Proof. See [Reference Varzaneh and Riedel34, Thm. 1.21(v)].

Proof. It goes along the lines of the proof of Theorem 3.5 in [Reference Mierczyński and Shen22].

Proof. The inequality $\widetilde{\lambda} _1\le \lambda_{\mathrm{top}}$ is straightforward. In order to prove the other one, observe that

\begin{align*} {\lVert U_{\omega}(t)\rVert} & = \sup\{\, \| U_{\omega}(t) (u^{+} - u^{-}) \|: {\lVert u^{+} - u^{-}\rVert} \le 1 \,\} \\ & \le 2\, K \sup\{\, \| U_{\omega}(t) \, u \| : u \in X^{+}, {\lVert u\rVert} \le 1 \,\}, \end{align*}

where $K \ge 1$ is a constant in lemma 3.3 and apply theorem 2.5(vi).

Proof. It is sufficient to show that $w({\omega}) \in E({\omega})$ for $\mathbb{P}$-a.e. ${\omega} \in \Omega$. By theorem 3.4, $\mathbb{P}$-a.e. on Ω, for each $n \in {\mathbb{N}}$ we can find $u_n \in E(\theta_{-n}{\omega}) \cap X^{+}$, with ${\lVert u_n\rVert} = 1$. The invariance of E gives that $U_{\theta_{-n}{\omega}}(n) \, u_n/{\lVert U_{\theta_{-n}{\omega}}(n) \, u_n\rVert}$ belongs to $E({\omega})$. By theorem 2.5(vii), $U_{\theta_{-n}{\omega}}(n) \, u_n/{\lVert U_{\theta_{-n}{\omega}}(n) \, u_n\rVert}$ converges, as $n \to \infty$, to $w({\omega})$.

Proof. Let $\widetilde{\Omega}_1$ be the invariant subset of Ω of theorem 2.5 and without loss of generality, assume that the set $\Omega'$ of theorem 3.1 is valid for theorem 3.4, that is, $E({\omega}) \cap X^{+} \varsupsetneq \{0\}$ for all ${\omega} \in \Omega'$.

Suppose to the contrary that $l \ge 2$. Fix ${\omega} \in \Omega' \cap \widetilde{\Omega}_1$ such that $E_1({\omega}) \subset E({\omega})$, and let $\{u_1, \ldots, u_{l - 1}, w({\omega}) \}$ be a basis of unit vectors for $E({\omega})$. We will look at the action of $U_{\omega}(n)$ on l-dimensional volume. We have, by (3.1),

(3.2)\begin{equation} \begin{array}{l} \text{vol}(U_{\omega}(n) \, u_1, \ldots, U_{\omega}(n) \, u_{l - 1}, U_{\omega}(n) \, w({\omega}) )\\ = {\lVert U_{\omega}(n) \, w({\omega})\rVert} \, \prod\limits_{i = 1}^{l - 1} \text{dist}(U_{\omega}(n) \, u_i, \text{span}{\{U_{\omega}(n) \, u_{i + 1}, \ldots, U_{\omega}(n) \, u_{l - 1}, U_{\omega}(n) \, w({\omega})\}}) \\ \qquad\qquad \le {\lVert U_{\omega}(n) \, w({\omega})\rVert} \, \prod\limits_{i = 1}^{l - 1} \text{dist}(U_{\omega}(n) \, u_i, E_1(\theta_{n}{\omega})) \end{array} \end{equation}

for all $n \in {\mathbb{N}}$.

By lemma 3.3, there is $K \ge 1$ such that for each u_i, $1 \le i \le l - 1$, one can find $u_{i}^{+}, u_{i}^{-} \in X^{+}$ with $u_{i} = u_{i}^{+} - u_{i}^{-}$, ${\lVert u_{i}^{+}\rVert} \le K$, ${\lVert u_{i}^{-}\rVert} \le K$.

It follows from theorem 2.5(vii) that for any $\rho \in (0, \sigma)$, one can find N ₀ such that for $n \ge N_0$ and $i \in \{1, \ldots, l - 1 \}$ there holds

\begin{equation*}\begin{array}{l} \Bigl\lVert U_{\omega}(n) \, u_{i}^{+} - {\lVert U_{\omega}(n) \, u_{i}^{+}\rVert} \, w(\theta_{n}{\omega}) \Bigr\rVert \le \exp(- n \rho) {\lVert U_{\omega}(n) \, u_{i}^{+}\rVert},\\ \Bigl\lVert U_{\omega}(n) \, u_{i}^{-} - {\lVert U_{\omega}(n) \, u_{i}^{-}\rVert} \, w(\theta_{n}{\omega}) \Bigr\rVert \le \exp(- n \rho) {\lVert U_{\omega}(n) \, u_{i}^{-}\rVert}, \end{array}\end{equation*}

consequently,

(3.3)\begin{equation} \begin{array}{l} \Bigl\lVert U_{\omega}(n) \, u_{i} - \bigl( {\lVert U_{\omega}(n) \, u_{i}^{+}\rVert} - {\lVert U_{\omega}(n) \, u_{i}^{-}\rVert} \bigr) \, w(\theta_{n}{\omega}) \Bigr\rVert \\ = \Bigl\lVert \bigl(U_{\omega}(n) \, u_{i}^{+} - {\lVert U_{\omega}(n) \, u_{i}^{+}\rVert} \, w(\theta_{n}{\omega}) \bigr) - \bigl( U_{\omega}(n) \, u_{i}^{-} - {\lVert U_{\omega}(n) \, u_{i}^{-}\rVert} \, w(\theta_{n}{\omega}) \bigr) \Bigr\rVert \\ \qquad \qquad \le \exp(- n \rho) \bigl( {\lVert U_{\omega}(n) \, u_{i}^{+}\rVert} + {\lVert U_{\omega}(n) \, u_{i}^{-}\rVert} \bigr). \end{array} \end{equation}

Pick some µ satisfying

\begin{equation*} \lambda_{\mathrm{top}} \lt \mu \lt \lambda_{\mathrm{top}} + \frac{l - 1}{l} \,\rho. \end{equation*}

There is N ₁ such that for $n \ge N_1$ and $i \in \{1, \ldots, l - 1 \}$, there holds

(3.4)\begin{equation} {\lVert U_{\omega}(n) \, u_{i}^{+}\rVert} \le \exp(n \mu) {\lVert u_{i}^{+}\rVert}, \quad {\lVert U_{\omega}(n) \, u_{i}^{-}\rVert} \le \exp(n \mu) {\lVert u_{i}^{-}\rVert}. \end{equation}

