}\to r_0$ for some $r_0\in M_0$ .

Show the density of the stable and unstable sets. Next, we show that the intersection of the stable and unstable sets implies their density.

Proof of Claim 3.5.

For convenience, we define $\widehat {p}_0 := \pi _1(p_0)$ and $\widehat {p}_1 := \pi _0(p_1)$ . Recall that $W^s(p_0)\pitchfork W^u(p_1)$ contains a central interval. Assume that the interval lies in a central leaf with endpoints $z_0\in M_0$ and $z_1\in M_1$ , then there exists an open central interval ${J \subseteq \mathcal {F}^c(z_0)}$ with an endpoint $z_0$ , lying in $W^s(p_0)$ and containing an interval in $W^u(p_1)$ , see Figure 4.

Figure 4

Density of $W^s(p_0)$ (color online).

Figure 4 Long description

The diagram consists of two rectangular planes meeting at a central vertical line.

Central Axis:

At the top of the central vertical line is the label z sub 1, and at the bottom is z sub 0. A red vertical segment on this axis is labeled J.

Left Plane:

The top-left corner is labeled p hat sub 0 and the bottom-left corner is p sub 0. A horizontal line with a left-pointing arrow bisects this plane, labeled gamma sub p sub 0 (s sub p sub 0). The region below this line is labeled W super s (p sub 0).

Right Plane:

The top-right corner is labeled p sub 1 and the bottom-right corner is p hat sub 1. A horizontal line with a left-pointing arrow bisects this plane, labeled gamma sub p sub 1 (s sub p sub 1). The region above this line is labeled W super u (p sub 1). A red vertical line segment on the far right edge is labeled F super minus N (J).

Directional Arrows:

Arrows on the outer boundaries point inward toward the central axis or downward along the vertical edges. Arrows on the horizontal bisecting lines point toward the left.

Now, consider $F^{-N}(J)$ . One of its endpoints is $F_0^{-N}(z_0)$ , which tends to $\widehat {p}_1$ ; the other endpoint tends to $p_1$ , since it lies in $W^u(p_1)$ . Moreover, $F^{-N}(J)$ is a central interval, and hence, $F^{-N}(J)$ tends to $\mathcal {F}^c(p_1)$ , in the sense that

$$ \begin{align*} & \text{for all }\, x\in\mathcal{F}^c(p_1),\ \text{for all } \varepsilon> 0,\ \text{there exists }\, N\in\mathbb{N},\ \text{there exists }\, x'\in F^{-N}(J),\\ &\qquad \text{such that}\ d(x',\, x) < \varepsilon. \end{align*} $$

Since $F_0$ is an Anosov diffeomorphism on a nilmanifold, the stable foliation restricted to $M_0$ is minimal, in the sense that each stable leaf is dense in $M_0$ . It follows that $\mathcal {F}^{cs}$ is minimal, because $\mathcal {F}^{cs}(\cdot ) = \pi _0^{-1}(\mathcal {F}^s(\pi _0(\cdot )))$ .

Now, $\mathcal {F}^{cs}(p_1)$ is dense in M, for any non-empty open subset $U\subseteq M$ , there exists ${x\in \mathcal {F}^c(p_1)}$ such that $\mathcal {F}^s(x)\cap U\neq \varnothing $ . Note that

$$ \begin{align*} \mathcal{F}^s(x) = \bigcup_{k\in\mathbb{N}} F^{-k}(\mathcal{F}^s_\varepsilon({F^k(x)})) \quad \text{for all } \varepsilon> 0. \end{align*} $$

Therefore, there exists $K = K(\varepsilon )\in \mathbb {N}$ such that $\mathcal {F}^s_\varepsilon ({F^K(x)})\cap F^K(U)\neq \varnothing $ . Recall that $F^{-N}(J)$ tends to $\mathcal {F}^c(p_1)$ , hence, there exist $N\in \mathbb {N}$ and $x'\in F^{-N}(J)$ such that $\mathcal {F}^s_\varepsilon ({F^K(x')})\cap F^K(U)\neq \varnothing $ . Now, we have that $\mathcal {F}^s(F^N({F^K(x')}))\cap F^{K + N}(U)\neq \varnothing $ . Since ${F^N(F^K(x'))\in F^K(J)}\subseteq W^s(p_0)$ , we conclude that

$$ \begin{align*}W^s(p_0) \cap U = F^{-(K + N)}(W^s(p_0))\cap U \neq\varnothing.\end{align*} $$

This shows the density of $W^s(p_0)$ . By the same argument, $W^u(q_0), W^u(p_1)$ , and $W^s(q_1)$ are also dense in M.

Move from $p_0$ to $p_{0,m}$ . We first show that the points in $U^*$ appear near the contracting central interval $I_{p_{0,m}}^c$ .

