Proof of Theorem 1.6

We recall that the extension of $E^{cu}_\Lambda$ to $U$ is not necessarily $D\phi_t$-invariant and note that, given any $2$-subspace $F_x$ of $E^{cu}_x$, the map $t\mapsto |\det (D\phi_t\mid_{F_x})|$ is a multiplicative function $ |\det (D\phi_{t+s}\mid_{F_x})| = |\det (D\phi_s\mid_{D\phi_t\cdot F_x})| \cdot |\det (D\phi_t\mid_{F_x})|$, for all $t,s\ge0, x\in U$.

In addition, by Proposition 2.3, we get $D\phi_t\cdot E^{cu}_x\subset C^{cu}_{\kappa \lambda^t a}(\phi_tx)$ for constants $\kappa,a \gt 0$. Then the stable direction $E^s_{\phi_tx}$ is complementary to both the $D\phi_t\cdot E^{cu}_x$ and $E^{cu}_{\phi_tx}$ directions at $\phi_tx$. Therefore there exists a natural isomorphism $\pi^s:(D\phi_t\cdot E^{cu}_x)\to E^{cu}_{\phi_tx}$ given by the projection parallel to $E^s_{\phi_tx}$. Hence, $\pi^s(D\phi_t\cdot F_x) = {\widehat F}_{\phi_tx}\subset E^{cu}_{\phi_tx}$ and since the width $\kappa\lambda^t a$ of the center-unstable cone around $E^{cu}_{\phi_tx}$ is small for large $t \gt 0$, then we obtain $\xi_t\to 1$ when $t\nearrow\infty$ such that for any fixed $s\ge0$

(25)\begin{align} \xi_t^{-1} |\det (D\phi_s\mid_{{\widehat F}_{\phi_t x}})| &\le \frac{|\det (D\phi_{t+s}\mid_{F_x})|}{|\det (D\phi_t\mid_{F_x})|} \le \xi_t |\det (D\phi_s\mid_{{\widehat F}_{\phi_t x}})|. \end{align}

We first show that (wASE), given by (15), implies the (wNU2SE) condition (11), i.e., (i) $\implies$(A), when $d_{cu}=2$. By assumption, given any $2$-subspace $F_x\subset E^{cu}_x$ and sufficiently small $\varepsilon \gt 0$, there exists a strictly increasing sequence $t_k$ such that $t_k\nearrow\infty$ and $ \log|\det(D\phi_{t_k}\mid_{F_x})| \ge (c_0-\varepsilon)t_k$. Since $|\det (D\phi_{[t_k]}\mid_{F_x})|=\prod_{i=0}^{[t_k]-1} \frac{|\det (D\phi_{i+1}\mid_{F_x})|}{|\det (D\phi_i\mid_{F_x})|}$ we getFootnote ⁸ from (25)

\begin{align*} \left|\log|\det (D\phi_{t_k}\mid_{F_x})^{-1}| +\sum\nolimits_{i=0}^{[t_k]-1}\log|\det(Df\mid_{{\widehat F}_{f^ix}})|\right| \le \sum\nolimits_{i=0}^{[t_k]}|\log\xi_i|. \end{align*}

Hence, for all $k$ and each $2$-subspace $F_x\subset E^{cu}_x$, we can write

\begin{align*} \sum\nolimits_{i=0}^{[t_k]-1}\log|\det(Df\mid_{F_{f^ix}})^{-1}| \le (-c_0+\varepsilon) t_k + \sum\nolimits_{i=0}^{[t_k]-1}|\log\xi_i|. \end{align*}

Since $\varepsilon \gt 0$ was arbitrary, we obtain $ \liminf\frac1n\sum\nolimits_{i=0}^{n-1} \log\|\wedge^2(Df\mid_{E^{cu}_{f^ix}})^{-1}\| \le -c_0, $ that is, (wNU2SE) if $\dim E^{cu}=2$. Analogously, we deduce that (ASE) $\implies$(NU2SE), i.e., (i) $\implies$(A) after exchanging $\liminf$ with $\limsup$, and note that no assumption on non-convergence to equilibria was needed here.

We now easily show both (NU2SE) $\implies$ (NUSE) and (wNU2SE) $\implies$ (wNUSE), i.e., (A) $\implies$(B), without restrictions on $d_{cu}$. For a regular point $x\in U$, any $2$-subspace $F_x\subset E^{cu}_x$ containing $G(x)$ admits an orthonormal basis $\{G(x)/\|G(x)\|,n(x)\}$Footnote ⁹ and $\widehat F_{f^ix}=\pi^s(Df^i_x\cdot F_x)$ admits an orthonormal basis $\{G(f^ix)/\|G(f^ix)\|,n(f^ix)\}$ for $i\ge1$, where $n(f^ix)$ is a unit vector on the line $\widehat F_{f^ix}\cap G(f^ix)^\perp$Footnote ¹⁰. These choices ensure that

(26)\begin{align} |\det(D\phi_1\mid_{{\widehat F}_{f^ix}})| = \det \begin{pmatrix} \|G(f^{i+1}x)\|/\|G(f^ix)\| & \star \\ 0 & \|P^1\cdot n(f^ix)\| \end{pmatrix}. \end{align}