Gathering (3.3) and (3.4) we obtain, in view of lemma 3.3, the following:

\begin{equation*} \text{dist}(U_{\omega}(n) \, u_i, E_1(\theta_{n}{\omega})) \le 2 K \exp(n (\mu - \rho)), \quad 1 \le i \le l - 1, \end{equation*}

for $n \ge \max\{N_0, N_1\}$, which gives, via (3.2),

\begin{equation*}\begin{array}{l} \text{vol}(U_{\omega}(n) \, u_1, \ldots, U_{\omega}(n) \, u_{l - 1}, U_{\omega}(n) \, w({\omega})) \\ \qquad \qquad \le 2^{l-1} K^{l-1} \exp((l-1) n (\mu - \rho)) \, \exp(n \mu), \end{array}\end{equation*}

consequently,

\begin{equation*} \limsup\limits_{n \to \infty} \frac{\ln{\text{vol}(U_{\omega}(n) \, u_1, \ldots, U_{\omega}(n) \, u_{l - 1}, U_{\omega}(n) \, w({\omega}) )}}{n} \le l \mu - (l - 1) \rho, \end{equation*}

which contradicts proposition 3.2 and finishes the proof.

Proof. We claim that $F({\omega})$ of theorem 3.1 can serve as $F_1({\omega})$ in the definition of generalized exponential separation. Indeed, parts (ii) and (iii) in definition 2.6 are direct consequences of theorems 3.1 and 2.5 combined with theorem 3.7. In order to prove (i), let $u \in X^{+}$ be such that $U_{\omega}(1) \, u = 0$. Then, $\lim_{t \to \infty} \ln{{\lVert U_{\omega}(t) \, u\rVert}}/t = - \infty$, and hence, by the characterization given in theorem 3.1 (iii)–(iv), $u \in F({\omega}) = F_1({\omega})$. Finally, suppose to the contrary that there is a $u \in F({\omega}) \cap X^{+}$ such that $U_{\omega}(1) \, u \ne 0$. Then, by remark 2.2, $U_{\omega}(t) \, u \in X^{+} \setminus \{0\}$ for all $t \ge 1$. Hence, by theorem 2.5 (ii) and lemma 3.5, $\lim_{t \to \infty} \ln{{\lVert U_{\omega}(t) \, u\rVert}}/t = \lambda_{\mathrm{top}}$, which contradicts theorem 3.1 (iv), shows that definition 2.6 (i) holds and finishes the proof.

Proof. Let $t^*$ be the point of $[0,1]$ and j ₁ be the index in which the maximum is attained, that is, $c_1=z_{j_1}(t^*)\geq z_i(t)$ for each $i\in \{1,2,\ldots N\}$ and $t\in[0,1]$. Consider the path and the constant M given in (S4)(i), and denote

(4.12)\begin{equation} K_i({\omega})= \exp\bigg(-\int_0^{N+H}\!\!\!|a_{j_ij_i}(\theta_s{\omega})|\,ds \bigg),\, \text{for}\, i=1,2,\ldots,N \end{equation}

where $H=(N-1)M+1$. From $z'_{j_1}(t)\geq a_{j_1j_1} (\theta_{t}{\omega}) \,z_{j_1}(t)$, and using a comparison result for Carathéodory differential equations [Reference Walter35, Theorem 2], we deduce that

\begin{equation*}z_{j_1}(t)\geq \exp\bigg(\int_{t^*}^{t}\!\!\!a_{j_ij_i}(\theta_s{\omega})\,ds\bigg)z_{j_1}(t^*)\geq c_1K_1({\omega}) \,\text{for}\, t\in[1,N+H].\end{equation*}

Analogously, if $t\in[2+M,N+H]$, from

\begin{align*} z_{j_2}'(t) &\geq a_{j_2j_2}(\theta_t{\omega})\,z_{j_2}(t)+ a_{j_2j_1}(\theta_t{\omega}) z_{j_1}(t)+b_{j_2j_1}(\theta_t{\omega}) z_{j_1}(t-1) \\ & \geq a_{j_2j_2}(\theta_t{\omega})\,z_{j_2}(t)+ \left[a_{j_2j_1}(\theta_t{\omega}) +b_{j_2j_1}(\theta_t{\omega}) \right] c_1 K_1({\omega})\, \end{align*}

we obtain

\begin{align*} z_{j_2}(t)&\geq c_1K_1({\omega})K_2({\omega})\int_2^{t} \left[a_{j_2j_1} +b_{j_2j_1}\right](\theta_{s}{\omega}) \,ds\\ &\geq c_1 K_1({\omega})K_2({\omega})\int_0^{M}\left[a_{j_2j_1} +b_{j_2j_1}\right](\theta_{s+2}{\omega}) \,ds \end{align*}

and (S4)(i) provides

\begin{equation*} z_{j_2}(t)\geq c_1 K_1({\omega}) K_2({\omega})\,\delta(\theta_2{\omega})\,\text{for}\, t\in[2+M,N+H].\end{equation*}

Similarly,

\begin{equation*} z_{j_3}(t)\geq c_1 K_1({\omega}) K_2({\omega})K_3({\omega})\,\delta(\theta_2{\omega})\,\delta(\theta_{3+M}{\omega})\,\text{for}\, t\in[3+2\,M,N+H],\end{equation*}

and finally, in a recursive way we prove that for $k=2,3,\ldots N$

(4.13)\begin{equation} z_{j_k}(t)\geq c_1 \prod_{j=1}^{k} K_j({\omega})\prod_{j=2}^{k}\delta(\theta_{j+(j-2)M} {\omega})\ \,\text{for}\, t\in[k+(k-1)M,N+H], \end{equation}

which finishes the proof.