Since $\unicode{x3bb} ^c(p_0) < 0$ , we have

$$ \begin{align*} W^s(p_0) = \bigcup_{k\in\mathbb{N}}F^{-k}(\mathcal{F}^{cs}_{\mathrm{loc}}(p_0)). \end{align*} $$

Therefore, by the density of $W^s(p_0)$ , we have

$$ \begin{align*}U^*\cap \mathcal{F}^{cs}_{\mathrm{loc}}(p_0)\neq \varnothing,\end{align*} $$

and hence, we can take

$$ \begin{align*}\widetilde{p}_0\in \pi_0(U^*)\cap \mathcal{F}^s_{\mathrm{loc}}(p_0) \subseteq M_0.\end{align*} $$

Moreover, $U^*\cap \mathcal {F}^c(\widetilde {p}_0)$ contains a closed central interval $J_{\widetilde {p}_0}^c$ , satisfying $\mathcal {P} := \mathcal {F}^u_{\mathrm {loc}}(J_{\widetilde {p}_0}^c)\subseteq U^*$ , see Figure 5.

Figure 5

Holonomy map from $p_0$ to $p_{0,m}$ (color online).

Figure 5 Long description

The diagram depicts a geometric mapping between two regions.

* On the left, a circular region labeled U super asterisk contains a vertical rectangular plane labeled J sub p tilde sub 0 super c. A point p tilde sub 0 is located on a red line below this circle. Further down, a black dot labeled p sub 0 marks the intersection of the red line and a green line.

* A long, narrow rectangular strip labeled F super N sub U (P) extends diagonally from the left region toward the right.

* On the right side, the strip terminates at a vertical blue line segment labeled J tilde sub p sub 0 comma m super c.

* Below this, a red line and a green line intersect at a black dot labeled p sub 0 comma m.

* A vertical blue line segment above this intersection is labeled J sub p sub 0 comma m super c.

* Dashed lines connect the upper blue segment on the strip to the lower blue segment near the intersection point.

* A point on the red line between the strip and the final intersection is labeled z sub 0 comma m.

* Arrows on the red and green lines indicate the direction of flow toward the intersection points.

Similarly, since $\unicode{x3bb} ^c(q_0)> 0$ and $W^u(q_0)$ is dense in M, we have

$$ \begin{align*}V^*\cap \mathcal{F}^{cu}_{\mathrm{loc}}(q_0)\neq \varnothing,\end{align*} $$

and, hence, we can take

$$ \begin{align*}\widetilde{q}_0\in \pi_0(V^*)\cap \mathcal{F}^u_{\mathrm{loc}}(q_0) \subseteq M_0.\end{align*} $$

We can further require that $V^*\cap \mathcal {F}^c(\widetilde {q}_0)$ contains a closed central interval $J_{\widetilde {q}_0}^c$ satisfying $\mathcal {Q} := \mathcal {F}^s_{\mathrm {loc}}(J_{\widetilde {q}_0}^c)\subseteq V^*$ , see Figure 6.

Figure 6

Holonomy map near $q_0$ (color online).

Note that $\mathcal {F}^u(p_0)$ is dense in $M_0$ . Hence, we can choose $z_{0,m}\in \mathcal {F}^u(p_0)\pitchfork \mathcal {F}^s_{\mathrm {loc}}(p_{0,m})$ . Since $p_{0,m}$ converges, we may further assume that $z_{0,m}$ also converges.

In fact, by taking a subsequence, we assume that $p_{0,m}$ lies in the local $su$ -box of $r_0$ , then it suffices to ensure that $\mathcal {F}^u(p_0)\pitchfork \mathcal {F}^s_{\mathrm {loc}}(r_0)\neq \varnothing $ .

It follows that there exists $N_U\in \mathbb {N}$ independent of $m\in \mathbb {N}$ such that

$$ \begin{align*}F^{N_U}(\mathcal{P})\pitchfork \mathcal{F}^c(\mathcal{F}^s_{\mathrm{loc}}(p_{0,m}))\neq\varnothing.\end{align*} $$

In particular, $F^{N_U}(\mathcal {P})\pitchfork \mathcal {F}^c(\mathcal {F}^s_{\mathrm {loc}}(p_{0,m}))$ contains a central interval $\widetilde {J}_{p_{0,m}}^c$ , whose length has a positive lower bound with respect to $m\in \mathbb {N}$ .

Let $J_{p_{0,m}}^c \subseteq \mathcal {F}^c(p_{0,m})$ be the image of $\widetilde {J}_{p_{0,m}}^c$ under the holonomy map along the stable leaves, restricted to $\mathcal {F}^{c}(\mathcal {F}^{s}_{\mathrm {loc}}(p_{0,m}))$ . By Lemma 2.1, the length of $J_{p_{0,m}}^c$ also has a positive lower bound with respect to $m\in \mathbb {N}$ . Moreover, since $p_{0,m}$ has a uniform size of contracting center and $J_{\widetilde {p}_0}^c$ lies in $W^s(p_0)$ , by taking a larger $N_U$ , we can assume further that ${J_{p_{0,m}}^c \subseteq I_{p_{0,m}}^c}$ .

