Proof of Lemma 2.2.

With the help of the normal coordinate chart, we deal with $W_{\delta (le^\epsilon )}^{u}(\omega ,x)$ and $W^{u}_{\delta (l)}(\theta \omega ,f_\omega (x))$ in Euclidean space. We let $\delta ^\prime (l)<\delta (le^\epsilon )$ be determined later. By Lemma 2.1, there exist $ \xi _y\in E^{u}(\omega ,x)(\delta ^\prime (l)) $ and $ \xi _{f_\omega (y)}\in E^{u}(\theta \omega ,f_\omega (x))(\delta (l))$ such that

(2.1) $$ \begin{align} y & =x+\xi_y+h^{u}_{(\omega,x)}(\xi_y), \end{align} $$

(2.2) $$ \begin{align} f_\omega (y) & =f_\omega(x)+\xi_{f_\omega(y)}+h^{u}_{(\theta\omega,f_\omega(x))}(\xi_{f_\omega(y)}), \end{align} $$

and $T_yW^u_{\delta (le^\epsilon )}(\omega ,x)=\mathrm {Graph}((Dh_{(x,\omega )}^{uu})_{\xi _y})$ , $T_{f_\omega (y)}W^u_{\delta (l)}(\theta \omega ,f_\omega (x))= \mathrm {Graph} ((Dh^{uu}_{(f_\omega (x),\theta \omega )})_{\xi _{f_\omega (y)}})$ . By (2.1) and (2.2), we have

$$ \begin{align*} |x-y|=|\xi_y+h_{(\omega,x)}^{u}(\xi_y)|\geq |\xi_y|-L(le^\epsilon)|\xi_y| \end{align*} $$

and

$$ \begin{align*} (1-L(l))|\xi_{f_\omega(y)}| & \leq |\xi_{f_\omega(y)}+h^{u}_{(\theta\omega,f_\omega(x))}(\xi_{f_\omega(y)})|=|f_\omega(y)-f_\omega(x)|\\ & \leq R|y-x|\leq R(1+L(le^\epsilon))|\xi_y|, \end{align*} $$

so $|\xi _{f_\omega (y)}|\leq R(({1+L(le^\epsilon )})/({1-L(l)}))|\xi _y|$ , where $R\geq 1$ is a constant such that

(2.3) $$ \begin{align} \max\bigg\{\!\sup_{(x,\omega)\in \mathcal{M}\, \times\,\Omega}|D_xf_\omega|,\sup_{(x,\omega)\in \mathcal{M}\, \times\,\Omega}|D_x^2f_\omega|\bigg\}\leq R. \end{align} $$

Now, we define the following linear maps $L_{(\omega ,x)}, \tilde {L}_{(\omega ,y)}:E^{u}(\omega ,x)\rightarrow E^{u}(\theta \omega ,f_\omega (x))$ by

$$ \begin{align*} L_{(\omega ,x)} & =D_xf_\omega|_{E^{u}(\omega,x)}, \\ \tilde{L}_{(\omega,y)} &= \pi_{(\theta\omega, f_\omega(x))}^{u}D_yf_\omega|_{T_yW^u_{\delta(le^\epsilon)}(\omega,x)}(I+(Dh_{(\omega,x)}^{u})_{\xi_y}), \end{align*} $$

where $\pi ^u_{(\theta \omega ,f_\omega (x))}$ is the projection from $T_{f_\omega (x)}\mathcal {M}$ to $E^u(\theta \omega ,f_\omega (x))$ with respect to the splitting $E^u(\theta \omega ,f_\omega (x))\oplus E^s(\theta \omega ,f_\omega (x))$ . Note that $(\theta \omega ,f_\omega (x))\in \Lambda _l$ and $\angle (E^u(\theta \omega ,f_\omega (x)),E^s(\theta \omega ,f_\omega (x)))\geq l^{-1}$ , there exists $\theta (l)>0$ such that $\|\pi ^u_{(\theta \omega ,f_\omega (x))}\|\leq \theta (l)$ . Then,

$$ \begin{align*} \|L_{(\omega,x)}\|,\|\tilde{L}_{(\omega,y)}\|\leq R\theta(l)(1+L(le^\epsilon)). \end{align*} $$

Now, we have

$$ \begin{align*} & \sup_{v\in E^{u}(\omega,x),\ \|v\|=1} |D_xf_\omega v-\pi_{(\theta\omega , f_\omega(x))}^{u}D_yf_\omega|_{T_yW^u_{\delta(le^\epsilon)}(\omega,x)}(I+(Dh_{(\omega,x)}^{u})_{\xi_y}) v|\\ &\quad \leq \theta(l)(\|D_xf_\omega-D_yf_\omega\|+\|D_yf_\omega (Dh_{(\omega,x)}^{u})_{\xi_y}\|)\\ &\quad \leq R\theta(l)|y-x|+R\theta(l)L(l)|\xi_y|\\ &\quad \leq (R\theta(l)+R\theta(l)L(l))d^u(x,y). \end{align*} $$

Hence, $\|L_{(\omega ,x)}-\tilde {L}_{(\omega ,y)}\|\leq C_0(R\theta (l)+R\theta (l)L(l))d^u(x,y)$ , where $C_0$ only depends on the normal coordinate chart. Then, by the property of the determinant,

(2.4) $$ \begin{align} |\text{det}(L_{(\omega,x)})-\text{det}(\tilde{L}_{(\omega,y)})|\leq C_1(l)d^u(x,y), \end{align} $$

where $C_1(l)$ is a polynomial of $R\theta (l)+R\theta (l)L(l)$ and $\dim E^{u}(\omega ,x)$ .