Proof. As in [Reference Mierczyński, Novo and Obaya18, Proposition 5.7], concerning the first part of (A1) with T note that

\begin{equation*}\sup\limits_{0 \le s \le T} {ln^{+}{\|U^{(L)}_{\omega}(s)\|}}\leq \sum_{k=0}^{T-1}\sup\limits_{k\le s \le k+1} {ln^{+}{\|U^{(L)}_{\omega}(s)\|}}\end{equation*}

and from cocycle property (2.1)

\begin{align*} \sup\limits_{k\le s \le k+1} ln^{+}{\|U^{(L)}_{\omega}(s)\|}&=\sup\limits_{0 \le s \le 1} {ln^{+}{\|U^{(L)}_{\omega}(k+s)\|}} \\ &\leq \sup\limits_{0 \le s \le 1} \bigl(ln^{+}\|U^{(L)}_{\theta_s{\omega}}(k)\|+ ln^{+} \|U^{(L)}_{\omega}(s)\|\bigr). \end{align*}

Therefore,

\begin{equation*}\sup\limits_{0 \le s \le T} {ln^{+}{\|U^{(L)}_{\omega}(s)\|}}\leq T\,\sup\limits_{0 \le s \le 1} {ln^{+}{\|U^{(L)}_{\omega}(s)\|}}+ \sum_{k=1}^{T-1}\sup\limits_{0 \le s \le 1} {ln^{+}{\|U^{(L)}_{\theta_s{\omega}}(k)\|}}\end{equation*}

and we have to check that both terms belong to $L_1{(\Omega,\mathfrak{F},\mathbb{P})}$. As shown in [Reference Mierczyński, Novo and Obaya19, Proposition 4.6], ${\lVert U^{(L)}_{\omega}(t)\rVert}\leq 3\,c({\omega})\,(1+d({\omega}))$ for each $t\in[0,1]$ and ${\omega}\in\Omega$, where $c({\omega})$ and $d({\omega})$ are defined in (4.3) and $c({\omega})\geq 1$. Thus, from inequality (4.4) and [Reference Mierczyński, Novo and Obaya18, Lemma 5.6], we deduce that

\begin{equation*}\sup\limits_{0 \le s \le 1}{ln^{+}{\|U^{(L)}_{\omega}(s)\|}}\leq \ln 6 + ln^{+} c({\omega})+ ln^{+} d({\omega}) \leq \ln 6 + \int_{0}^{1} a(\theta_{s}{\omega}) \, ds + ln^{+} d({\omega}),\end{equation*}

which belongs to $L_1{(\Omega,\mathfrak{F},\mathbb{P})}$ because of the definition of d, (S1), (S2), Fubini’s theorem and the invariance of $\mathbb{P}$. We omit the proof of the second term, as well as that of the second part of (A1) because they are analogous. In addition, it is also immediate to check that (A2) follows from (S3).

We will finish by verifying that (A3) holds for time T. First we claim that for each ${\omega}\in\Omega$ and $u \in C^+$ with $U_{\omega}^{(C)}(T)\, u\neq 0$

(4.14)\begin{equation} \widetilde\beta({\omega},u)\,\widetilde{\mathbf{e}}\leq U_{\omega}^{(C)}(T)\,u\leq \varkappa({\omega})\,\widetilde \beta({\omega},u)\,\widetilde{\mathbf{e}} \end{equation}

where $\widetilde{\mathbf{e}}\in C^+$ is the constant unit function $\widetilde{\mathbf{e}}(s)=1$ for each $s\in[-1,0]$,

(4.15)\begin{align} \widetilde\beta({\omega},u)&={\lVert U_{\omega}^{(C)}(1)\,u\rVert_{C}}\,k_\delta({\omega}),\nonumber\\ \varkappa({\omega})&=k_\delta^{-1}({\omega})\,3^{T-1} \prod\limits_{j=0}^{T-2}c(\theta_{j+1}{\omega})(1+d(\theta_{j+1}{\omega})), \end{align}

(4.16)\begin{equation} k_\delta({\omega})=\prod_{j=1}^{N} K_j({\omega})\prod_{j=2}^{N}\delta(\theta_{j+(j-2)M} {\omega}), \end{equation}

and $K_j({\omega})$ are defined in (4.12) ( $N+H=T$). Notice that ${\lVert U_{\omega}^{(C)}(1)\,u\rVert_{C}}$ is strictly positive because of $U_{\omega}^{(C)}(T)\, u\neq 0$ and the cocycle property (2.1), so that $\widetilde\beta({\omega},u)$ is strictly positive. Moreover, since $(U_{\omega}^{(C)}(T)\,u)(s)= z(T+s,{\omega},u)$ for each $s\in[-1,0]$, the lower inequality of (4.14) follows from (4.13) with k = N and t = T and (4.16).

Concerning the upper inequality, we first claim that for each $n\geq 1$

(4.17)\begin{equation} {\lVert U_{\omega}^{(C)}(t)\,u\rVert_{C}}\leq 3^n \prod_{j=0}^{n-1}c(\theta_{j}{\omega})(1+d(\theta_{j}{\omega})) \,{\lVert u\rVert_{C}}\,\text{whenever}\, t\in[n-1,n].\end{equation}

From [Reference Mierczyński, Novo and Obaya19, Proposition 4.4], we deduce that

\begin{equation*} {\lVert U^{(C)}_{\omega}(t)\,u\rVert_{C}}\leq 3\,c({\omega})\,(1+d({\omega}))\,{\lVert u\rVert_{C}} \quad\text{for each}\, t\in[0,1] \text{and}\, {\omega}\in\Omega, \end{equation*}

which together with the cocycle property (2.1) yields

\begin{align*} {\lVert U_{\omega}^{(C)}(t)\,u\rVert_{C}}&={\lVert U_{\theta_1{\omega}}^{(C)}(t-1)\big(U_{\omega}^{(C)}(1)\,u\big)\rVert_{C}}\\ &\leq 3\,c(\theta_1{\omega})(1+d(\theta_1{\omega})\,3\,c({\omega})(1+d({\omega}))\,{\lVert u\rVert_{C}} \end{align*}

for each $t\in[1,2]$ and ${\omega}\in\Omega$. Thus (4.17) is easily checked in a recursive way.

Finally, from $U_{\omega}^{(C)}(T)\,u\leq {\lVert U_{\omega}^{(C)}(T)\,u\rVert_{C}}\, \widetilde{\mathbf{e}} $ and (4.17) for $t=T-1$, we obtain

\begin{align*} U_{\omega}^{(C)}(T)\,u&\leq{\lVert U_{\theta_1{\omega}}^{(C)}(T-1) (U_{\omega}^{(C)}(1)\,u)\rVert_{C}} \,\widetilde{\mathbf{e}} \\ & \leq \,{\lVert U_{\omega}^{(C)}(1)\,u)\rVert_{C}}\,3^{T-1} \prod\limits_{j=0}^{T-2}c(\theta_{j+1}{\omega})(1+d(\theta_{j+1}{\omega}))\,\widetilde{\mathbf{e}}\\ &=\varkappa({\omega})\,\widetilde\beta({\omega},u)\,\widetilde{\mathbf{e}}, \end{align*}

as claimed.