Recall that in Lemma 3.2, we have a $C^{1+}$ parameterization $\gamma _{p_{0,m}}^s : [0, 1] \to \bar {I}_{p_{0,m}}^c$ for each $m\in \mathbb {N}$ . Since $I_{p_{0,m}}^c$ has a uniform size, we can further require that there exists ${0 < \sigma _0 < 1}$ such that

$$ \begin{align*}\sigma_0 < |(\gamma_{p_{0,m}}^s)'(t)| < \sigma_0^{-1}\quad \text{for all } t\in[0, 1],\ \text{for all } m\in\mathbb{N}.\end{align*} $$

However, let $J_{q_0}^c\subseteq \mathcal {F}^c_{\mathrm {loc}}(q_0)$ be the image of $J_{\widetilde {q}_0}^c$ under the holonomy map along the unstable leaves, restricted to $\mathcal {F}^{cu}_{\mathrm {loc}}(q_0)$ . Since $\unicode{x3bb} ^c(q_0)> 0$ , we may assume further that ${J_{q_0}^c\subseteq I_{q_0}^c}$ . Similarly, we have a $C^{1+}$ parameterization $\gamma ^s_{q_0}: [0, 1] \to \bar {I}_{q_0}^c$ satisfying

$$ \begin{align*}\sigma_0 < |(\gamma^s_{q_0})'(t)| < \sigma_0^{-1}\quad \text{for all } t\in[0, 1].\end{align*} $$

Exhibit central intersection. From now on, we fix $m\in \mathbb {N}$ so large that

$$ \begin{align*} L(\varepsilon_m) < \min\{\inf_{m\in\mathbb{N}}|(\gamma_{p_{0,m}}^s)^{-1}(J_{p_{0,m}}^c)|, \ |(\gamma_{q_0}^s)^{-1}(J_{q_0}^c)|\}. \end{align*} $$

Here, $L(\cdot )$ is given by Proposition 2.9 and $\varepsilon _m$ is defined in Claim 3.3 satisfying

$$ \begin{align*}(S_\varphi F_0(p_{0,m}),\ S_\varphi F_0(q_0)) < \varepsilon_m.\end{align*} $$

Without loss of generality, we assume that $p_{0,m}$ is a fixed point. Fix $r_{0,m} \in \mathcal {F}^u(p_{0,m}) \pitchfork \mathcal {F}^s(q_0)$ . For simplicity, we denote by $\mathrm {Hol}_x^y$ the holonomy map from x to y along stable or unstable manifolds defined on $cu$ -leaves or $cs$ -leaves, see Figure 7.

Figure 7

Central intersections (color online).

Figure 7 Long description

A three-dimensional diagram illustrating central intersections. At the central vertex is point r sub 0,m. Extending from this center are three colored axes. A green line extends to the bottom-left, ending at point p sub 0,m. A red line extends to the bottom-right, ending at point q sub 0. A vertical black line extends upward.

Along the green axis, a blue rectangular prism is labeled F super k sub n (J super c sub p sub 0,m) at its base and F super k sub n (tilde J super c sub p sub 0,m) at its top. Along the red axis, a similar blue rectangular prism is labeled F super minus l sub n (J super c sub q sub 0) at its base and F super minus l sub n (tilde J super c sub q sub 0) at its top.

Near the central intersection r sub 0,m, there are additional labeled points on the intersecting grid lines. Point u sub n is located on a green line slightly above the main axis. Point v sub n is on a red line to the right of the center. Point r super n sub 0,m is located vertically above the central r sub 0,m point where the upper blue frames intersect. Arrows on the red and green lines indicate directional flow toward and away from the central intersection points.

Claim 3.6. There exist two strictly increasing subsequences $\{k_n\},\ \{l_n\}$ of $\mathbb {N}$ such that

$$ \begin{align*}H\circ F^{k_n}(J_{p_{0,m}}^c)\cap F^{-l_n}(J_{q_0}^c)\neq\varnothing,\end{align*} $$

where

$$ \begin{align*}H := \mathrm{Hol}_{r_{0,m}}^{q_0}\circ\mathrm{Hol}_{p_{0,m}}^{r_{0,m}}.\end{align*} $$

Moreover, there exists a constant $c> 0$ such that

$$ \begin{align*} |H\circ F^{k_n}(J_{p_{0,m}}^c)\cap F^{-l_n}(J_{q_0}^c)| \geq c\cdot\max\{e^{k_n\unicode{x3bb}^c(p_{0,m})},\ e^{-l_n\unicode{x3bb}^c(q_0)}\}\quad \text{for all } n\in\mathbb{N}. \end{align*} $$

Proof of Claim 3.6.