Note that

$$ \begin{align*} & \|(\pi_{(\theta\omega,f_\omega(x))}^{u}|_{T_{f_\omega(y)}W^u_{\delta(l)}(\theta\omega,f_\omega(x))}-I)(I+(Dh^{u}_{(\theta\omega,f_\omega(x))})_{\xi_{f_\omega(y)}})\| \\&\quad \leq \|(Dh^{u}_{(\theta\omega,f_\omega(x))})_{\xi_{f_\omega(y)}}\|. \end{align*} $$

Therefore,

$$ \begin{align*} \|\pi_{(\theta\omega,f_\omega(x))}^{u}|_{T_{f_\omega(y)}W^u_{\delta(l)}(\theta\omega,f_\omega(x))}-I\| &\leq \frac{\|(Dh^{u}_{(\theta\omega,f_\omega(x))})_{\xi_{f_\omega(y)}}\|}{1-\|(Dh^{u}_{(\theta\omega,f_\omega(x))})_{\xi_{f_\omega(y)}}\|}\\ &\leq \frac{L(l)|\xi_{f_\omega(y)}|}{1-L(l)|\xi_{f_\omega(y)}|}\\ &\leq \frac{L(l)\cdot R(({1+L(le^\epsilon)})/({1-L(l)}))|\xi_y|}{1-L(l)\cdot R(({1+L(le^\epsilon)})/({1-L(l)}))|\xi_y|}\\ &\leq \frac{L(l)\cdot R(({1+L(le^\epsilon)})/({1-L(l)}))|\xi_y|}{1-L(l)\cdot R(({1+L(le^\epsilon)})/({1-L(l)}))\delta^\prime(l)}\\ &\leq 2L(l)\cdot R\frac{1+L(le^\epsilon)}{1-L(l)}|d^u(x,y), \end{align*} $$

where $\delta ^\prime (l)$ is chosen so that $L(l)\cdot R(({1+L(le^\epsilon )})/({1-L(l)}))\delta ^\prime (l)<1/2.$ So, we have

(2.5) $$ \begin{align} |\!\det(\pi_{(\theta\omega,f_\omega(x))}^{u}|_{T_{f_\omega(y)}W^u_{\delta(l)}(\theta\omega,f_\omega(x))})-1|\leq C_2(l)d^u(x,y) \end{align} $$

where $C_2(l)$ is a polynomial of $L(l)\cdot R(({1+L(le^\epsilon )})/({1-L(l)}))$ and $\dim E^{u}(\omega ,x)$ . In addition,

$$ \begin{align*} \|I+(Dh_{(\omega,x)}^{u})_{\xi_y}-I\|\leq L(le^\epsilon)|\xi_y|\leq L(le^\epsilon) d^u(x,y) \end{align*} $$

implies that there exists $C_3(l)$ such that

(2.6) $$ \begin{align} |\text{det}(I+(Dh_{(\omega,x)}^{u})_{\xi_y})-1|\leq C_3(l) d^u(x,y). \end{align} $$

Combining (2.4), (2.5), and (2.6), we conclude that there exists $C_4(l)>0$ ,

$$ \begin{align*} |\!\det D_xf_\omega|_{E^u(\omega,x)}-\det D_yf_\omega|_{T_yW^u_{\delta(le^\epsilon)}(\omega,x)}|\leq C_4(l)d^u(x,y). \end{align*} $$

Then, the proof of Lemma 2.2 is complete by letting $C_l=C_4(l)/\sup _{(\omega ,x,E)}|\!\det D_xf_\omega |_{E}|$ .

Proof of Lemma 2.3.

For any $(\omega ,x)\in \Lambda _l$ and $[0,n]$ that is N- $\{(\omega ,x)\}$ -admissible, then by definition, there exists $0=a_0<a_1<\cdots <a_k=n$ with $a_{i+1}-a_i\leq N$ such that $(\theta ^{a_i}\omega ,f_\omega ^{a_i}(x))\in \Lambda _l$ for $i=0,\ldots ,k$ . Therefore, for any $t\in [0,n]$ , we have $(\theta ^t\omega ,f_\omega ^t(x))\in \Lambda _{le^{N\epsilon }}$ . As a consequence, for any

(2.7) $$ \begin{align} \delta<\delta_0(l,N,\gamma):=\frac{\min\{\delta^\prime(l)/4,\delta(l)\}e^{-N(\epsilon+R)}}{8}\leq \delta(le^{N\epsilon})e^{-NR}/8, \end{align} $$

$W_{8\delta }^u(\theta ^t\omega ,f_\omega ^t(x))$ is well defined, where $\delta ^\prime (l)$ is given in Lemma 2.2, and R is a constant as in (2.3).

For $i\in [a_0,a_1]$ , we have

$$ \begin{align*} \text{diam}(f_\omega^iW_{4\delta}^u(\omega,x))\leq 4\delta\cdot e^{iR}\leq 4\frac{\delta(l)e^{-N(\epsilon+R)}}{8}\cdot e^{iR}\leq \frac{\delta(l)e^{-N\epsilon}}{2}\leq \delta(le^{N\epsilon}). \end{align*} $$

As a consequence, $f_\omega ^iW_{4\delta }^u(\omega ,x)$ belongs to the local stable manifolds. Therefore, for $i\in [a_0,a_1]$ , we have

$$ \begin{align*} f_\omega^i(W_{4\delta}^u(\omega,x)\cap B_n(\omega,x,\delta))\subset f_\omega^iW_{4\delta}^u(\omega,x)\cap B(\theta^i\omega,f_\omega^i(x),\delta)\subset W_{4\delta}^u(\theta^i\omega,f_\omega^i(x)). \end{align*} $$

Inductively, one can prove statement (1) holds. Next, we prove statement (2).