Next we will prove that for any $u\in L^+$ such that $U_{\omega}^{(L)}(T)\,u\not=0$

(4.18)\begin{equation} \beta({\omega},u)\,\mathbf e\leq U_{\omega}^{(L)}(T)\,u\leq \varkappa({\omega})\, \beta({\omega},u)\,\mathbf e, \end{equation}

where $\mathbf e=(1/(2\sqrt{N}))\,(\widetilde{\mathbf{e}}(0),\widetilde{\mathbf{e}})\in L^+$ is a unitary vector of L, i.e ${\lVert\mathbf e\rVert_{L}}=1$ and

(4.19)\begin{equation} \beta({\omega},u)=2\sqrt{N} \,{\lVert U_{\omega}^{(L,C)}(1)\,u\rVert_{C}}\, k_\delta({\omega}) \gt 0. \end{equation}

As shown in [Reference Mierczyński, Novo and Obaya19], the following relations hold

\begin{equation*} U^{(L)}_{\omega}(t) = J \circ U^{(L, C)}_{\omega}(t), \quad U^{(L, C)}_{\omega}(t)=U_{\theta_1{\omega}}^{(C)}(t-1)\circ U_{\omega}^{(L,C)}(1) \end{equation*}

for any $t \ge 1$ and any ${\omega} \in \Omega$, where J is the linear map defined in (4.2). Hence

\begin{equation*} U_{\omega}^{(L)}(T)\,u=(z(T,{\omega},u),z_T({\omega},u))=J\big( U_{\theta_1{\omega}}^{(C)}(T-1)(U_{\omega}^{(L,C)}(1)\,u)\big) \end{equation*}

and notice that $U_{\omega}^{(L,C)}(1)\,u=z_1({\omega},u)\in C$. Therefore, as in proposition 4.2,

\begin{equation*} z(T+s,{\omega},u)\geq {\lVert U_{\omega}^{(L,C)}(1)\,u\rVert_{C}} \,k_\delta({\omega})\, \widetilde{\mathbf{e}} \end{equation*}

for each $s\in[-1,0]$ and the lower inequality of (4.18) holds. Concerning the upper inequality, as above

\begin{equation*}z_T({\omega},u)\leq \,{\lVert U_{\omega}^{(L,C)}(1)\,u)\rVert_{C}}\,3^{T-1} \prod_{j=0}^{T-2}c(\theta_{j+1}{\omega})(1+d(\theta_{j+1}{\omega}))\,\widetilde{\mathbf{e}}\end{equation*}

from which the upper inequality of (4.18) can be easily checked.

In order to finish the proof we have to check that the $(\mathfrak{F},\mathfrak{B}(\mathbb{R}))$-measurable function $\varkappa$ defined by (4.15) satisfies $ln^{+}\ln\varkappa\in L_1{(\Omega,\mathfrak{F},\mathbb{P})}$. From (4.12) and the definition of $a({\omega})={\lVert A({\omega})\rVert}$ we deduce that there is a constant c ₀ such that

(4.20)\begin{equation} \Bigg(\prod_{j=1}^{N} K_j({\omega}))\Bigg)^{\!\!-1}\!\!\!= \exp\int_0^T \sum_{j=1}^N|a_{jj}(\theta_s{\omega})|\,ds\leq \exp\int_0^T c_0\,a(\theta_s{\omega}) \,ds. \end{equation}

Moreover, from (4.20), (4.4) and $\ln(1+x)\leq x$ if $x\geq 0$

\begin{equation*}ln^{+} \ln c(\theta_{j+1}{\omega})\leq \int_0^1 a(\theta_{j+1+s}{\omega})\,ds\, ,\quad ln^{+} \ln(1+d(\theta_{j+1}{\omega}))\leq ln^{+} d(\theta_{j+1}{\omega}),\end{equation*}

\begin{equation*} 0\leq ln^{+} \ln\Bigg(\prod_{j=1}^{N} K_j({\omega}))\Bigg)^{\!\!-1}\!\!\!\leq c_0 \int_0^T a(\theta_s{\omega}) \,ds, \end{equation*}

and we deduce from [Reference Mierczyński, Novo and Obaya18, Lemma 5.6] that there is an integer number n ₀ such that

\begin{align*} ln^{+} \ln\varkappa ({\omega})\leq & \, c_0 \int_0^T a(\theta_s{\omega})\,ds+ \sum_{j=0}^{T-2} \left[\int_0^1 a(\theta_{j+1+s} {\omega})\,ds+ \ln ^+d(\theta_{j+1}{\omega})\,\right] \\ &\quad + \sum_{j=2}^Nln^{+} \ln \left(1/\delta\right)(\theta_{j+(j-2)M}{\omega})+ ln^{+} \ln(3^{T-1})\,+\ln (n_0). \end{align*}

As before, from the definition of d, (S1), (S2), (S4)(ii), Fubini’s theorem and the invariance of $\mathbb{P}$ we conclude that $ln^{+} \ln \varkappa\in L_1{(\Omega,\mathfrak{F},\mathbb{P})}$ and (A3) holds for T, as stated.

Proof. As $L=\mathbb{R}^N\times L_p([-1,0],\mathbb{R}^N)$ is an ordered separable Banach space with separable dual $L^*=\mathbb{R}^N\times L_q([-1,0],\mathbb{R}^N)$ and positive cone $L^+$ normal and reproducing, the claim follows from theorems 2.9, 4.3 and 4.4, once we check that $\widetilde\lambda_1^{(L)} \gt -\infty$.