Note that

$$ \begin{align*}(S_\varphi F_0(p_{0,m}),\ S_\varphi F_0(q_0)) < \varepsilon_m,\end{align*} $$

which means that

$$ \begin{align*}(\ln\|DF|_{E^c(p_{0,m})}\|,\ \ln\|DF|_{E^c(q_0)}\|) < \varepsilon_m,\end{align*} $$

since $p_{0,m}$ and $q_0$ are assumed to be fixed points. Equivalently, we have

$$ \begin{align*}(\ln\|DF|_{E^c(p_{0,m})}\|,\ \ln\|DF^{-1}|_{E^c(q_0)}\|) < \varepsilon_m.\end{align*} $$

The conclusion then follows from Proposition 2.9, by taking

$$ \begin{align*} f &= (\gamma_{p_{0,m}}^s)^{-1}\circ F\circ\gamma_{p_{0,m}}^s,\\ g &= (\gamma^s_{q_0})^{-1}\circ F^{-1}\circ \gamma^s_{q_0},\\ h &= (\gamma^s_{q_0})^{-1}\circ H\circ\gamma_{p_{0,m}}^s. \end{align*} $$

The initial intervals are $(\gamma _{p_{0,m}}^s)^{-1}(J_{p_{0,m}}^c)$ and $(\gamma ^s_{q_0})^{-1}(J_{q_0}^c)$ .

As a result, there exists a constant $c> 0$ such that

$$ \begin{align*} |{J_{p_{0,m}}^n}\cap {J_{q_0}^n}| \geq c\cdot\max\{e^{k_n\unicode{x3bb}^c(p_{0,m})},\ e^{-l_n\unicode{x3bb}^c(q_0)}\}\quad \text{for all }\, n\in\mathbb{N}, \end{align*} $$

where

$$ \begin{align*} {J_{p_{0,m}}^n} &:= \mathrm{Hol}_{p_{0,m}}^{r_{0,m}}\circ F^{k_n}(J_{p_{0,m}}^c);\\ {J_{q_0}^n} &:= (\mathrm{Hol}_{r_{0,m}}^{q_0})^{-1}\circ F^{-l_n}(J_{q_0}^c). \end{align*} $$

Now, consider the intersection between $F^{k_n}(\mathcal {F}^u_{\mathrm {loc}}(\widetilde {J}_{p_{0,m}}^c))$ and $F^{-l_n}(\mathcal {F}^s_{\mathrm {loc}}(J_{\widetilde {q}_0}^c))$ . Since $F_0$ is an Anosov diffeomorphism on a nilmanifold, for n sufficiently large, their projections on $M_0$ , $\pi _0(F^{k_n}(\mathcal {F}^u_{\mathrm {loc}}(\widetilde {J}_{p_{0,m}}^c)))$ , and $\pi _0(F^{-l_n}(\mathcal {F}^s_{\mathrm {loc}}(J_{\widetilde {q}_0}^c)))$ , intersect at $r_{0,m}^n\in M_0$ , which converges to $r_{0,m}$ as $n \to +\infty $ . Moreover, we have

$$ \begin{align*} u_n &\in \mathcal{F}^u_{\mathrm{loc}}(r_{0,m}^n)\pitchfork \mathcal{F}^s_{\mathrm{loc}}(r_{0,m}),\\ v_n &\in \mathcal{F}^s_{\mathrm{loc}}(r_{0,m}^n)\pitchfork \mathcal{F}^u_{\mathrm{loc}}(r_{0,m}). \end{align*} $$

For simplicity, we denote the intersections by

$$ \begin{align*} {\widehat{J}_{p_{0,m}}^n} &:= F^{k_n}(\mathcal{F}^u_{\mathrm{loc}}(\widetilde{J}_{p_{0,m}}^c))\cap \mathcal{F}^c(r_{0,m}^n);\\ {\widehat{J}_{q_0}^n} &:= F^{-l_n}(\mathcal{F}^s_{\mathrm{loc}}(J_{\widetilde{q}_0}^c)) \cap \mathcal{F}^c(r_{0,m}^n). \end{align*} $$

Our aim is to show that ${\widehat {J}_{p_{0,m}}^n}\cap {\widehat {J}_{q_0}^n}\neq \varnothing $ , see Figure 8.

Figure 8

Dynamics near $r_{0,m}$ (color online).

By Lemma 2.1 (the $C^1$ property of holonomy maps) and Lemma 2.2 (the estimate of central twists), there exists a constant $C\geq 1$ such that

$$ \begin{align*} &\mathrm{Hol}_{u_n}^{r_{0,m}}\circ \mathrm{Hol}_{r_{0,m}^n}^{u_n} ({\widehat{J}_{p_{0,m}}^n}) \text{ is a } Ce^{k_n\unicode{x3bb}_{p,m}}\text{-perturbation of } {J_{p_{0,m}}^n};\\&\quad \mathrm{Hol}_{v_n}^{r_{0,m}}\circ \mathrm{Hol}_{r_{0,m}^n}^{v_n} ({\widehat{J}_{q_0}^n}) \text{ is a } Ce^{-l_n\unicode{x3bb}_q} \text{-perturbation of } {J_{q_0}^n}, \end{align*} $$