Let $\mathcal{G}_2(f^ix)$ be the compact family of all $2$-subspaces of $E^{cu}_{f^ix}$ which contain $G(f^ix)$. From (26) we get

\begin{align*} \sup\nolimits_{F\in\mathcal{G}_2(f^ix)}|\det(Df\mid_{F})^{-1}| = \big(\|G(f^{i}x)\|/\|G(f^{i+1}x)\|\big)\cdot\sup\nolimits_{n\in N^{cu1}_{f^ix}} \|P^1\cdot n\|^{-1}, \end{align*}

where $N^{cu1}_{f^ix}$ is the unit sphere on the normal bundle $N^{cu}_{f^ix}$ and $i\ge1$. Since this is a subset of all $2$-subspaces of $E^{cu}_x$, we get

(27)\begin{align} \frac{\|G(f^{i}x)\|}{\|G(f^{i+1}x)\|} \cdot \|(P^1\mid_{N^{cu}_{f^ix}})^{-1}\| \le \|\wedge^2(Df\mid_{E^{cu}_{f^ix}})^{-1}\|. \end{align}

Let $\Gamma(x):= \log\|\wedge^2(Df\mid_{E^{cu}_{x}})^{-1}\|$ and $\psi(x):= \log \|(P^1\mid_{N^{cu}_{x}})^{-1}\|$ in what follows. Therefore

\begin{align*} \sum\nolimits_{i=0}^{n-1} \Gamma(f^ix) + \sum\nolimits_{i=0}^{n-1}\log\frac{\|G(f^{i+1}x)\|}{\|G(f^ix)\|} &\ge \sum\nolimits_{i=0}^{n-1} \psi(f^ix), \quad n\ge1. \end{align*}

Since we assume that the future trajectory of $x$ does not converge to any equilibrium, we can use the following.

Hence, dividing by $n$ and taking $\limsup$ when $n\nearrow\infty$, Lemma 2.8 together with (27) ensures that (NU2SE) $\implies$(NUSE). Analogously, using $\liminf$ provides (wNU2SE) $\implies$(wNUSE) since, for all $\varepsilon \gt 0$ there exists $N \gt 1$ so that $\frac1n\sum_{i=0}^{n-1} \log\frac{\|G(g^{i+1}x)\|}{\|G(g^{i}x)\|} \lt \varepsilon$ for all $n\ge N$, by Lemma 2.8. Thus, for any $\varepsilon \gt 0$ we get

\begin{align*} \liminf_{n\nearrow\infty} \frac1n \sum\nolimits_{i=0}^{n-1} \psi(f^i x) \le \varepsilon+\liminf_{n\nearrow\infty} \frac1n\sum\nolimits_{i=0}^{n-1} \Gamma(f^ix) \le\varepsilon -c_0. \end{align*}

Both (wMASE) $\implies$ (wASEP) and (MASE) $\implies$ (ASEP), i.e., (a) $\implies$(b), follow from the analogous to (26) and (27): we get for the family of $2$-subspaces $F$ of $E^{cu}_x$ with orthonormal basis $\{G(x),n(x)\}$ for some $n(x)\in N^{cu1}_x$

(28)\begin{align} |\det(D\phi_t\mid_F)| = \det \begin{pmatrix} \|G(\phi_tx)\|/\|G(x)\| & \star \\ 0 & \|P^t\cdot n(x)\| \end{pmatrix} \end{align}

and it follows, as before, that $ \frac{\|G(x)\|}{\|G(\phi_tx)\|} \cdot \|(P^t\mid_{N^{cu}_x})^{-1}\| \le \|\wedge^2(D\phi_t\mid_{E^{cu}_x})^{-1}\|. $ Then, for each $t \gt 0$, we get $ \frac1t\left(\log\|\wedge^2(D\phi_t\mid_{E^{cu}_x})^{-1}\| + \log\frac{\|G(\phi_tx)\|}{\|G(x)\|}\right) \ge \frac1t\log\|(P^t\mid_{N^{cu}_x})^{-1}\|, $ and the claimed relations follow just like above using Lemma 2.8.

Analogously, we obtain (b) $\implies$(i), that is, (wASEP) $\implies$ (wASE) and (ASEP) $\implies$ (ASE). For any $2$-subspace $F$ of $E^{cu}_x$ containing $G(x)$ and each fixed $t \gt 0$, we have (28). For $F$ not containing $G(x)$, we choose a basis $\{v_i+\omega_i G(x), i=1,2\}$ of $F$, with $\omega_1\omega_2\neq0$; and $\{v_i^t+\omega_i^tG(\phi_tx), i=1,2\}$ of $F_t:=D\phi_t(x)F$, such that $\{v_1^t , v_2^t\}$ is orthonormal. Thus, $P^{-t}\big(v_i^t+\omega_i^tG(\phi_tx )\big)=v_i, i=1,2$ and $ |\det(D\phi_t\mid_F)^{-1}|$ is given by

\begin{align*} \frac{\| \big(v_1+\omega_1G(x)\big) \wedge \big(v_2+\omega_2G(x)\big)\|}{\|\big(v_1^t+\omega_1^tG(\phi_tx)\big)\wedge\big(v_2^t+\omega_2^tG(\phi_tx)\big)\|} = \|(P^{-t}v_1^t) \wedge (P^{-t}v_2^t)\| \le \|(P^t\mid_{N^{cu}_x})^{-1}\|^2. \end{align*}