Let $\mathbf e$ be the unitary vector of L defined for the focusing inequality (4.18). From proposition 4.2, we deduce that $U_{\omega}^{(L)}(T)\,\mathbf e\neq 0$ and hence $U_{\omega}^{(L)}(T)\,\mathbf e \geq \beta({\omega},\mathbf e)\,\mathbf e$ for each ${\omega}\in\Omega$. Thus, from the cocycle property (2.1), (A2), and the monotonicity of the norm, it is easy to check that

\begin{equation*} {\lVert U_{\omega}^{(L)}(n\,T)\,\mathbf e\rVert_{L}}\geq \prod_{k=0}^{n-1} \beta(\theta_{kT}{\omega},\mathbf e) \quad \text{for each}\, n\in\mathbb{N}, \end{equation*}

and therefore

\begin{equation*} \frac{\ln {\lVert U_{\omega}^{(L)}(n\,T)\,\mathbf e\rVert_{L}}}{nT}\geq \frac{1}{nT}\sum_{k=0}^{n-1} \ln \beta(\theta_{kT}{\omega},\mathbf e). \end{equation*}

Finally, note that the function $-\ln \beta(\cdot,\mathbf e)$ for β defined in (4.19) satisfies

\begin{equation*} 0\leq -\ln \beta({\omega},\mathbf e)\leq -\sum_{j=1}^N \ln K_j({\omega})+\sum_{j=2}^N\ln \left(1/\delta)(\theta_{j+(j-2)M}{\omega}\right)+ c_1 \end{equation*}

for some constant c ₁, and thus belongs to $L_1{(\Omega,\mathfrak{F},\mathbb{P})}$ from (4.20) and (S5). Then, an application of Birkhoff ergodic theorem gives that for $\mathbb{P}$-a.e. ${\omega}\in\Omega$, there holds

\begin{equation*} \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1} \ln \beta(\theta_{kT}{\omega},\mathbf e)=\int_\Omega \ln\beta(\theta_T{\omega}',\mathbf e)\,d\mathbb{P}({\omega}') \gt -\infty, \end{equation*}

hence

\begin{equation*} \limsup_{t\to\infty} \frac{\ln {\lVert U_{\omega}^{(L)}(t)\,\mathbf e\rVert_{L}}}{t} \gt -\infty, \end{equation*}

and (2.2) finishes the proof.

Proof. From [Reference Aliprantis and Border2, Lemma 4.51, pp. 153], it is enough to check that it is a Carathéodory function, i.e. for each fixed ${\omega}\in\Omega$, the map L → C, $u\mapsto U_{\omega}^{(L,C)}(1)\,u=z_1({\omega},u)$ is continuous, which is well known, and for each fixed $u\in L$, the map

(4.21)\begin{equation} [\Omega\ni {\omega} \mapsto U_{\omega}^{(L,C)}(1)\,u=z_1({\omega},u)\in C] \text{is} (\mathfrak{F}, \mathfrak{B}(C))\text{-measurable}. \end{equation}

In order to prove this, denoting by $u=(c,v)\in \mathbb{R}^N\times L_p([-1,0],\mathbb{R}^N)$, we can find a sequence of continuous functions $v_n\in C$ such that, $v_n(0)$ converges to c in $\mathbb{R}^N$ and v_n converges to v in $L_p([-1,0],\mathbb{R}^N)$ as $n \uparrow\infty$. Therefore, the measurability of the maps $[\Omega\ni {\omega} \mapsto z_1({\omega},v_n(0),v_n)=z_1({\omega},v_n)\in C]$ (see [Reference Mierczyński, Novo and Obaya19, Lemma 4.11]) for each $n\in\mathbb{N}$, the convergence of them to the map (4.21) as $n \uparrow\infty$ and [Reference Aliprantis and Border2, Corollary 4.29] show the measurability and finishes the proof.

Proof. Theorem 4.4 states the existence of a family $\{E_1^{(L)}({\omega})\}_{{\omega} \in \widetilde\Omega_1}$ of generalized principal Floquet subspaces for $\Phi^{(L)}$. For ${\omega} \in \widetilde{\Omega}_1$, put $E_1^{(L)}({\omega}) := \text{span}\{w^{(L)}({\omega})\}$ with ${\lVert w^{(L)}({\omega})\rVert_{L}} = 1$. First notice that from (4.9) the function

\begin{equation*} U_{\theta_{-1}w}^{(L,C)}(1)w^{(L)}(\theta_{-1}{\omega}) =(z_1(\theta_{-1}{\omega},w^{(L)}(\theta_{-1}{\omega}))\in \ C, \end{equation*}

from (4.11), we deduce that

\begin{equation*} U_{\theta_{-1}{\omega}}^{(L)}(1)w^{(L)}(\theta_{-1}{\omega})=(z_1(\theta_{-1}{\omega},w^{(L)}(\theta_{-1}{\omega}))(0), z_1(\theta_{-1}{\omega},w^{(L)}(\theta_{-1}{\omega}))\in \mathbb{R}^N\times C, \end{equation*}

which is proportional to $w^{(L)}({\omega})$ because $U_{\theta_{-1}w}^{(L)}(1)E_1^{(L)}(\theta_{-1} w)=E_1^{(L)}({\omega})$, and hence, can be denoted by $w^{(L)}({\omega})=(w({\omega})(0),w({\omega}))\in \mathbb{R}^N\times C$. In particular, $w({\omega})$ is proportional to $U_{\theta_{-1}{\omega}}^{(L,C)}(1)w^{(L)}(\theta_{-1}{\omega})\neq 0$.

Next, we define $w^{(C)}({\omega}) := U_{\theta_{-1}{\omega}}^{(L,C)}(1) \, w^{(L)}(\theta_{-1}{\omega}) / {\lVert U_{\theta_{-1}{\omega}}^{(L,C)}(1) \, w^{(L)}(\theta_{-1}{\omega})\rVert_{C}} $, unitary and also proportional to $w({\omega})$. We consider the one-dimensional subspace of C defined by $E_1^{(C)}({\omega}):=\text{span}\{w^{(C)}({\omega})\}$ and we claim that $\{E_1^{(C)}({\omega})\}_{{\omega} \in \widetilde\Omega_1}$ is a family of generalized principal Floquet subspaces of $\Phi^{(C)}$, i.e. conditions (i)–(iv) of definition 2.4 hold. First, it is easy to deduce from lemma 4.6 that $w^{(C)}\colon\widetilde\Omega_1\to C^+\setminus \{0\}$ is $(\mathfrak{F}, \mathfrak{B}(C))$-measurable and thus, (i) holds. Condition (ii), that is, the invariance of $\{E_1^{(C)}({\omega})\}_{{\omega} \in \widetilde\Omega_1}$, follows from $E_1^{(C)}({\omega}) = \text{span}\{w({\omega})\}$, $w^{(L)}({\omega})=(w({\omega})(0),w({\omega}))$ and the invariance of $\{E_1^{(L)}({\omega})\}_{{\omega} \in \widetilde\Omega_1}$. In order to check (iii), notice that from [Reference Mierczyński, Novo and Obaya19, Proposition 5.2(2)], we deduce that