as intervals in $\mathcal {F}^c(r_{0,m})$ , where

$$ \begin{align*} \unicode{x3bb}_{p,m} &:= \frac{\unicode{x3bb}^u(p_{0,m}) - \unicode{x3bb}^c(p_{0,m})}{\unicode{x3bb}^u(p_{0,m}) - \unicode{x3bb}^s(p_{0,m})}\unicode{x3bb}^s(p_{0,m}) < \unicode{x3bb}^c(p_{0,m}) < 0,\\ \unicode{x3bb}_q &:= \frac{-\unicode{x3bb}^s(q_0) + \unicode{x3bb}^c(q_0)}{-\unicode{x3bb}^s(q_0) + \unicode{x3bb}^u(q_0)}\unicode{x3bb}^u(q_0)> \unicode{x3bb}^c(q_0) > 0. \end{align*} $$

Again, by the argument of central twist in Lemma 2.2, for n sufficiently large and $x\in \widehat {J}_q^n$ , we have

$$ \begin{align*} &d_c( \mathrm{Hol}_{v_n}^{r_{0,m}}\circ \mathrm{Hol}_{r_{0,m}^n}^{v_n}(x),\, \mathrm{Hol}_{u_n}^{r_{0,m}}\circ \mathrm{Hol}_{r_{0,m}^n}^{u_n}(x))\\ &\quad\leq C[d_u(r_{0,m}^n,\, u_n) + d_s(u_n,\, r_{0,m}) + d_s(r_{0,m}^n,\, v_n) + d_u(v_n,\, r_{0,m})]\\ &\quad\leq C^2[e^{k_n\unicode{x3bb}^s(p_{0,m})} + e^{-l_n\unicode{x3bb}^u(q_0)}]. \end{align*} $$

Compared with the length of the intersection which is controlled by

$$ \begin{align*} |{J_{p_{0,m}}^n}\cap {J_{q_0}^n}| \geq c\cdot\max\{e^{k_n\unicode{x3bb}^c(p_{0,m})},\ e^{-l_n\unicode{x3bb}^c(q_0)}\} \triangleq \rho_n, \end{align*} $$

and the preceding perturbation errors, which are $o(\rho _n)$ , implied by the strict inequalities among the corresponding Lyapunov exponents, we conclude together with the injectivity of the maps $\mathrm {Hol}_{u_n}^{r_{0,m}}\circ \mathrm {Hol}_{r_{0,m}^n}^{u_n}$ and $\mathrm {Hol}_{v_n}^{r_{0,m}}\circ \mathrm {Hol}_{r_{0,m}^n}^{v_n}$ that there exists $n_0\in \mathbb {N}$ such that ${\widehat {J}_{p_{0,m}}^n}\cap {\widehat {J}_{q_0}^n}\neq \varnothing $ for any $n \geq n_0$ . Since ${\widehat {J}_{p_{0,m}}^n}\subseteq U^*$ and ${\widehat {J}_{q_0}^n} \subseteq V^*$ , we have $U^*\cap V^*\neq \varnothing $ .

This ends the proof of Theorem 3.1.

Proof. Notice that the boundary interconnection property is $C^1$ -open in $\mathrm {PHI}^1(M)$ . The conclusion then follows directly from Theorem 3.1.

Proof. By Corollary 2.7, we have $\unicode{x3bb} ^c(x) \leq 0$ whenever $\unicode{x3bb} ^c(x)$ exists. To show that $M_0$ is an attractor of G, it suffices (see [Reference CaoCao03, Lemma 2]) to check that there exists a constant $\varepsilon> 0$ such that

$$ \begin{align*} \{x\in M_0: \unicode{x3bb}^c_G(x) \leq - \varepsilon\} \end{align*} $$

has full measure with respect to any ergodic measure of $G_0$ .

Let $X(s) := s(s - 1)$ be a $C^\infty $ vector field on $[0,1]$ and $\Phi _t(s)$ be the $C^\infty $ flow generated by X. Then, by abuse of notation, we define

$$ \begin{align*}\Phi_t(z, s) := (z, \Phi_t(s)): N\times[0, 1] \to N\times [0,1].\end{align*} $$

It follows that $\Phi _t$ converges to identity as $t\to 0^+$ , in any $C^k$ topology.

Now, consider $G := \Phi _t\circ F$ . Since G acts in the same way as F on $M_0$ and $M_1$ , we have $G\in \mathrm {Diff}_\partial ^k(M)$ and, hence, when $t> 0$ is sufficiently small, $G\in \mathrm {PHI}^k(M)$ .