Then $ |\det(D\phi_t\mid_F)^{-1}| \le \max\left\{ \frac{\|G(x)\|}{\|G(\phi_tx)\|}\|(P^t\mid_{N^{cu}_x})^{-1}\| , \|(P^t\mid_{N^{cu}_x})^{-1}\|^2 \right\}=\max\{a_t,b_t\}$. From (wASEP) and Lemma 2.8, for any $\varepsilon \gt 0$ we have $T \gt 0$ so that $\|G(\phi_tx)\| \gt e^{-\varepsilon t}$ for all $t \gt T$, and then $\liminf\frac1t\log a_t$ satisfies

\begin{align*} \liminf_{t\nearrow\infty}\frac1t\log\Big(\frac{\|G(x)\|}{\|G(\phi_tx)\|} \|(P^t\mid_{N^{cu}_x})^{-1}\|\Big) \le \varepsilon+ \liminf_{t\nearrow\infty}\frac1t\log\|(P^t\mid_{N^{cu}_x})^{-1}\| \le -c_0+\varepsilon. \end{align*}

Since $\varepsilon \gt 0$ is arbitrary, we see that for any sequence $t_n\nearrow\infty$ there exists $N \gt 1$ so that for all $n \gt N$ we have $|\det(D\phi_{t_n}\mid_F)^{-1}|\le a_{t_n}$ and so

\begin{align*} \limsup_{n\nearrow\infty}\frac1n\log |\det(D\phi_t\mid_F)| &= \liminf_{n\nearrow\infty}\frac1n\log |\det(D\phi_t\mid_F)^{-1}| \le -c_0. \end{align*}

Since $F$ is an arbitrary $2$-subspace of $E^{cu}_x$, we have obtained (wASE), as claimed. If we assume (ASEP), then from Lemma 2.8 we have

\begin{align*} \limsup_{t\nearrow\infty}\frac1t\log a_t &\le \liminf_{t\nearrow\infty}\frac1t\log\frac{\|G(\phi_tx)\|}{\|G(x)\|} + \limsup_{t\nearrow\infty}\frac1t\log\|(P^t\mid_{N^{cu}_x})^{-1}\|\le -c_0. \end{align*}

Hence, by the same reasoning as above, we conclude (ASE)

\begin{align*} \liminf_{n\nearrow\infty}\frac1n\log |\det(D\phi_t\mid_F)| &= \limsup_{n\nearrow\infty}\frac1n\log |\det(D\phi_t\mid_F)^{-1}| \le -c_0. \end{align*}

We can now show that (i) $\implies$(ii), that is, (wASE) $\implies$(wPSL) and (ASE) $\implies$(PSL). Indeed, given any $v\in E^{cu}_x\setminus {\mathbb R}\cdot G(x)$, we let $F$ be the $2$-subspace of $E^{cu}_x$ generated by $\{G(x),n(x)\}$, where $n(x)\in N^{cu}_x$ is the orthogonal projection of $v$. Then, assuming (wASE), given $\varepsilon \gt 0$, there exists $t_n\nearrow\infty$ so that

\begin{align*} e^{\varepsilon t_n}\|P^{t_n}n(x)\| \ge \frac{\|G(\phi_{t_n}x)\|}{\|G(x)\|}\|P^{t_n}n(x)\| = |\det(D\phi_{t_n}\mid_F)| \ge e^{(c_0-\varepsilon)t_n} \end{align*}

from (28) and Lemma 2.8. By definition of the Linear Poincaré Flow, since $v=n(x)+\omega G(x)$ with $\omega\in{\mathbb R}$, then there exits $\eta \gt 0$ so that

\begin{align*} \|D\phi_{t_n}(x)v\|^2 = \|P^{t_n}n(x)+\eta G(\phi_{t_n}x)\|^2 = \|P^{t_n}n(x)\|^2+\|\eta G(\phi_{t_n}x)\|^2 \ge \|P^{t_n}n(x)\|^2. \end{align*}

Altogether we obtain $\|D\phi_{t_n}(x)v\|\ge e^{(c_0-2\varepsilon)t_n}$. Since $\varepsilon \gt 0$ is arbitrary, we get (wPSL). Assuming now (ASE), that is $\liminf\frac1T\log|\det(D\phi_T\mid F)|\ge c_0$, then we obtain the same estimates above for all times $t$ larger than some $T \gt 0$ in the place of $t_n$. We thus conclude that $\|D\phi_{t}(x)v\|\ge e^{(c_0-2\varepsilon)t}$ for all $t \gt T$. Hence, we get $\liminf\frac1t\log\|D\phi_{t}(x)v\|\ge c_0$, i.e, (PSL).