(4.22)\begin{equation} \widetilde\lambda_1^{(L)}=\lim_{t\to\infty} \frac{\ln {\lVert U_{\omega}^{(L)}(t)\,w^{(L)}({\omega})\rVert_{L}}}{t}=\lim_{t\to\infty} \frac{\ln {\lVert U_{\omega}^{(L,C)}(t)\,w^{(L)}({\omega})\rVert_{C}}}{t}. \end{equation}

However, as stated in (4.11), $U_{\omega}^{(C)}(t)=U_{\omega}^{(L,C)}(t)\circ J$ for $t\geq 1$, then

(4.23)\begin{equation} \widetilde\lambda_1^{(L)}= \lim_{t\to\infty} \frac{\ln {\lVert U_{\omega}^{(C)}(t)\,w({\omega})\rVert_{C}}}{t}=\lim_{t\to\infty} \frac{\ln {\lVert U_{\omega}^{(C)}(t)\,w^{(C)}({\omega})\rVert_{C}}}{t}, \end{equation}

as needed, and in particular the generalized principal Lyapunov exponents of $\Phi^{(L)}$ and $\Phi^{(C)}$ coincide. Finally, for each ${\omega}\in\widetilde\Omega_1$ with $U_{\omega}^{(C)}(1)\,u\neq 0$ we know that $U_{\omega}^{(L)}(1)\,Ju\neq 0$. Then, as in [Reference Mierczyński, Novo and Obaya19, Proposition 5.2(1)], it can be shown that

(4.24)\begin{equation} \limsup_{t\to\infty} \frac{\ln {\lVert U_{\omega}^{(C)}(t)\,u\rVert_{C}}}{t}=\limsup_{t\to\infty} \frac{\ln {\lVert U_{\omega}^{(L)}(t)\,Ju\rVert_{L}}}{t}\leq \widetilde\lambda_1^{(L)}=\widetilde\lambda_1^{(C)}. \end{equation}

Therefore, (iv) holds and $\{E_1^{(C)}({\omega})\}_{{\omega} \in \widetilde\Omega_1}$ is a family of generalized principal Floquet subspaces of $\Phi^{(C)}$, as claimed.

Proof. In order to prove that $\Phi^{(C)}$ admits a generalized exponential separation of type II we need to check that the family $\{E_1^{(C)}({\omega})\}_{{\omega} \in \widetilde\Omega_1}$ of theorem 4.7, defined by $E_1^{(C)}({\omega})=\text{span}\{w({\omega})\}$, satisfies conditions (i)–(iii) of definition 2.6.

First notice that from theorems 4.4 and 2.9, we know, among other things, that there are an invariant set $\widetilde\Omega_1$ of full measure $\mathbb{P}(\widetilde\Omega_1)=1$ and a family of generalized principal Floquet subspaces $\{E_1^{(L)}({\omega})\}_{{\omega} \in \widetilde\Omega_1}$ with $E_1^{(L)}({\omega}) = \text{span}\{w^{(L)}({\omega})\}$ satisfying that the family of projections $\{P^{(L)}({\omega})\}_{{\omega}\in\widetilde\Omega_1}$, associated with the invariant decomposition $L=E_1^{(L)}({\omega})\oplus F_1^{(L)}({\omega})$, is strongly measurable and tempered, i.e.

(4.25)\begin{equation} \lim\limits_{t \to \pm\infty} \frac{\ln{\|P^{(L)}(\theta_{t}{\omega})\|}}{t} = 0 \qquad \mathbb{P}\text{-a.e. on}\, \widetilde\Omega_1, \end{equation}

and $F_1^{(L)}({\omega})\cap L^+=\{u\in L^+ : U_{\omega}^{(L)}(1)\,u=0\}$ for any ${\omega}\in\widetilde\Omega_1$. We also know that ${\lVert w^{(L)}({\omega})\rVert_{L}}=1$ for any ${\omega}\in\widetilde\Omega_1$. As in remark 2.1, the family of projections onto $E_1^{(L)}({\omega})$ will be denoted by $\{\widetilde P^{(L)}({\omega})\}_{{\omega}\in\widetilde\Omega_1}$ (remember that $\widetilde{P}^{(L)}({\omega}) = \mathrm{Id}_L - P^{(L)}({\omega})$). As $\widetilde{P}^{(L)}({\omega})$ is a projection on $\text{span}\{w^{(L)}({\omega})\}$, for each $\widehat u\in L$ we have $ \widetilde P^{(L)}({\omega})\,\widehat u=\lambda({\omega},\widehat u)\, w^{(L)}({\omega})$.

Next, for any $u\in C$, we know that $Ju=(u(0),u)\in L$ so that u can be decomposed as $u= \lambda({\omega},Ju)\,w({\omega})+ \bigl( u-\lambda({\omega},Ju)\,w({\omega}) \bigr)$, with $u-\lambda({\omega},Ju)\,w({\omega})\in C$. For ${\omega} \in \widetilde{\Omega}_1$, put $F_1^{(C)}({\omega}) := \{\, u \in C : J u \in F_1^{(L)}({\omega}) \,\}$. The closedness of $F_1^{(C)}({\omega})$ in C is a consequence of the continuity of J and the closedness of $F_1^{(L)}({\omega})$ in L.

We have thus obtained an invariant decomposition $C = E_1^{(C)}({\omega}) \oplus F_1^{(C)}({\omega})$. We claim that for any ${\omega} \in \widetilde{\Omega}_1$ there holds $F_1^{(C)}({\omega}) \cap C^+ = \{u\in C^+ : U_{\omega}^{(C)}(1)\,u=0 \}$. Indeed, let $u \in C^{+}$ be such that $J u \in F_1^{(L)}({\omega})$. Then, by (4.11) and the invariance of $F_1^{(L)}$, $J U_{\omega}^{(C)}(1)\, u = J U_{\omega}^{(L,C)}(1) J u = U_{\omega}^{(L)}(1) J u \in F_1^{(L)}(\theta_{1}{\omega})$. Further, it follows from (A2) and definition 2.6(i) for $\Phi^{(L)}$ that $J U_{\omega}^{(C)}(1)\, u = 0$. As J is injective, $U_{\omega}^{(C)}(1) \,u = 0$ holds. On the other hand, if $u \in C^{+}$ is such that $U_{\omega}^{(C)}(1)\, u = 0$, then, by (4.11), $0 = J U_{\omega}^{(L,C)}(1) J u = U_{\omega}^{(L)}(1) J u$, and, since $J u \in L^{+}$, again by definition 2.6(i) for $\Phi^{(L)}$, $J u \in F_1^{(L)}({\omega})$, that is, $u \in F_1^{(C)}({\omega})$, our claim is true and (i) of definition 2.6 holds for $\Phi^{(C)}$.