For any Borel measurable set $\Delta \subset W_{4\delta }^u(\omega ,x)\cap B_n(\omega ,x,\delta )$ , by statement (1), $f_\omega ^n(\Delta )\subset W_{4\delta }^u(\theta ^n\omega ,f_\omega ^n(x))$ . Therefore,

$$ \begin{align*} \lambda^u(f^n\Delta)& =\int_{\Delta}\det(D_yf_\omega^n|_{T_yW_{4\delta}^u(\omega,x)})\,d\lambda^u(y)\\ & = \int_{\Delta}\frac{\det(D_yf_\omega^n|_{T_yW_{4\delta}^u(\omega,x)})}{\det(D_xf_\omega^n|_{T_xW_{4\delta}^u(\omega,x)})}\,d\lambda^u(y)\cdot \det(D_xf_\omega^n|_{T_xW_{4\delta}^u(\omega,x)}). \end{align*} $$

By statement (1) and Lemma 2.2, we have estimate

$$ \begin{align*} e^{-\sum_{i=0}^{n-1}C_{le^{N\epsilon}}\cdot d^u(f_\omega^i(x),f_\omega^i(y))}\leq\bigg|\frac{\det(D_yf_\omega^n|_{T_yW_{4\delta}^u(\omega,x)})}{\det(D_xf_\omega^n|_{T_xW_{4\delta}^u(\omega,x)})}\bigg| \leq e^{\sum_{i=0}^{n-1}C_{le^{N\epsilon}}\cdot d^u(f_\omega^i(x),f_\omega^i(y))}, \end{align*} $$

where $C_{le^{N\epsilon }}$ is given in the statement of Lemma 2.2. This implies

$$ \begin{align*} \frac{\lambda^u(\Delta)\det(D_xf_\omega^n|_{T_xW_{4\delta}^u(\omega,x)})}{\lambda^u(f^n\Delta)} \in (e^{-k\sum_{i=1}^{N}C_{le^{N\epsilon}}\cdot e^{-\chi i}\cdot 4\delta},e^{k\sum_{i=1}^{N}C_{le^{N\epsilon}}\cdot e^{-\chi i}\cdot 4\delta} ). \end{align*} $$

The proof is complete by shrinking $\delta $ .

Proof of Lemma 2.4.

By the Hölder continuity of the fiber unstable subbundle (see e.g. [Reference Liu and Qian20, Ch. III, Theorem 4.1]), there exists a constant $C(l)$ such that for any $(\omega ,z_1),(\omega ,z_2)\in \Lambda _{le^{N\epsilon }}$ , one has

$$ \begin{align*} \bigg|\log \frac{\det (D_{z_1}f_\omega|_{E^u(\omega,z_1)})}{\det (D_{z_2}f_\omega|_{E^u(\omega,z_2)})}\bigg|<C(l)e^{N\epsilon}d(z_1,z_2)^\alpha \end{align*} $$

for some exponent $\alpha \in (0,1)$ . Note that ${}[\![ {} 0,n{}]\!] {}$ is N- $\{(\omega ,x),(\omega ,y)\}$ -admissible, by the same argument in the proof of Lemma 2.3, for any $i\in {}[\![ {} 0,n{}]\!] {}$ , $(\theta ^i\omega ,f_\omega ^i(x))$ and $(\theta ^i\omega ,f_\omega ^i(y))$ belong to $\Lambda _{le^{N\epsilon }}$ . Then, the conclusion follows from the above inequality by letting

$$ \begin{align*} \delta_1(l,N,\gamma)=\min\bigg\{\delta_0(l,N,\gamma),\bigg(\frac{\gamma}{C(l)e^{N\epsilon}}\bigg)^{1/\alpha}\bigg\}. \end{align*} $$

This completes the proof of this lemma.

Proof of Lemma 2.5.

We first consider the case where $F_n={}[\![ {} n_0,n_1 {}]\!] {}$ is an interval of integers. Fix a natural number $q\in \mathbb {N}$ . We consider the case where $1<q\leq n_1-n_0$ . For $1\leq j\leq q-1$ , we define $a(j)=\lceil ({n_1-n_0-j}/{q})\rceil $ . For any fixed $j\in {}[\![ {} 0,q-1{}]\!] {}$ , we have decomposition

$$ \begin{align*} \bigvee_{i=0}^{n_1-n_0}G^{-i}G^{-n_0}\mathcal{Q}=\bigvee_{r=0}^{a(j)-1}G^{-(rq+j+n_0)}\bigg(\bigvee_{i=0}^{q-1}G^{-i}\mathcal{Q}\bigg)\vee\bigvee_{l\in S}G^{-l}\mathcal{Q}, \end{align*} $$

where S has cardinality at most $2q$ . Therefore, we have

$$ \begin{align*} H_{\mu_n}\bigg(\bigvee_{i=n_0}^{n_1-1}G^{-i}\mathcal{Q}|\mathfrak{F}_{\Omega\, \times\, X}\bigg) & \leq \sum_{r=0}^{a(j)-1}H_{\mu_n}(G^{-(rq+j+n_0)}\bigvee_{i=0}^{q-1}G^{-i}\mathcal{Q}|\mathfrak{F}_{\Omega\, \times\, X})+2q\log\#\mathcal{Q}\\ & =\sum_{r=0}^{a(j)-1}H_{\mu_n\circ G^{-(rq+j+n_0)}}\bigg(\bigvee_{i=0}^{q-1}G^{-i}\mathcal{Q}|\mathfrak{F}_{\Omega\, \times\, X}\bigg)+2q\log\#\mathcal{Q}. \end{align*} $$

Summing the above over j from $0$ to $q-1$ and dividing it by $ \#F_n$ , we obtain

$$ \begin{align*} &\frac{q}{\#F_n}H_{\mu_n}\bigg(\bigvee_{i=n_0}^{n_1}G^{-i}\mathcal{Q}|\mathfrak{F}_{\Omega\, \times\, X}\bigg)\\& \quad \leq \frac{1}{\#F_n}\sum_{k=0}^{n_1-n_0}H_{\mu_n\circ G^{-(k+n_0)}}\bigg(\bigvee_{i=0}^{q-1}G^{-i}\mathcal{Q}|\mathfrak{F}_{\Omega\, \times\, X}\bigg) +\frac{2q^2\log \#\mathcal{Q}}{\#F_n} \\& \quad \leq H_{\mu_n^{F_n}}\bigg(\bigvee_{i=0}^{q-1}G^{-i}\mathcal{Q}|\mathfrak{F}_{\Omega\, \times\, X}\bigg)+\frac{2q^2\log \#\mathcal{Q}}{\#F_n}, \end{align*} $$

where the second inequality is due to the concavity of the conditional entropy. Then, dividing it by q, we obtain the desired inequality. Now, we consider the case where $q> n_1-n_0$ . By direct computation, we have

$$ \begin{align*} \frac{1}{\# F_n} H_{\mu_n}\bigg(\bigvee_{i\in F_n}(G^{i})^{-1}\mathcal{Q}|\mathfrak{F}_{\Omega\, \times\, X}\bigg)& \leq \log \#\mathcal{Q}\leq \frac{q}{\#F_n}\log\#\mathcal{Q}\\ & \leq \frac{1}{q}H_{\mu_n^{F_n}}\bigg(\bigvee_{i=0}^{q-1}(G^i)^{-1}\mathcal{Q}|\mathfrak{F}_{\Omega\, \times\, X}\bigg)+\frac{q}{\#F_n}\log\#\mathcal{Q}. \end{align*} $$