Finally, for (A) $\implies$(a), that is, (wNU2SE) $\implies$ (wMASE) and (NU2SE) $\implies$ (MASE): these follow from the subadditivity of the continuous map $(t,x)\in{\mathbb R}^+\times U \mapsto\psi_t(x)=\log\|\wedge^2(D\phi_t\mid_{E^{cu}_x})^{-1}\|$ together with the relative compactness of $U$. Indeed, for any real $T \gt 0$ we have

\begin{align*} \frac{\log\|\wedge^2(D\phi_T\mid_{E^{cu}_x})^{-1}\|}T &\le \frac{\log\|\wedge^2(D\phi_{T-[T]}\mid_{E^{cu}_{f^{[T]}x}})^{-1}\|}T + \sum_{i=0}^{[T]-1}\frac{\log\|\wedge^2(Df\mid_{E^{cu}_{f^ix}})^{-1}\|}T \\ &\le \frac{[T]}T\cdot\frac1{[T]}\sum\nolimits_{i=0}^{[T]-1} \Gamma(f^ix) + \frac2T\log L, \end{align*}

where $L:=\sup\{\log\|\wedge^2(D\phi_t\mid_{E^{cu}_x})^{-1}\| : x\in U, 0\le t\le1\}$. So letting $T\nearrow\infty$ shows that the discrete time average of $\Gamma$ is larger than the asymptotic average, as claimed.

This completes the proof of Theorem 1.6, except for the proof Lemma 2.8, which we present next.

Proof of Lemma 2.8

The sum in the statement is telescopic, so it becomes

(29)\begin{align} \limsup\nolimits_{n\nearrow\infty} (\log\|G(f^nx)\|-\log\|G(x)\|)/n = \limsup\nolimits_{n\nearrow\infty} \log\|G(f^nx)\|^{1/n}. \end{align}

Since $\|G\|$ is bounded above, we get (29) $\le\limsup_{t\nearrow\infty}\frac1t\log\|G(\phi_tx)\|\le0$. Hence, it is enough to find the same lower bound.

Let us assume that the future trajectory of $x$ does not converge to an equilibrium. We argue by contradiction: if there exists $\varepsilon_0 \gt 0$ so that $ \limsup_{t\nearrow\infty} \log\|G(\phi_tx)\|^{1/t} \lt -\varepsilon_0$, then we can find $T=T(x,\varepsilon_0) \gt 1$ so that for all $t \gt T$ we get $ \|G(\phi_tx)\| \le e^{-\varepsilon_0 t}$. This implies that $\phi_tx$ is converging to the finite subset $\operatorname{Sing}_\Lambda(G)$, and so converges to some equilibrium of this set. This contradiction shows that $\limsup\frac1t\log\|G(\phi_tx)\|=0$. The same argument replacing a real $t\nearrow\infty$ with an integer $n\nearrow\infty$ completes the proof of the first statement of the lemma.

Let $S(\Omega)\subset\operatorname{Sing}_\Lambda(G)$ be the collection of equilibria accumulated by the future trajectories of points of $\Omega$. Then $\sigma\in S(\Omega)$ cannot be a sink (otherwise trajectories converging to $\sigma$ cannot have sectional expansion) nor a source. Since $S(G)$ consists of finitely many hyperbolic fixed points of saddle-type, then every approach of the trajectory to any element $\sigma\in S(G)$ is followed by a departure from $\sigma$, except for the points of the stable manifold of $\sigma$. But, in this setting, the union $\cup_{\sigma\in S(G)}W^s_\sigma$ of the local stable manifolds has zero volume. Indeed, the stable manifold of a hyperbolic critical element in an immersed submanifold [Reference Palis and de Melo51], so it has zero volume as a subset of the ambient manifold. Hence, $U \setminus\cup \{W^s_\sigma:\sigma\in\operatorname{Sing}_\Lambda(G)\}$ has full volume in $U$.

Thus, the previous convergence of $f^nx$ or $\phi_tx$ to $\operatorname{Sing}_\Lambda(G)\supset S(\Omega)$ is impossible on a positive Lebesgue measure subset. This completes the proof of the lemma.

Proof of Theorem 1.8

Let $\Gamma(x):=\log\| \wedge^2(Df\mid_{E^{cu}_x})^{-1}\|$ and set $E:=\{x\in U:$ (19) holds for $x\}$. If $\mu(E)=1$ for some ergodic invariant measure $\mu$ for the flow, we obtain $ \mu(\Gamma) = \lim_{t\nearrow\infty}\frac1t\int\Gamma(\phi_tx)\,d\mu(x) = \int \lim_{t\nearrow\infty}\frac1t\Gamma(\phi_tx)\,d\mu(x) \lt -\omega $ from Kingman’s Subadditive Ergodic Theorem, since the limit exists and coincides with the limit inferior for $\mu$-a.e. $x$. But by subadditivity $\int\log\|\wedge^2(D\phi_t\mid_{E^{cu}_x})^{-1}\|\,d\mu(x)$ is bounded above by

\begin{align*} &\int\log\| \wedge^2(D\phi_{t-[t]}\mid_{E^{cu}_{\phi_{[t]}x}})^{-1}\| \,d\mu(x) + \int\log\|\wedge^2(D\phi_{[t]}\mid_{E^{cu}_x})^{-1}\|\,d\mu(x) \\ &\le \sup_{x\in U}\sup_{s\in[0,1]}\log\| \wedge^2(D\phi_s\mid_{E^{cu}_x})^{-1}\| + \sum\nolimits_{i=0}^{[t]-1}\int\Gamma(f^ix)\,d\mu(x) = C + [t]\mu(\Gamma) \end{align*}

where $f:=\phi_1$, $C$ is a constant depending on the flow, and we used that $\mu$ is $f$-invariant. Therefore, again by the Subadditive Ergodic Theorem

\begin{align*} \int \lim_{t\nearrow\infty}\frac1t \log\|\wedge^2(D\phi_t\mid_{E^{cu}_x})^{-1}\|\,d\mu(x) &= \lim_{t\nearrow\infty}\frac1t \int\log\|\wedge^2(D\phi_t\mid_{E^{cu}_x})^{-1}\|\,d\mu(x) \\ &\le \limsup_{t\nearrow\infty}\frac{[t]}t\cdot\mu(\Gamma) \lt -\omega. \end{align*}