Next, in view of remark 2.1, we prove that the family $\{\widetilde P^{(C)}({\omega})\}_{{\omega}\in\widetilde\Omega_1}$ of projections onto $E_1^{(C)}({\omega})$, defined as $\widetilde{P}^{(C)}({\omega})\,u=\lambda({\omega},Ju)\,w({\omega})$ for each $u\in C$, is strongly measurable and tempered. Concerning the strong measurability, we have to check that for each $u \in C$ the mapping $[\widetilde \Omega_1 \ni {\omega} \mapsto \widetilde P^{(C)}({\omega})u \in C ]$ is $(\mathfrak{F}, \mathfrak{B}(C))$-measurable. Once we fix $u\in C$, the strong measurability of $\{\widetilde P^{(L)}({\omega})\}_{{\omega}\in\widetilde\Omega_1}$ together with $ \widetilde P^{(L)}({\omega})\, Ju=\lambda({\omega},Ju)\, w^{(L)}({\omega})$ and ${\lVert w^{(L)}({\omega})\rVert_{L}}=1$ show that $[\widetilde \Omega_1 \ni {\omega} \mapsto \lambda({\omega},Ju) \in \mathbb{R} ]$ is $(\mathfrak{F}, \mathfrak{B}(\mathbb{R}))$-measurable, and the result follows from the composition of the maps

\begin{equation*} \widetilde\Omega_1 \to \mathbb{R}\times C \to C,\quad {\omega} \mapsto (\lambda({\omega},Ju),w({\omega}))\mapsto \lambda({\omega},Ju)\,w({\omega}).\end{equation*}

In order to show that the family is tempered, first notice that from the definition of $\widetilde P^{(L)}$ and ${\lVert w^{(L)}({\omega})\rVert_{L}}=1$ we deduce that

\begin{equation*} {\lVert\widetilde P^{(L)}(\theta_t{\omega})\rVert} = \sup_{{\lVert\widehat u\rVert_{L}}\leq 1}{\lVert \widetilde P^{(L)}(\theta_t{\omega})\,\widehat u\rVert_{L}}= \sup_{{\lVert\widehat u\rVert_{L}}\leq 1} |\lambda(\theta_t{\omega},\widehat u)| \,\text{for any}\, {\omega}\in\widetilde\Omega_1. \end{equation*}

Therefore, from $\{Ju\in L : u\in C \,\text{and}\, {\lVert u\rVert_{C}}\leq 1\} \subset \{\widehat u\in L: {\lVert\widehat u\rVert_{L}}\leq 2\}$ it holds

\begin{align*} 1\leq {\lVert\widetilde P^{(C)}(\theta_t{\omega})\rVert} & =\sup_{{\lVert u\rVert_{C}}\leq 1}{\lVert \widetilde P^{(C)}(\theta_t{\omega})\,u\rVert_{C}}= \sup_{{\lVert u\rVert_{C}}\leq 1} |\lambda(\theta_t{\omega},Ju)\,|{\lVert w(\theta_t{\omega})\rVert_{C}}\\ &\leq \sup_{{\lVert\widehat u\rVert_{L}}\leq 2} |\lambda(\theta_t{\omega},\widehat u)| \,{\lVert w(\theta_t{\omega})\rVert_{C}}\leq 2 \,{\lVert\widetilde P^{(L)}(\theta_t{\omega})\rVert} \,{\lVert w(\theta_t{\omega})\rVert_{C}}, \end{align*}

and consequently,

\begin{equation*} 0\leq \lim\limits_{t \to \pm\infty} \frac{\ln{\|\widetilde P^{(C)}(\theta_{t}{\omega})\|}}{t}\leq \lim\limits_{t \to \pm\infty} \frac{\ln{\|\widetilde P^{(L)}(\theta_{t}{\omega})\|}}{t}+ \lim\limits_{t \to \pm\infty} \frac{\ln{{\lVert w(\theta_t{\omega})\rVert_{C}}}}{t}. \end{equation*}

Thus, because of (4.25), in order to prove that $\{\widetilde P^{(C)}({\omega})\}_{{\omega}\in\widetilde\Omega_1}$ is tempered, it is enough to check that

(4.26)\begin{equation} \lim\limits_{t \to \pm\infty} \frac{\ln{{\lVert w(\theta_t{\omega})\rVert_{C}}}}{t}=0. \end{equation}

From $U_{\omega}^{(L)}(t)\,w^{(L)}({\omega})=\alpha(t,{\omega})\,w^{(L)}(\theta_t {\omega})$ and ${\lVert w^{(L)}(\theta_tw)\rVert_{L}}=1$, we obtain

\begin{equation*} w^{(L)}(\theta_t{\omega})=\frac{U_{\omega}^{(L)}(t)\,w^{(L)}({\omega})}{{\lVert U_{\omega}^{(L)}(t)\,w^{(L)}({\omega})\rVert_{L}}} \, , \,\text{and thus,}\, \,w(\theta_t{\omega})=\frac{U_{\omega}^{(L,C)}(t)\,w^{(L)}({\omega})}{{\lVert U_{\omega}^{(L)}(t)\,w^{(L)}({\omega})\rVert_{L}}} \end{equation*}

for $t\geq 0$. Hence, $\ln{{\lVert w(\theta_t{\omega})\rVert_{C}}}= \ln{{\lVert U_{\omega}^{(L,C)}(t)\,w^{(L)}({\omega})\rVert_{C}}}-\ln {\lVert U_{\omega}^{(L)}(t)\,w^{(L)}({\omega})\rVert_{L}}$ and (4.26) as t goes to $+\infty$ follows from (4.22). Concerning the limit as t goes to  $-\infty$, as in theorem 2.5 (see [Reference Mierczyński, Novo and Obaya18, Theorem 3.10] for the complete proof), we consider the negative semiorbit for $\Phi^{(L)}$ defined  as