Note that the partially hyperbolic splitting of G restricted to $M_0$ has the same stable and unstable subbundles as F. We denote it by

$$ \begin{align*}T_xM = E^s(x)\oplus E^c_G(x)\oplus E^u(x)\quad \text{for all } x\in M_0.\end{align*} $$

Define $\alpha _t := \Phi ^{\prime }_t(0){=\partial _s\Phi _t(0)} \in (0, 1)$ and $\beta (z) := f^{\prime }_z(0)=\partial _s f_z(0)> 0$ , where we denote $F(z,s)=(F_s(z),f_z(s))$ . Here, $\alpha _t=e^{-t}$ , which can be solved by $({d}/{dt})\alpha _t=-\alpha _t$ and $\alpha _0=1$ , and hence, $\alpha _t \in (0, 1)$ when $t>0$ . Then, we have

$$ \begin{align*} D\Phi_t(z, 0) &= \bigg( \begin{array}{@{}cc@{}} I&0\\ 0&\alpha_t \end{array}\bigg);\\ DF(z, 0) &= \bigg( \begin{array}{@{}cc@{}} DF_0(z)&*\\ 0&\beta(z) \end{array} \bigg);\\ DG(z, 0) &= \bigg( \begin{array}{@{}cc@{}} DF_0(z)&* \\ 0&\alpha_t\beta(z) \end{array} \bigg). \end{align*} $$

Let $v^c = (*, 1)^T$ and $v^c_G = (*, 1)^T$ be vectors of $E^c(z, 0)$ and $E^c_G(z, 0)$ , respectively. Then,

$$ \begin{align*}DF(z, 0)(v^c) = (*,\, \beta(z))^T \quad\text{and}\quad DG(z, 0)(v^c_G) = (*,\, \alpha_t\beta(z))^T.\end{align*} $$

Let $\|\cdot \|$ be the original induced norm in $E^c$ and $E^c_G$ , and define

$$ \begin{align*} &\|(Z, S)^T\|_F := |S|\quad \text{for all } (Z,S)^T\in E^c;\\ &\|(Z, S)^T\|_G := |S|\quad \text{for all } (Z,S)^T\in E^c_G. \end{align*} $$

Since $E^c$ and $E^c_G$ are both transverse to $M_0$ , which is compact, the norms $\|\cdot \|_F$ and $\|\cdot \|_G$ are both equivalent with $\|\cdot \|$ . That is, there exists a constant $C \geq 1$ such that

$$ \begin{align*} C^{-1}\|\cdot\| \leq \|\cdot\|_F \leq C\|\cdot\|;\\ C^{-1}\|\cdot\| \leq \|\cdot\|_G \leq C\|\cdot\|. \end{align*} $$

Moreover, by the above discussion, we have

$$ \begin{align*}\|DF(z, 0)(v^c)\|_F = \beta(z)\|v^c\|_F\quad \text{and}\quad \|DG(z, 0)(v^c_G)\|_G = \alpha_t\beta(z)\|v^c_G\|_G.\end{align*} $$

As a result, for any $x\in M_0$ ,

$$\begin{align*}\|DG^n|_{E^c_G(x)}\|_G = \alpha_t^n\|DF^n|_{E^c(x)}\|_F \end{align*}$$

and, hence,

$$\begin{align*}C^{-4}\alpha_t^n\|DF^n|_{E^c(x)}\|\leq \|DG^n|_{E^c_G(x)}\|\leq C^{ 4}\alpha_t^n\|DF^n|_{E^c(x)}\|. \end{align*}$$

By the classical Oseledets theorem, for each ergodic measure $\mu $ of ${F_0 {\kern-1pt}={\kern-1pt} G_0: M_0 {\kern-1pt}\to{\kern-1pt} M_0}$ , there exists a $\mu $ -full measure subset $B_\mu \subseteq M_0$ such that for any $x\in B_\mu $ , the Lyapunov exponents of F and G at x both exist. Consider

$$ \begin{align*}B_0 := \bigcup_{\mu \text{ ergodic}}B_\mu,\end{align*} $$

then $\mu (B_0) = 1$ for any ergodic measure $\mu $ .

Fix $x\in B_0$ . We have

$$ \begin{align*} &\frac{1}{n}\ln\|DG^n|_{E^c_G(x)}\|\\ &\quad= \frac{1}{n}(\ln\|DG^n|_{E^c_G(x)}\| - \ln\|DF^n|_{E^c(x)}\|) + \frac{1}{n}\ln\|DF^n|_{E^c(x)}\|\\ &\quad\in \bigg[\frac{1}{n}\ln\|DF^n|_{E^c(x)}\| + \ln\alpha_t - \frac{ 4}{n}\ln C,\, \frac{1}{n}\ln\|DF^n|_{E^c(x)}\| + \ln\alpha_t + \frac{ 4}{n}\ln C\bigg]. \end{align*} $$

Let $n \to +\infty $ and we have

$$ \begin{align*}\unicode{x3bb}^c_G(x) = \unicode{x3bb}^c(x) + \ln\alpha_t \leq \ln\alpha_t < 0\quad \text{for all } x\in B_0.\end{align*} $$

This completes the proof of Lemma 4.2.

Proof of Theorem 4.1.