Since we assume that the above holds for any invariant ergodic probability measure $\mu$ for the flow, we are in the conditions of the following result.

If we set $f_t(x):=\log\|\wedge^2(D\phi_t\mid_{E^{cu}_x})^{-1}\|$ we obtain $T_u,\lambda_u \gt 0$ such that

\begin{align*} \|\wedge^2(D\phi_t\mid_{E^{cu}_x})^{-1}\|\le e^{-\lambda_u t} \quad\text{for}~t\ge T_u~\text{and all}~x\in \Lambda. \end{align*}

Analogously, if we instead set $f_t(x)=\log\|D\phi_t\mid_{E^{cs}_x}\|$ and reapply the same reasoning, we conclude the existence of $T_s,\lambda_s \gt 0$ so that $\|D\phi_t\mid_{E^{cs}_x}\|\le e^{-\lambda_s t}$ for all $x\in\Lambda$ and $t\ge T_s$.

This ensures that the compact set $\Lambda$ with the invariant continuous splitting $E^{cs}\oplus E^{cu}$ is a sectional-hyperbolic set, and concludes the proof of the first statement of the theorem.

For the second statement, we write $\psi_\delta(x):=\int_0^\delta\Gamma(\phi_tx) \, dt$ where $\delta=1/m \gt 0$ for some $m \gt 1$ so large that for $0 \lt \zeta \lt \omega/4$ we get $ \exp\big(\Gamma(\phi_\delta x)-\Gamma(x)\big) \le 1+\zeta$ and so $|\Gamma(\phi_\delta x)- \Gamma(x)|\le \log(1+\zeta) \lt \omega/4 $ for all $x\in U$. Then $\psi_\delta(x)=\Gamma(x)+\xi(x)$ where $\xi:U\to(-\zeta,\zeta)$ is a continuous function.

Lemma 2.10. Let us consider a point $x\in U$ and a sequence $T_n\nearrow\infty$ satisfying

(30)\begin{align} \int_0^{T_n}\Gamma(\phi_tx) \, dt \lt -\omega T_n \end{align}

for all large $n \gt 1$. Then there exists $0\le \widehat{j} \lt m$ so that for $n$ large enough, with $\ell=\ell_n=[T_n/\delta]/m$ and $g:=\phi_\delta$, we have $ \sum\nolimits_{i=0}^{\ell-1}\Gamma(f^ig^{\widehat{j}}x) \lt -\ell \omega/4m. $

Proof. We rewrite the integral in (30) as follows

(31)\begin{align} \int_0^{T_n}\Gamma(\phi_tx)\,dt = \int_{[T_n]}^{T_n-\delta[T_n/\delta]}\Gamma(\phi_tx) \, dt +\sum\nolimits_{i=0}^{[T_n/\delta]-1}\psi_\delta(g_\delta^ix). \end{align}

Since the integrand $\Gamma$ is uniformly bounded by a constant $C$ and $T_n-\delta[T_n/\delta] \lt \delta$, can rewrite (30) as $ \sum\nolimits_{i=0}^{[T_n/\delta]-1}\psi_\delta(g^ix) \lt -\omega T_n + C\delta \lt (-\omega + C\delta/T_n)T_n \lt - [T_n]\omega/2 $ for all $n$ large enough. We group iterates that are time- $1$ apart rewriting the summation as follows: we let $[T_n/\delta]=\ell m$ for some $\ell=\ell_n \gt 1$ and

\begin{align*} \sum\nolimits_{j=0}^{m-1} \sum\nolimits_{i=0}^{\ell-1}\psi_\delta(f^ig^jx) \lt - [T_n/\delta]\delta \omega/2 = - \ell \omega/2. \end{align*}

So, one of the inner sums is negative. More precisely, there exists $\widehat{j}=\widehat{j}(x)\in\{0,\dots,m-1\}$ so that $ \sum_{i=0}^{\ell-1}\psi_\delta(f^ig^{\widehat{j}}x) \lt -\ell \omega/2m. $ By the choice of $\delta$ we obtain

\begin{align*} &\sum\nolimits_{i=0}^{\ell-1} \big(\Gamma(f^ig^{\widehat{j}}x)\pm\xi(f^ig^{\widehat{j}}x)\big) \lt -\ell \omega/2m\\ &\quad \implies \sum\nolimits_{i=0}^{\ell-1}\Gamma(f^ig^{\widehat{j}}x) \lt -\ell \omega/2m+\zeta \lt -\ell \omega/4m \end{align*}

for $n$ large enough, with $\ell=\ell_n=[T_n/\delta]/m$, as in the statement of the lemma.