Now, we consider a general set $F_n\in \mathcal {P}_n$ , which can be decomposed into connected components $F_n=\bigsqcup _{j\in J_n}F_n^j$ . Noticing that $\#J_n\leq \#\partial F_n$ , then we obtain

$$ \begin{align*} \frac{1}{q}H_{\mu_n^{F_n}}\bigg(\bigvee_{i=0}^{q-1}G^{-i}\mathcal{Q}|\mathfrak{F}_{\Omega\, \times\, X}\bigg) & \geq \sum_{j\in J_n}\frac{\# F_n^j}{q\# F_n}H_{\mu_n^{F_n^j}}\bigg(\bigvee_{i=0}^{q-1}G^{-i}\mathcal{Q}|\mathfrak{F}_{\Omega\, \times\, X}\bigg)\\ & \geq \sum_{j\in J_n}\bigg(H_{\mu_n}\bigg(\bigvee_{i\in F_n^j}G^{-i}\mathcal{Q}|\mathfrak{F}_{\Omega\, \times\, X}\bigg)-\frac{2q\log\#\mathcal{Q}}{\# F_n}\bigg)\\ & \geq H_{\mu_n}\bigg(\bigvee_{i\in F_n}G^{-i}\mathcal{Q}|\mathfrak{F}_{\Omega\, \times\, X}\bigg)-\frac{2q\#J_n\log\#\mathcal{Q}}{\# F_n}, \end{align*} $$

where the first inequality is due to the concavity of the conditional entropy function and the second inequality is due to the first case for intervals. Note that $(F_n)_{n\in \mathfrak {n}}$ is a Følner sequence with $\underline {d}^{\mathfrak {n}}(\mathcal {F})>0$ . Therefore, we have

$$ \begin{align*} \limsup_{n\in\mathfrak{n}\to\infty}\frac{2q\#J_n\log\#\mathcal{Q}}{\#F_n}\leq \limsup_{n\in\mathfrak{n}\to\infty}\frac{2q\#\partial F_n\log\#\mathcal{Q}}{\#F_n}=0.\\[-41pt] \end{align*} $$

Proof of Lemma 2.6.

We denote $E_k(x):=\Lambda ^k(T_x\mathcal {M})$ and $E_k=\bigsqcup _{x\in \mathcal {M}}E_k(x)$ , where $\Lambda ^k$ is the kth exterior power. Let $A_k:\mathcal {G}^0(T\Omega )\, \times \, E_k\to \mathcal {G}^0(T\Omega )\, \times \, E_k$ be a vector bundle endomorphism over $F:\Omega \, \times \, \mathcal {M}\to \Omega \, \times \, \mathcal {M}$ and be defined by $(A_k)_{(\omega ,x)}(0,e_1\wedge \cdots \wedge e_k)=(0,D_xf_\omega e_1\wedge \cdots \wedge D_xf_\omega e_k)$ . Note that for any two bases $\{e_1,\ldots ,e_k\}$ and $\{w_1,\ldots ,w_k\}$ spanning the same k-dimensional subspaces of $T_x\mathcal {M}$ if and only if $w_1\wedge \cdots \wedge w_k=ae_1\wedge \cdots \wedge e_k$ for some $a\in \mathbb {R}$ , it gives an equivalence relation $\sim $ . Let $D_k(x)=\{e_1\wedge \cdots \wedge e_k|\ e_1,\ldots ,e_k\text { be linearly independent in }T_x\mathcal {M}\}$ and set $G_k(x)=D_k(x)/\sim $ , $G_k=\bigsqcup _{x\in \mathcal {M}}G_k(x)$ . We note that

$$ \begin{align*} G_k(x)\subset P(E_k(x)):=\{[u]|\ v\in [u]\text{ iff }v=au,\ a\not=0,\ u,v\in E_k(x)\} \end{align*} $$

is a subbundle whose fiber over x is the projectivization of the set of decomposable k-vectors in $E_k(x)$ . We note that $\mathcal {G}^k(T_x\mathcal {M})$ is isomorphic to $G_k(x)$ by

$$ \begin{align*} E\subset \mathcal{G}^k(T_x\mathcal{M})\mapsto \{[e_1\wedge\cdots \wedge e_k]|\ e_1,\ldots,e_k\text{ forms a basis of }E\}. \end{align*} $$

Moreover, this isomorphism conjugates $\tilde {F}$ to $P(A_k):P(\mathcal {G}^0(T\Omega ))\, \times \, G_k\subset P(\mathcal {G}^0(T\Omega )\, \times \, E_k)\to P(\mathcal {G}^0(T\Omega ))\, \times \, G_k$ over F, which is defined by $P(A_k)_{(\omega ,x)}([0]\, \times \, [e_1\wedge \cdots \wedge e_k])=[0]\, \times \, [D_xf_\omega e_1\wedge \cdots \wedge D_xf_\omega e_k]$ . Now, for any $\mathbb {F}$ -invariant Borel probability measure $\mu $ on $\Omega \, \times \, \mathcal {G}^k(T\mathcal {M})$ , let $\hat {\mu }$ be the corresponding $\tilde {F}$ -invariant Borel probability measure on $\mathcal {G}^0(T\Omega )\, \times \,\mathcal {G}^k(T\mathcal {M})$ , and $\hat {\mu }^*$ be the $P(A_k)$ -invariant Borel probability measure on $P(\mathcal {G}^0(T\Omega ))\, \times \, G_k$ by conjugation. Then, the projection of $\hat {\mu }^*$ on $\Omega \, \times \, \mathcal {M}$ is $\pi _*\mu $ . By [Reference Sun and Vargas27, Theorem 3.7], we have $ h_{\mu }(\mathbb {F})=h_{\hat {\mu }^*}(P(A_k))=h_{\pi _*\mu }(F).$

Proof of Proposition 3.1.