(4.27)\begin{equation} w_{\omega}^{(L)}(s) =\frac{w^{(L)}(\theta_s{\omega})}{{\lVert U_{\theta_s{\omega}}^{(L)}(-s) \, w^{(L)}(\theta_s{\omega} )\rVert_{L}}} \quad \text{for} s\leq 0, \end{equation}

which satisfies $w_{\omega}^{(L)}(0)= w^{(L)}({\omega})$ and

(4.28)\begin{equation} \widetilde\lambda_1^{(L)} = \lim_{s\to- \infty} \frac{1}{s}\ln{\lVert w_{\omega}^{(L)}(s)\rVert_{L}} =-\lim_{s\to-\infty} \frac{\ln {\lVert U_{\theta_s {\omega}}^{(L)}(-s)\,w^{(L)}(\theta_s{\omega})\rVert_{L}}}{s}. \end{equation}

Then, as in [Reference Mierczyński, Novo and Obaya19, Theorem 5.6], denoting by $w_{\omega}(s)$ the second component of $w_{\omega}^{(L)}(s)$, which belongs to C, defines a negative semiorbit for $\Phi^{(C)}$, $J w_{\omega}(s)= w_{\omega}^{(L)}(s)$ and

(4.29)\begin{equation} \lim_{s\to-\infty}\frac{\ln{{\lVert w_{\omega}(s)\rVert_{C}}}}{s}=\widetilde\lambda_1^{(L)}. \end{equation}

Therefore, from (4.27), we deduce that

\begin{equation*} w(\theta_s{\omega})={\lVert U_{\theta_s {\omega}}^{(L)}(-s)\,w^{(L)}(\theta_s{\omega})\rVert_{L}}\,w_{\omega}(s) \quad \text{for each}\, s\leq 0, \end{equation*}

and $\ln{\lVert w(\theta_s{\omega})\rVert_{C}}= \ln{\lVert U_{\theta_s {\omega}}^{(L)}(-s)\,w^{(L)}(\theta_s{\omega})\rVert_{L}}+\ln {\lVert w_{\omega}(s)\rVert_{C}}$ holds. Thus from (4.28) and (4.29) the limit (4.26) holds as t goes to $-\infty$, $\{\widetilde P^{(C)}({\omega})\}_{{\omega}\in\widetilde\Omega_1}$ is tempered, as claimed, and (ii) of definition 2.6 holds.

Now we check that (iii) of this definition also holds, that is, there exists $\widetilde{\sigma}\in (0,\infty]$ such that

\begin{equation*} \lim_{t\to\infty}\frac{1}{t}\ln \frac{\|U_{\omega}^{(C)}(t)|_{F_1^{(C)}({\omega})}\|}{\|U_{\omega}^{(C)}(t)\,w({\omega})\|_C}=-\widetilde{\sigma} \quad \text{for each}\, {\omega}\in\widetilde\Omega_1.\end{equation*}

Again, from theorems 4.5 and 2.9, we know that there exists $\widetilde\sigma\in(0,\infty]$ such that $\widetilde\lambda^{(L)}_2=\widetilde\lambda^{(L)}_1-\widetilde\sigma$ and

\begin{equation*} \lim_{t\to\infty}\frac{1}{t}\ln \frac{\|U_{\omega}^{(L)}(t)|_{F_1^{(L)}({\omega})}\|}{\|U_{\omega}^{(L)}(t)\,w^{(L)}({\omega})\|_L}=-\widetilde{\sigma} \quad \text{and} \quad \lim_{t\to\infty}\frac{1}{t}\ln\|U_{\omega}^{(L)}(t)|_{F_1^{(L)}({\omega})}\|= \widetilde\lambda^{(L)}_2 \end{equation*}

for each ${\omega}\in\widetilde\Omega_1$, which together with (4.23) shows that it is enough to prove that

(4.30)\begin{equation} \lim_{t\to\infty}\frac{1}{t}\ln\|U_{\omega}^{(C)}(t)|_{F_1^{(C)}({\omega})}\| = \widetilde\lambda^{(L)}_2 \quad \text{for each}\, {\omega}\in\widetilde\Omega_1. \end{equation}

As in [Reference Mierczyński, Novo and Obaya19, Proposition 5.2(2)], it is easy to check that

(4.31)\begin{equation} \widetilde\lambda^{(L)}_2 =\lim_{t\to\infty}\frac{1}{t}\ln\|U_{\omega}^{(L)}(t)|_{F_1^{(L)}({\omega})}\| = \lim_{t\to\infty}\frac{1}{t}\ln\|U_{\omega}^{(L,C)}(t)|_{F_1^{(L)}({\omega})}\| \end{equation}

for each ${\omega}\in\widetilde\Omega_1$. Moreover, from (4.10) and the construction of $F^{(C)}_1({\omega})$, we obtain

(4.32)\begin{equation} U_{\omega}^{(L,C)}(1)\big(F_1^{(L)}({\omega})\big)\subset F_1^{(C)}(\theta_1{\omega}) \quad\text{and}\quad J(F_1^{(C)}({\omega}))\subset F_1^{(L)}({\omega}). \end{equation}

Thus, from (4.10), (4.11), (4.31) and (4.32), we deduce the following chain of inequalities

\begin{align*} \widetilde\lambda^{(L)}_2&=\lim\limits_{t\to\infty}\frac{1}{t+1}\ln\|U_{\theta_{-1}{\omega}}^{(L,C)}(t+1)|_{F_1^{(L)}(\theta_{-1}{\omega})}\| \\ &= \lim\limits_{t\to\infty}\frac{1}{t+1}\ln\|U_{\omega}^{(C)}(t)\circ U_{\theta_{-1}{\omega}}^{(L,C)}(1)|_{F_1^{(L)}(\theta_{-1}{\omega})}\|\\ & \leq \liminf\limits_{t\to\infty}\frac{1}{t}\ln\|U_{\omega}^{(C)}(t)|_{F_1^{(C)}({\omega})}\|\leq \limsup\limits_{t\to\infty}\frac{1}{t}\ln\|U_{\omega}^{(C)}(t)|_{F_1^{(C)}({\omega})}\| \\ & = \limsup\limits_{t\to\infty}\frac{1}{t}\ln\|U_{\omega}^{(L,C)}(t)\circ J|_{F_1^{(C)}({\omega})}\| \leq \lim\limits_{t\to\infty}\frac{1}{t}\ln\|U_{\omega}^{(L,C)}(t)|_{F_1^{(L)}({\omega})}\| = \widetilde\lambda^{(L)}_2, \end{align*}

which shows (4.30) and (iii) of definition 2.6 holds. Finally, from $\widetilde{\lambda}_1^{(L)} \gt - \infty$ and  (4.24), we deduce that $\widetilde{\lambda}_1^{(C)} \gt - \infty$, which finishes the proof.