We define $ E_j := \left\{x\in U: \liminf\nolimits_{\ell\nearrow\infty} \sum\nolimits_{i=0}^{\ell-1}\Gamma(f^ig^jx)^{1/\ell} \lt -\omega/4m \right\} $ for each $0\le j \lt m$. If we assume that $\operatorname{Leb}(E) \gt 0$, then from Lemma 2.10 we have that $E\subset\cup_{j=0}^{m-1}E_j$ and so there exists $\widehat{j}\in\{0,\dots,m-1\}$ such that $\operatorname{Leb}(E_{\widehat{j}}) \gt 0$. By smoothness of $f$, the set $\Omega=g^{-\widehat{j}}E_{\widehat{j}}$ satisfies $\operatorname{Leb}(\Omega) \gt 0$ and every $x\in \Omega$ is such that

\begin{align*} \liminf_{n\nearrow\infty} \frac1n\sum\nolimits_{i=0}^{n-1} \Gamma(f^ix) \le -\omega/4m. \end{align*}

This is (wNU2SE) with $c_0=\omega/4m$, completing the proof of the first part of the second statement of Theorem 1.8.

Exchanging $\liminf$ by $\limsup$ in $E$ and $E_j$ enables us to follow the same reasoning to show that (20), on a positive volume subset of $U$, implies (NU2SE) on a positive volume subset of $U$ with a slightly smaller rate $c_0=\omega/m$. This completes the proof of Theorem 1.8.

Proof. Since all equilibria in $\operatorname{Sing}_\Lambda(G)$ are hyperbolic, there are at most finitely many and those accumulated by the orbit of $x\in\Omega$ are of saddle type. Thus, there exists $b \gt 0$ so that $\|(DG(\sigma))^{-1}\| \lt 1/b$ for all $\sigma\in\omega_G(x)\cap\operatorname{Sing}_\Lambda(G)$.

Moreover, we have $\|DG(y)-DG(\sigma)\|\le \kappa_0 d(y,\sigma)^\beta$ for all $y\in B(\sigma,2\delta_0)$ and some constants $\kappa_0 \gt 0$ and $0 \lt \beta\le1$ since $G$ is of class $C^{1+}$ — here and in the following estimates, we identify $B(\sigma,\rho_0)$ with the $\rho_0$-ball on $T_\sigma M$. Hence, if $2\kappa_0\delta_0^\beta \lt b$, then by the Mean Value Inequality

\begin{align*} \|G(x)-G(\sigma)-DG(\sigma)\cdot(x-\sigma)\| \le \kappa d(x,\sigma)^\beta\|x-\sigma\| \le \kappa_0\delta_0^\beta d(x,\sigma) \end{align*}

and so $ \|G(x)\| \ge \|DG(\sigma)\cdot(x-\sigma)\|-\kappa_0\delta_0^\beta d(x,\sigma) \ge (b -\kappa_0\delta_0^\beta) d(x,\sigma)$. On the other hand, by the smoothness of $G$ we have that $\|G(x)\|=\|G(x)-G(\sigma)\|\le L \cdot d(x,\sigma)$. Finally, if $2d(y,x) \lt d(x,\sigma)$, then

\begin{align*} \|G(y)\| \ge (b -\kappa_0\delta_0^\beta) d(y,\sigma) \ge \frac{b -\kappa_0\delta_0^\beta}2 d(x,\sigma) \ge \frac{b -\kappa_0\delta_0^\beta}{L} \|G(x)\| \end{align*}

which completes the proof after setting $b_*=(b-\kappa_0\delta_0^\beta)/L$.

Proof. First choose $s_0 \gt 0$ small enough so that $d(z,\phi_sz) \lt \rho_1$ for all $z\in U$ and $|s| \lt s_0$, and use (34) to obtain

\begin{align*} \prod\nolimits_{i=n-k}^{n-1}\|\big(P^1\mid_{N^{cu}_{\phi_sf^ix}}\big)^{-1}\| &= \prod\nolimits_{i=n-k}^{n-1}\left(\frac{\|\big(P^1\mid_{N^{cu}_{\phi_sf^ix}}\big)^{-1}\|}{\|\big(P^1\mid_{N^{cu}_{f^ix}} \big)^{-1}\|} \cdot \|\big(P^1\mid_{N^{cu}_{f^ix}}\big)^{-1}\|\right) \\ &\le e^{kc_0/8}\cdot e^{-kc_0/4}=e^{-kc_0/8}; \quad k=1,\dots,n; |s| \lt s_0. \end{align*}

Then, if $d(f^ix,\operatorname{Sing}_\Lambda(G)) \lt \delta$, we note that $d(\phi_sf^ix,\operatorname{Sing}_\Lambda(G))$ is bounded from below by

\begin{align*} d(f^ix,\operatorname{Sing}_\Lambda(G))-d(\phi_sf^ix,f^ix) \ge d(f^ix,\operatorname{Sing}_\Lambda(G))-|s|\sup\nolimits_{|t| \lt s}\|G({\phi_tf^ix})\|. \end{align*}