Denote $\mathbb {A}_n^{\omega ^*}=\{(\omega ^*,x,E^u(\omega ^*,x)):\ x\in A_n^{\omega ^*}\}$ , $\mathbb {E}_{\omega ^*}=\{(\omega ^*,x,E^u(\omega ^*,x)):\ x\in E_{\omega ^*}\}$ , and let $\hat {\mu }_n$ be the measure on $\mathbb {A}_n^{\omega ^*}$ corresponding to $\mu _n$ on $A_n^{\omega ^*}$ .

Let $\mathfrak {n}^\prime $ be a subsequence of $\mathfrak {n}$ such that $\{\hat {\mu }_n^{F_n}:=({1}/{\# F_n})\sum _{k\in F_n}\mathbb {F}_*^k\hat {\mu }_n\}_{n\in \mathfrak {n}^\prime }$ is converging to $\hat {\mu }$ under the weak $^*$ topology. By the property of the Følner sequence and a standard argument, one can see that $\hat {\mu }$ is $\mathbb {F}$ -invariant. Since $F:\Omega \, \times \, \mathcal {M}\to \Omega \, \times \, \mathcal {M}$ is a factor system of $\mathbb {F}$ , the projection of $\hat {\mu }$ is F-invariant. The projection of $\hat {\mu }$ has marginal $\mathbb {P}$ on $\Omega $ since $\omega ^*$ is a generic point of $\mathbb {P}$ .

For $(\omega ,x,E)\in \mathcal {G}^0(T\Omega )\, \times \, \mathcal {G}^k(T\mathcal {M})$ , we define $\phi _n(\omega ,x,E)=\log m(D_xf_\omega ^n|_E)$ for any $n\in \mathbb {N}$ , where m denotes the conorm. Then, $\phi _n$ is a continuous function on $\mathcal {G}^0(T\Omega )\, \times \, \mathcal {G}^k(T\mathcal {M})$ . Moreover, one has

$$ \begin{align*} \phi_{n+m}(\omega,x,E)\geq \phi_n(\omega,x,E)+\phi_m(\mathbb{F}^n(\omega,x,E)). \end{align*} $$

Recall the definition of $I(\omega ^*,x):=I_{T}(\omega ^*,x)$ . We can pick T sufficiently large such that for any $q>p\in I(\omega ^*,x)$ and $(\omega ^*,x,E^u(\omega ^*,x))\in \mathbb {E}_{\omega ^*}$ ,

(3.1) $$ \begin{align} \phi_{q-p}(\mathbb{F}^p(\omega^*,x,E^u(\omega^*,x)))\geq (q-p)\frac{\chi}{2}. \end{align} $$

In fact, for any $q>p\in I(\omega ^*,x)$ , then $q-p\geq T$ and $F^p(\omega ^*,x)\in \Lambda _l$ ; therefore, we have

$$ \begin{align*} \phi_{q-p}(\mathbb{F}^p(\omega^*,x,E^u(\omega^*,x))) & =\log m(D_{f_{\omega^*}^p(x)}f_{\theta^p\omega^*}^{q-p}|_{E^u(\theta^p\omega^*,f_{\omega^*}^p(x))}) \\ & \geq \log (C(F^p(\omega^*,x))^{-1}e^{(q-p)\chi})\geq \log(e^{-l}e^{(q-p)\chi}) \\ & \geq (q-p)\cdot \frac{\chi}{2} \end{align*} $$

by picking T sufficiently large.

For convenience, define $\phi _n^\prime (\omega ^*,x,E)=\phi _n(\omega ^*,x,E)-n\cdot ({\chi }/{2})$ for $n\in \mathbb {N}$ . For a finite subset of integers J, we denote $\phi _J(\omega ^*,x,E)$ (respectively $\phi _J^\prime (\omega ^*,x,E)$ ) to be the sum of $\phi _{q-p}(\mathbb {F}^p(\omega ^*,x,E))$ (respectively $\phi _{q-p}^\prime (\mathbb {F}^p(\omega ^*,x,E))$ ), where ${}[\![ {} p,q{}[\![ {}$ runs over all maximal intervals of integers contained in J. Finally, define $\| \phi ^\prime _1\|_{\sup }:=\sup _{(\omega ,x,E)\in \mathcal {G}^0(T\Omega )\, \times \, \mathcal {G}^k(T\mathcal {M})}|\phi ^\prime _1(\omega ,x,E)|$ .

Fix $\alpha <0$ , $M<N\in \mathbb {N}$ . For $x\in A_n^{\omega ^*}$ , $n\in \mathfrak {n}^\prime $ , we define

$$ \begin{align*} F_n^\alpha(\omega^*,x)=\bigg\{k\in F_n,\ \frac{\phi_N^\prime(\mathbb{F}^k(\omega^*,x,E^u(\omega^*,x)))}{N}\leq \alpha\bigg\}. \end{align*} $$

For $l\in \mathbb {N}$ and $k\in \{0,\ldots ,N-1\}$ with $k+lN\in F_n$ , define $J_{k,l}= [k+lN,k+(l+1)N)\cap \mathbb {N}$ . The interval of integers $J_{k,l}$ can be written as $J_{k,l}=I_{k,l}^1(x)\sqcup I_{k,l}^2(x)\sqcup I_{k,l}^3(x)\sqcup I_{k,l}^4(x)$ , where we have the following:

• • $I_{k,l}^1(x)$ is the union of disjoint $I(\omega ^*,x)$ irreducible interval of length less than M contained in $J_{k,l}$ ; by (3.1), we have $\phi _{I_{k,l}^1}^\prime (\omega ^*,x,E^u(\omega ^*,x))\geq 0$ ;

• • $I_{k,l}^2(x)$ consists of subintervals of an $I(\omega ^*,x)$ irreducible interval of length less than M, which does not lie in $J_{k,l}$ , but may contain the right extreme of $J_{k,l}$ ; in particular, $\#I_{k,l}^2\leq 2M$ and, therefore, $\phi _{I_{k,l}^2}^\prime (\omega ^*,x,E^u(\omega ^*,x))\geq -2M\| \phi ^\prime _1\|_{\sup }$ ;

• • $I_{k,l}^3(x)\subset F_n\backslash I_M(\omega ^*,x)$ ;

• • $I_{k,l}^4(x)\subset \mathbb {N}\backslash F_n$ .