Since F is $C^k$ -robustly transitive, there exists an open neighborhood $\mathcal {U}\subseteq \mathrm {PHI}^k(M)$ of F such that every $G\in \mathcal {U}$ is topologically transitive.

We first prove that there exist $p_0,\ q_0\in \mathrm {Per}(F_0)$ such that $\unicode{x3bb} ^c(p_0) < 0 < \unicode{x3bb} ^c(q_0)$ . A similar conclusion holds for $F_1$ . For contradiction, without loss of generality, we assume that

$$ \begin{align*}\unicode{x3bb}^c(p)\leq 0\quad \text{for all } p\in \mathrm{Per}(F_0).\end{align*} $$

Then, by Lemma 4.2, there exists a $C^k$ -perturbation $G\in \mathcal {U}$ such that $M_0$ is an attractor of G, which contradicts the topological transitivity of G.

Now, take $p_0,q_0\in M_0$ and $p_1,q_1\in M_1$ such that

$$ \begin{align*}\unicode{x3bb}^c(p_0) < 0 < \unicode{x3bb}^c(q_0) \quad\text{and} \quad \unicode{x3bb}^c(q_1) < 0 < \unicode{x3bb}^c(p_1).\end{align*} $$

It remains to show that $W^s(p_0)\cap W^u(p_1)\neq \varnothing $ and, similarly, ${W^u(q_0)}\cap W^s(q_1)\neq \varnothing $ .

First, notice that

$$ \begin{align*} W^s(p_0) = \bigcup_{n\in\mathbb{N}}F^{-n}(\mathcal{F}^{cs}_{\mathrm{loc}}(p_0))\quad \text{and}\quad W^u(p_1) = \bigcup_{n\in\mathbb{N}} F^n (\mathcal{F}^{cu}_{\mathrm{loc}}(p_1)). \end{align*} $$

Since F is topologically transitive, there exists $x\in M$ such that

$$ \begin{align*}\overline{\{F^i(x): i \geq i_0\}} = M\quad \text{for all } i_0\in\mathbb{Z}.\end{align*} $$

In particular, there exists $i_0\in \mathbb {Z}$ such that $F^{i_0}(x)\in \mathcal {F}^s_{\varepsilon _0}(\mathcal {F}^{cu}_{\mathrm {loc}}(p_1))$ for some $\varepsilon _0> 0$ . Assume that $F^{i_0}(x)\in \mathcal {F}^s_{\varepsilon _0}(z_1)$ , where $z_1\in \mathcal {F}^{cu}_{\mathrm {loc}}(p_1)$ .

By the density of $\{F^i(x): i\geq i_0\}$ , for any $\varepsilon> 0$ , there exists $N\in \mathbb {N}$ such that $F^{i_0 + N}(x)\in \mathcal {F}^u_\varepsilon (\mathcal {F}^{cs}_\varepsilon (p_0))$ . Moreover, for fixed $\varepsilon> 0$ , N can be arbitrarily large. By taking $\varepsilon> 0$ small, we may assume that

$$ \begin{align*}\mathcal{F}^s_\varepsilon(F^{i_0 + N}(x))\subseteq \mathcal{F}^u_{\varepsilon_0}(\mathcal{F}^{cs}_{\mathrm{loc}}(p_0)).\end{align*} $$

Now, take N large so that $F^N(z_1)\in \mathcal {F}^s_\varepsilon (F^{i_0 + N}(x))$ . It follows that $F^N(z_1)\in \mathcal {F}^u_{\varepsilon _0}(z_0)$ for some $z_0\in \mathcal {F}^{cs}_{\mathrm {loc}}(p_0)$ . Since $W^u(p_1)$ is F-invariant and u-saturated, we have $z_0\in W^s(p_0)\cap W^u(p_1)$ .

This completes the proof of Theorem 4.1.

Proof of Theorem 5.1.

We first prove the following claim.

Claim 5.2. For any $cu$ -box $\mathcal {B}$ , $\mathcal {B}\cap B(\mu _0)$ has positive Lebesgue measure in $\mathcal {B}$ , and further, for any $cu$ -box $\mathcal {B}'$ that is $C^1$ -close to $\mathcal {B}$ ,

$$ \begin{align*}\mathrm{Leb}(\mathcal{B}'\cap B(\mu_0)) \geq \tfrac{1}{2}\mathrm{Leb}(\mathcal{B}\cap B(\mu_0)).\end{align*} $$

A similar result holds for $B(\mu _1)$ . Therefore, we conclude that for any non-empty open subset $U\subseteq M$ , $U\cap B(\mu _0)$ and $U\cap B(\mu _1)$ both have positive Lebesgue measure.