Lemma 3.4 provides $\|G({\phi_tf^ix})\|\ge b_*\|G(x)\|$ whenever $2 d(\phi_tf^ix,f^ix) \lt d(f^ix,\operatorname{Sing}_\Lambda(G))$, which holds for $|t| \lt d(f^ix,\operatorname{Sing}_\Lambda(G))/(2\|G(x)\|)\le (2b_*L)^{-1}$. In this case, we obtain

\begin{align*} d(\phi_sf^ix,\operatorname{Sing}_\Lambda(G)) &\ge d(f^ix,\operatorname{Sing}_\Lambda(G))-|s|\cdot b_*\|G(x)\|\\ & \ge (1-b_*L\cdot|s|)\cdot d(f^ix,\operatorname{Sing}_\Lambda(G)). \end{align*}

Hence, choosing $s_0\in(0,(2b_*L)^{-1})$ small enough so that $d(z,\phi_sz) \lt \rho_1$ for all $z\in U$ we can assume without loss of generality that

\begin{align*} d_\delta(\phi_sf^ix,\operatorname{Sing}_\Lambda(G)) \ge e^{-L}d_\delta(f^ix,\operatorname{Sing}_\Lambda(G)) \ge e^{-L}e^{-(n-i)c_0/16}; \quad i=0,\dots, n; |s| \lt s_0. \end{align*}

The above conclusions show that, modulo a small change of rates, $n$ is still a hyperbolic time for $\phi_sx$ for each $|s| \lt s_0$.

Proof. We have $y\in\Sigma_{f^i x}^{n-i}$ by definition of $R_i$ and so $2\cdot d(y,f^ix) \le \rho_1 e^{-(n-i)c_0/16-L}$. Moreover, $d_{\delta_0}(f^ix,\operatorname{Sing}_\Lambda(G)) \gt e^{-(n-i)c_0/16}$ with $\delta_0 \lt \rho_1\le1$. Hence $2\cdot d(y,f^ix) \le d_{\delta_0}(f^ix,\operatorname{Sing}_\Lambda(G))$ and thus $2\cdot d(y,\operatorname{Sing}_\Lambda(G)) \gt d_{\delta_0}(f^ix,\operatorname{Sing}_\Lambda(G))$. At this point, we divide the argument into two cases.

Away from equilibria: If $d\big(f^ix,\operatorname{Sing}_\Lambda(G)\big)\ge\delta_0\ge\rho_1$, then $\Sigma_{f^ix}^{n-i}$ is away from $\operatorname{Sing}(G)$ and the Poincaré time from $\operatorname{dom}(\Sigma_{f^ix}^{n-i})$ to $\Sigma_{f^{i+1}x}^{n-i-1}$ is between $1-\xi$ and $1+\xi$ for some uniform small $\xi \gt 0$ depending on $\rho_1$.

This ensures that $R_iy=(\phi_s\circ f)y$ with $s=s(y)$ such that $|s-1| \lt \xi$ and so for $y\in\operatorname{dom}(R_i)$ and $v\in T_{R_iy}D_{i+1}^\perp$

(36)\begin{align} \|DR_i(R_iy)^{-1}v\| &= \|{\mathcal O}_y\cdot [D(\phi_s\circ f)(R_iy)]^{-1}v\| \cr &= \|{\mathcal O}_y\cdot Df^{-1}(\phi_{-s}R_iy)\cdot D(\phi_s)^{-1}(R_iy)v\| \cr &= \|{\mathcal O}_y\cdot Df^{-1}(fy)\cdot D(\phi_s)^{-1}(R_iy)v\|. \cr \end{align}

The time $s=s(y)$ can be seen as the Poincaré first visit time from the cross-section $S=f(\Sigma_{f^ix}^{n-i})$ to $\Sigma_{f^{i+1}x}^{n-i-1}$, and so $D(\phi_s)^{-1}(R_iy)v\in T_{fy}(S\cap f^{i+1}(D))\subset C^{cu}_a(fy)\cap C^\perp_a(fy)$ by the proximity between $R_iy, fy$ and $f^{i+1}x$. Then the statement of the lemma follows from (34).

Close to equilibria: Otherwise, $d\big(f^ix,\operatorname{Sing}_\Lambda(G)\big) \lt \delta_0$ and $\Sigma_{f^ix}^{n-i}$ is close to an equilibrium $\sigma\in\operatorname{Sing}_\Lambda(G)$.

We show that we can repeat the above argument by obtaining a flow box from $\operatorname{dom}(R_i)$ to $\Sigma_{f^{i+1}x}^{n-i-1}$ with flight time bounded from above.

Reducing $\delta_0$ if necessary, we may assume, without loss of generality, that the flow on $B(\sigma,2\delta_0)$ is topologically conjugated to the flow of $\dot X=DG(\sigma)\cdot X$, because $\sigma$ is hyperbolic. That is, there exists a bi-Hölder homeomorphism $h:B(\sigma,2\delta_0)\to {\mathbb R}^m$ so that $(h\circ \phi_t)(z)=e^{t\cdot DG(\sigma)}h(z)$ for $t \gt 0$ such that $\phi_{[0,t]}z\subset B(\sigma,2\delta_0)$; see e.g. [Reference Barreira and Valls22]. We arrange so that ${\mathbb R}^m={\mathbb R}^{u}\times{\mathbb R}^{s}$ is the decomposition into stable and unstable subspaces of $DG(\sigma)$, which decomposes in block form as $\operatorname{diag}\{A_u,A_s\}$. We identify $f^ix$ with $(v,w)\in{\mathbb R}^{u}\times{\mathbb R}^{s}$ and then $f^{i+1}x$ becomes $(v_1,w_1)=(e^{A_u}v,e^{A_s}w)$.