Fix $0\leq k<N$ . Since $k+lN\in F_n$ , if $I_{k,l}^4(x)\not =\emptyset $ , then $J_{k,l}$ must contain a point of $\partial F_n$ . By the intervals $\{J_{k,l}\}_l$ are pairwise disjoint, we have $\#(\bigcup _{l:~I_{k,l}^4(x)\not =\emptyset }J_{k,l})\leq N\cdot \#\partial F_n$ . Therefore, by the superadditivity of $\phi ^\prime $ , for the fixed k, we obtain

$$ \begin{align*} \sum_{\substack{l:~k+lN\in F_n^\alpha(\omega^*,x)\\ I_{k,l}^4(x)\not=\emptyset}}\phi_{J_{k,l}}^\prime(\omega^*,x,E^u(\omega^*,x)) &\geq -\| \phi^\prime_1\|_{\sup} \cdot \sum_{\substack{l:~k+lN\in F_n^\alpha(\omega^*,x)\\ I_{k,l}^4(x)\not=\emptyset}}\# J_{k,l} \\ &\geq -N\cdot \#\partial F_n\cdot\| \phi^\prime_1\|_{\sup}. \end{align*} $$

If $I_{k,l}^4(x)=\emptyset $ , by the superadditivity of $\phi ^\prime $ , we obtain

$$ \begin{align*} \phi_{J_{k,l}}^\prime(\omega^*,x,E^u(\omega^*,x)) & \geq \phi_{I_{k,l}^1}^\prime(\omega^*,x,E^u(\omega^*,x))+\sum_{i=2,3}\phi_{I_{k,l}^i}^\prime(\omega^*,x,E^u(\omega^*,x))\\ & \geq -(\# I_{k,l}^3+2M)\cdot\| \phi^\prime_1\|_{\sup}. \end{align*} $$

As $F_n\subset \{0,\ldots ,n-1\}$ , the cardinality of $\{l:\ k+lN\in F_n\}$ is less than or equal to $\lceil {n}/{N}\rceil $ . Also by $\{J_{k,l}\}_l$ are pairwise disjoint, $\sum _{\ell }\# I_{k,l}^3\leq \#(F_n\backslash I_M(\omega ^*,x))$ . Therefore, we have

$$ \begin{align*} \sum_{\substack{l:~k+lN\in F_n^\alpha(\omega^*,x)\\ I_{k,l}^4(x)=\emptyset}}\phi_{J_{k,l}}^\prime(\omega^*,x,E^u(\omega^*,x))\geq -\bigg(\#(F_n\backslash I_M(\omega^*,x))+2M\bigg\lceil \frac{n}{N}\bigg\rceil\bigg)\|\phi_1^\prime\|_{\sup}. \end{align*} $$

Combining these two cases, for each fixed $0\leq k<N$ , we have

(3.2) $$ \begin{align} & \sum_{l:~k+lN\in F_n^\alpha(\omega^*,x)}\phi_{J_{k,l}}^\prime(\omega^*,x,E^u(\omega^*,x)) \nonumber\\ &\quad \geq -\bigg(N\cdot \partial F_n+\#(F_n\backslash I_M(\omega^*,x)) +2M\bigg\lceil \frac{n}{N}\bigg\rceil\bigg)\cdot \|\phi_1^\prime\|_{\sup}. \end{align} $$

For each $(\omega ^*,x,E^u(\omega ^*,x))\in \mathbb {A}_n^{\omega ^*}$ , define $\delta _{(\omega ^*,x,E^u(\omega ^*,x))}^{F_n}=({1}/{\#F_n})\sum _{j\in F_n}\mathbb {F}_*^k \delta _{(\omega ^*,x,E^u(\omega ^*,x))}$ . Note that (recall that $\phi ^{\prime }_{J_{k,l}}=\phi ^{\prime }_N$ )

$$ \begin{align*} \int_{\{{\phi_N^\prime}/{N}\leq \alpha\}} \frac{\phi_N^\prime}{N} \,ad\delta_{(\omega^*,x,E^u(\omega^*,x))}^{F_n} =\frac{1}{N\# F_n}\sum_{(k,l):~k+lN\in F_n^\alpha(\omega^*,x)}\phi_{J_{k,l}}^\prime(\omega^*,x,E^u(\omega^*,x)). \end{align*} $$

By (3.2), after summing over $k\in \{0,\ldots ,N-1\}$ ,

$$ \begin{align*} & \sum_{(k,l):~k+lN\in F_n^\alpha(\omega^*,x)}\phi_{J_{k,l}}^\prime(\omega^*,x,E^u(\omega^*,x))\\ &\quad \geq -N\bigg(N\cdot \partial F_n+\#(F_n\backslash I_M(\omega^*,x)) +2M\bigg\lceil \frac{n}{N}\bigg\rceil\bigg)\cdot \|\phi_1^\prime\|_{\sup}. \end{align*} $$

Hence, dividing by $N\cdot \#F_n$ , we have

$$ \begin{align*} & \int_{\{{\phi_N^\prime}/{N}\leq \alpha\}} \frac{\phi_N^\prime}{N} \,d\delta_{(\omega^*,x,E^u(\omega^*,x))}^{F_n}\\ &\quad \geq-\frac{1}{\#F_n}\bigg(N\cdot \partial F_n+\#(F_n\backslash I_M(\omega^*,x)) +2M\bigg\lceil \frac{n}{N}\bigg\rceil\bigg)\cdot \|\phi_1^\prime\|_{\sup}. \end{align*} $$