To show that $B(\mu _0)\cup B(\mu _1)$ has full Lebesgue measure, assume for contradiction that the complement $\mathcal {C}$ has positive Lebesgue measure. Take a density point $x^*\in M$ of $\mathcal {C}$ . Since the basins are s-saturated, we have

$$ \begin{align*} \frac{\mathrm{Leb}(\mathcal{C} \cap \mathcal{F}_\varepsilon^{cu}(x^*))} {\mathrm{Leb}(\mathcal{F}_\varepsilon^{cu}(x^*))} \to 1\quad \text{as } \varepsilon \to 0. \end{align*} $$

By the above argument, there exists $N_k \to +\infty $ and a $cu$ -box $\mathcal {P}_k \subseteq F^{N_k}(\mathcal {F}_{\varepsilon _0}^{cu}(x^*))$ near $M_0$ , with the same size, such that $\mathcal {P}_k\cap B(\mu _0)$ consists of central leaves. Moreover, we have that $F^{-N_k}(\mathcal {P}_k)\subseteq \mathcal {F}^c_{\varepsilon _0}(\mathcal {F}^u_{\varepsilon _k}(x^*))$ , where ${\varepsilon _k \to 0^+}$ . Define

$$ \begin{align*}\mathcal{P}^*_k = F^{N_k}(F^{-N_k}(\mathcal{P}_k)\cap\mathcal{F}^{cu}_{\varepsilon_k}(x^*)).\end{align*} $$

Since for each $k\in \mathbb {N}$ , $\det (DF^{N_k}|_{E^{cu}})$ is nearly constant on $\mathcal {F}^{cu}_{\varepsilon _0}(x^*)$ when $\varepsilon _0$ is small, we conclude that

$$ \begin{align*}\frac{\mathrm{Leb}(\mathcal{C}\cap \mathcal{P}^*_k)}{\mathrm{Leb}(\mathcal{P}^*_k)} \to 1 \quad \text{as } k \to +\infty, \end{align*} $$

and consequently,

$$ \begin{align*}\frac{\mathrm{Leb}(B(\mu_0)\cap\mathcal{P}^*_k)}{\mathrm{Leb}(\mathcal{P}^*_k)} \to 0 \quad \text{as } k \to +\infty.\end{align*} $$

Noting that $\mathcal {P}_k\cap B(\mu _0)$ consists of central leaves, we also have

$$ \begin{align*}\frac{\mathrm{Leb}(B(\mu_0)\cap\mathcal{P}_k)}{\mathrm{Leb}(\mathcal{P}_k)} \to 0\quad \text{as } k \to +\infty.\end{align*} $$

By taking a subsequence, we may assume that $\mathcal {P}_k$ converges to a $cu$ -box $\mathcal {P}$ . This contradicts the fact that a $cu$ -box near $\mathcal {P}$ intersects $B(\mu _0)$ on a subset whose Lebesgue measure has a lower bound away from zero, see Claim 5.2.

Proof of Theorem 5.3.

Since F admits $C^k$ -robustly intermingled basins, there exists an open neighborhood $\mathcal {U}\subseteq \mathrm {PHI}_-^k(M)$ of F such that every $G\in \mathcal {U}$ admits intermingled basins.

We first show that there exist $q_0\in \mathrm {Per}(F_0)$ and $p_1\in \mathrm {Per}(F_1)$ admitting positive central Lyapunov exponents.

Assume for contradiction that $\unicode{x3bb} ^c(p) \leq 0$ for all $p\in \mathrm {Per}(F_0)$ . By Lemma 4.2, there exists $G\in \mathcal {U}$ such that $M_0$ is an attractor of G. Moreover, G acts in the same way as F on $M_0$ and $M_1$ , hence, the boundary SRB measures of G are also $\mu _0$ and $\mu _1$ . In this case, $B_G(\mu _1)$ cannot be dense near $M_0$ , which contradicts the fact that G admits intermingled basins.

Now, we have $q_0\in \mathrm {Per}(F_0)$ and $p_1\in \mathrm {Per}(F_1)$ with positive central Lyapunov exponents. Fix $p_0\in \mathrm {Per}(F_0)$ and $q_1\in \mathrm {Per}(F_1)$ with negative central Lyapunov exponents. Since F admits intermingled basins, $B(\mu _0)\cap \mathcal {F}^{cu}_{\mathrm {loc}}(p_1)$ has positive Lebesgue measure in $\mathcal {F}^{cu}_{\mathrm {loc}}(p_1)$ . By forward iteration, we know that there exists a $cu$ -box $\mathcal {P}\subseteq W^u(p_1)$ near $p_0$ , since we have $\mathrm {supp}(\mu _0) = M_0$ . Therefore, $\mathcal {P}\pitchfork \mathcal {F}^{cs}_{\mathrm {loc}}(p_0)\neq \varnothing $ , and hence, $W^u(p_1)\pitchfork W^s(p_0)\neq \varnothing $ . A similar argument shows that $W^s(q_1)\pitchfork W^u(q_0)\neq \varnothing $ . Hence, F is boundary interconnected.

Proof of Theorem B.

Theorem 5.1 shows that $(2) \implies (1)$ ; Theorem 5.3 shows that $(1) \implies (3)$ . Finally, Corollary 3.7 shows that $(3) \implies (2)$ .