For a point in $\Sigma_{f^ix}^{n-i}$ to arrive at $\Sigma_{f^{i+1}x}^{n-i-1}$, the flow time is close to the time $\tau$ it takes a point $(v',w)\in W_0={\mathbb R}^u\times\{w\}$, with $\|v\|-\|v'\|$ small but positive, to arrive at the segment $W_1=\{e^{A_u}v\}\times{\mathbb R}^s$; see Figure 2. For each $y\in\Sigma_{f^ix}^{n-i}$ we have $h(y)=(v',w)$ and for some $\kappa \gt 0$ and $\beta\in(0,1)$ we get $ \|h(y)-h(f^ix)\| \le \kappa d(y,f^ix)^\beta \le \frac{\kappa}{2^\beta} d(f^ix,\sigma)^\beta$ and $ d(f^ix,\sigma) \le \kappa d(h(f^ix),h(\sigma))^\beta = \kappa \|h(f^ix)\|^\beta$, then

\begin{align*} \|h(y)\| &\ge \|h(f^ix)\|-\frac{\kappa}{2^\beta} d(f^ix,\sigma)^\beta = \|h(f^ix)\| \left( 1-\frac{\kappa}{2^\beta}\frac{d(f^ix,\sigma)^\beta}{\|h(f^ix)\|} \right) \\ &\ge \|h(f^ix)\| \left( 1-\frac{\kappa^{1+\beta}}{2^\beta} d(f^ix,\sigma)^{\beta+\beta^{-1}} \right) \\ &\ge \|h(f^ix)\| \left( 1-\frac{\kappa^{1+\beta}}{2^\beta} \delta_0^{\beta+\beta^{-1}} \right) \ge \frac12 \|h(f^ix)\|, \end{align*}

Figure 2.

The estimation of the flight time in the linearized setting.

if $\delta_0 \gt 0$ is small enough, depending only on $G$. Using the previous bound, we estimate $ \|e^{\tau A_u} v'\| =\|v_1\| \ge e^{\tau\|A_u^{-1}\|^{-1}}\|v'\|$ since $\|v\|\le 2\|v'\|$ and $e^{\tau A}\cdot h(y)=(v_1,w')$ for some $w'$, and so we obtain

\begin{align*} \tau & \le \|A_u^{-1}\|\log\frac{\|v_1\|}{\|v'\|} = \|A_u^{-1}\| \log\left(\frac{\|v_1\|}{\|v\|}\cdot\frac{\|v\|}{\|v'\|}\right) \le \|A_u^{-1}\| \log\left(2\frac{\|v_1\|}{\|v\|}\right)\\ & = \|A_u^{-1}\| \log(2e^{\|A_u\|}) \end{align*}

because $v_1=e^{A_u}v'$ satisfies $\|v_1\|=\|e^{A_u}v'\|\le e^{\|A_u\|}\|v'\|$. This shows that $\tau$ is bounded depending only on $DG(\sigma)$.

Going back to the original coordinates, the cross-sections $h^{-1}(W_i), i=0,1$ touch $\Sigma_{f^ix}^{n-1}$ and $\Sigma_{f^{i+1}x}^{n-i-1}$ at $f^ix, f^{i+1}x$, respectively; see Figure 2. The vector field in between these sections has norm uniformly bounded away from zero and close to $G({f^ix})$, by the estimate (35). Hence, the flight time is also bounded above depending only on $G$ in a neighbourhood of $\sigma$.

We have recovered a flow box with bounded flight time from $\operatorname{dom}(R_i)$ to $\Sigma_{f^{i+1}x}^{n-i-1}$. We can thus finish repeating the argument as before, using (36) and (34).

Proof. We follow the proof of [Reference Alves, Bonatti and Viana4, Lemma 2.7]. Let $\gamma_0$ be a curve of minimal length in $D_n^\perp(x)$ from $f^nx$ to $g_ny$ and let $ \gamma_k=(g_k)^{-1}(\gamma_0), k=1,\dots,n$. Arguing by induction, let us assume that for some $k=1,\dots,n$ we have the following bound for the length: $\ell(\gamma_k)\le e^{-kc_0/8}\ell(\gamma_0)$. The choice of $\rho_1$ in (34) together with the definition of hyperbolic times and Lemma 3.6 ensure that

\begin{align*} \big\|D(R_{n-1}\circ\dots\circ R_{n-k-1})^{-1}\gamma_0'(z)\big\| &\le e^{(k+1)c_0/8}\|\gamma_0'(z)\| \prod_{j=n-k}^n \|(P^1\mid_{N_{f^jx}^u})^{-1}\| \\ &\le e^{-(k+1)c_0/8}\|\gamma_0'(z)\| \end{align*}

where $\gamma_0'(z)$ denotes the tangent vector to $\gamma_0$ at $z$. Therefore