Recall that $\hat {\mu }_n$ is supported on $\mathbb {A}_n^{\omega ^*}$ . We integrate with $\hat {\mu }_n^{F_N}$ , then

$$ \begin{align*} \int_{\{{\phi_N^\prime}/{N}\leq \alpha\}} \frac{\phi_N^\prime}{N}\, d\hat{\mu}_n^{F_n} & \geq -\frac{1}{\#F_n}\bigg(N\cdot \partial F_n+\#(F_n\backslash I_M(\omega^*,x))+2M\bigg\lceil \frac{n}{N}\bigg\rceil\bigg)\cdot \|\phi_1^\prime\|_{\sup}\\ & \geq -\frac{1}{d_n(F_n)}\bigg(Nd_n(\partial F_n)+\sup_{x\in A_n^{\omega^*}}d_n(F_n\backslash I_M(\omega^*,x))\\ &\quad +\frac{M}{n}\bigg\lceil \frac{n}{N}\bigg\rceil\bigg)\|\phi_1^\prime\|_{\sup}. \end{align*} $$

Note that $\{{\phi _N^\prime }/{N}\leq \alpha \}$ is closed, so by weak $^*$ convergence, we have

$$ \begin{align*} \alpha \hat{\mu}\bigg\{\frac{\phi_N^\prime}{N}\leq \alpha\bigg\}& \geq \int_{\{{\phi_N^\prime}/{N}\leq \alpha\}} \frac{\phi_N^\prime}{N}\, d\hat{\mu} \\ & \geq -\frac{1}{\beta}\limsup_{n\in\mathfrak{n}^\prime}\bigg(\sup_{x\in A_n^{\omega^*}}d_n(F_n\backslash I_M(\omega^*,x))+\frac{M}{N}\bigg)\|\phi_1^\prime\|_{\sup}. \end{align*} $$

Note that $\alpha <0$ ; therefore, we get for all M:

$$ \begin{align*} \lim_{N\to\infty}\hat{\mu}\bigg\{\frac{\phi_N^\prime}{N}\leq \alpha\bigg\}\leq -\frac{1}{\alpha\beta}\limsup_{n\in\mathfrak{n}^\prime}\bigg(\sup_{x\in A_n^{\omega^*}}d_n(F_n\backslash I_M(\omega^*,x))\bigg)\|\phi_1^\prime\|_{\sup}. \end{align*} $$

By taking M to infinity, we arrive at $\lim _{N\to \infty }\hat {\mu }\{{\phi _N^\prime }/{N}\leq \alpha \}=0$ by Lemma 3.1. Therefore, $\lim _{N\to \infty }({\phi _N^\prime }/{N})=\lim _{N\to \infty }({\phi _N}/{N})-{\chi }/{2}\geq \alpha $ for $\hat {\mu }$ -a.e $(\omega ,x,E)\in \mathcal {G}^0(T\Omega )\, \times \, \mathcal {G}^k(T\mathcal {M})$ , where the limit exists due to the Kingman’s subadditive ergodic theorem. Since $\alpha <0$ is arbitrary, we conclude that

(3.3) $$ \begin{align} \phi_*=\lim\nolimits _{N}\dfrac{\phi_N}{N}\geq \dfrac{\chi}{2}\quad \text{for }\hat{\mu}\text{-a.e. }(\omega,x,E)\in \mathcal{G}^0(T\Omega)\, \times\, \mathcal{G}^k(T\mathcal{M}). \end{align} $$

For $(\omega ,x,E)\in \mathcal {G}^0(T\Omega )\, \times \, \mathcal {G}^{(\dim \mathcal {M}-k)}(T\mathcal {M})$ , we define $\psi _n(\omega ,x,E)= \log \|D_x f_\omega ^n|_E\|^{-1}$ . Then, $\psi _n$ is a continuous function on $\mathcal {G}^0(T\Omega )\, \times \, \mathcal {G}^{(\dim \mathcal {M}-k)}(T\mathcal {M})$ . It is clear that $ \psi _{n+m}(\omega ,x,E)\geq \psi _n(\omega ,x,E)+\psi _m(\mathbb {F}^n(\omega ,x,E)).$ Denote $\bar {E}_{\omega ^*}=\{(\omega ^*,x,E^s (\omega ^*,x)):\ x\in E_{\omega ^*}\}$ and $\bar {A}_n^{\omega ^*}=\{(\omega ^*,x,E^s(\omega ^*,x)):\ x\in A_n^{\omega ^*}\}$ . We let $\bar {\mu }_n$ be the measure on $\bar {A}_n^{\omega ^*}$ corresponding to $\mu _n$ on $A_n^{\omega ^*}$ . Note that for any $q>p\in I(\omega ^*,x)$ , then $q-p\geq T$ and $F^p(\omega ^*,x)\in \Lambda _l$ ; therefore, we have

$$ \begin{align*} \psi_{q-p}(\mathbb{F}^p(\omega^*,x,E^s(\omega^*,x))) & =\log \|D_{f_{\omega^*}^p(x)}f_{\theta^p\omega^*}^{q-p}|_{E^s(\theta^p\omega^*,f_{\omega^*}^p(x))}\|^{-1} \\ & \geq \log (C(F^p(\omega^*,x))^{-1}e^{(q-p)\chi})\geq \log(e^{-l}e^{(q-p)\chi}) \\ & \geq (q-p)\cdot \frac{\chi}{2}. \end{align*} $$

By the same proof as (3.3), for any $\mathfrak {n}^{\prime \prime }$ being a subsequence of $\mathfrak {n}^{\prime }$ such that $\{({1}/{\# F_n})\sum _{k\in F_n}\mathbb {F}_*^k\bar {\mu }_n\}_{n\in \mathfrak {n}^{\prime \prime }}$ converges to $\bar {\mu }$ , one has

(3.4) $$ \begin{align} \psi_*=\lim\nolimits _{N}\dfrac{\psi_N}{N}\geq \dfrac{\chi}{2}\quad \text{for }\bar{\mu}\text{-a.e. }(\omega,x,E)\in \mathcal{G}^0(T\Omega)\, \times\, \mathcal{G}^{\dim \mathcal{M}-k}(T\mathcal{M}). \end{align} $$

By the construction, we have

$$ \begin{align*} Pro\hspace{-1pt}j_{\Omega\, \times\, \mathcal{M}}(\hat{\mu})=Pro\hspace{-1pt}j_{\Omega\, \times\, \mathcal{M}}(\bar{\mu})=\lim_{n\in\mathfrak{n}^{\prime\prime}\to\infty}\frac{1}{\#F_n}\sum_{k\in F_n}F_*^k(\delta_{\omega^*}\, \times\,\mu_n). \end{align*} $$

We name $Pro\hspace {-1pt}j_{\Omega \, \times \, \mathcal {M}}(\hat {\mu })$ by $\mu $ , and this measure has marginal $\mathbb {P}$ on $\Omega $ since $\omega ^*$ is a generic point of $\mathbb {P}$ . We conclude from (3.3) and (3.4) that for $\mu $ -a.e. $(\omega ,x)\in \Omega \, \times \, \mathcal {M}$ , there are k positive Lyapunov exponents greater than $\chi /2$ and $\dim \mathcal {M}-k$ negative Lyapunov exponents less than $-\chi /2$ . The proof of Proposition 3.1 is complete.

Proof of Proposition 3.2.

For any $x\in E_{\omega ^*}$ , $M>0$ , and any $n\gg M$ , let $L_n(\omega ^*,x)\subset I(\omega ^*,x)\cap {}[\![ {}0,n{}[\![ {}$ be the disjoint union of $\{a,b\}$ satisfying

(3.8) $$ \begin{align} a,b\in I(\omega^*,x),\quad b-a>M\quad \text{and}\quad {}]\!]{} a,b{}[\![{}\cap I(\omega^*,x)=\emptyset. \end{align} $$

Denote $n_{(\omega ^*,x)}$ to be the number of pairs in $L_n(\omega ^*,x)$ . By (3.8), we have $n_{(\omega ^*,x)}\leq \lceil {n}/{M}\rceil $ and, therefore,

$$ \begin{align*} \# L_n(\omega^*,x)\leq \bigg\lceil \frac{n}{M}\bigg\rceil. \end{align*} $$

We consider $S_n=\{\bigsqcup _{i=1}^k\{a_i,b_i\}|\ b_i-a_i>M,\ k\leq \lceil {n}/{m}\rceil \}$ , that is, all the possibilities of $L_n(\omega ^*,x)$ . When n is sufficiently large, one has

(3.9) $$ \begin{align} \# S_n\leq \sum_{k=1}^{2\lceil{n}/{M}\rceil} \binom{n}{l} \leq e^{3n/M}. \end{align} $$

For any $s\in S_n$ , we denote $A_{n,s}^{\omega ^*}=\{z\in A_n^{\omega ^*}|\ L_n(x)=s\in S_n\}$ . Then, for any $x\in A_{n}^{\omega ^*}$ ,

(3.10) $$ \begin{align} \lambda^u(A_n^{\omega^*}\cap \mathcal{P}_{\omega^*}^{F_n}(x))\leq \sum_{s\in S_n}\lambda^u(A_{n,s}^{\omega^*}\cap \mathcal{P}_{\omega^*}^{F_n}(x)). \end{align} $$

Next, we are going to estimate the right side of the above inequality for fixed $x\in A_{n}^{\omega ^*}$ .

Recall that $I_M(\omega ^*,x)$ is the union of the intervals ${}[\![ {} c,d{}[\![ {}$ with $c,d\in I(\omega ^*,x)$ and $0<d-c\leq M$ . Fix $s=\bigsqcup _{i=1}^k\{a_i,b_i\}$ and for any $x\in A_{n,s}^{\omega ^*}$ , we let $I_M(s)=I_M(\omega ^*,x)\cap {}[\![ {} 0,n{}[\![ {}$ . By the definition of $A_{n,s}^{\omega ^*}$ , we have

$$ \begin{align*} I_M(s)={}[\![{}0,a_1{}[\![{}\cup{}[\![{} b_1,a_2{}[\![{}\cup\cdots \cup{}[\![{} b_k,n{}[\![{}. \end{align*} $$

We decompose $F_n$ into the following two parts:

• • $F_{n,s}^2:=F_n\backslash I_M(s)$ ;

• • $F_{n,s}^1:= F_n\backslash F_{n,s}^1=\bigcup _{i=0}^{m_s-1}{}[\![ {} c_i,d_i{}]\!] {}$ .

Without loss of generality, we can assume $0\in F_n$ for all $n\in \mathfrak {n}$ and, therefore, $C_0=0$ . Note that $F_{n,s}^1=F_n\cap I_M(s)$ ; therefore, for any $0\leq i\leq m_s-1$ , ${}[\![ {} c_i,d_i{}]\!] {}$ is M- $\{(\omega ^*,x):\ x\in A_{n,s}^{\omega ^*}\}$ -admissible and

(3.11) $$ \begin{align} m_s\leq \#\partial F_n+\bigg\lceil\frac{\# (F_n\backslash I_M(s))}{M}\bigg\rceil. \end{align} $$

Claim 3.1. There exists a constant $\tilde {C}>0$ such that for any $x_s\in A_{n,s}^{\omega ^*}\cap P_{\omega ^*}^{F_n}(x)$ , one has

(3.12) $$ \begin{align} \lambda^u(A_{n,s}^{\omega^*}\cap P_{\omega^*}^{F_n}(x))&\leq \lambda^u(A_{n,s}^{\omega^*} \cap P_{\omega^*}^{F_{n,s}^1}(x))\nonumber\\& \leq (C^*_{M})^{m_s}\cdot e^{m_s\gamma+2(d_{m_s-1}-d_0)\gamma}\cdot e^{m_s\tilde{C}}\cdot e^{-\sum_{i\in F_{n,s}^1} J^u(f_{\omega^*}^i(x_s))}, \end{align} $$

where $\mathcal {P}_{\omega ^*}^{F_{n,s}^1}=\bigvee _{i\in F_{n,s}^1}(f_{\omega ^*}^i)^{-1}\mathcal {P}$ .

Proof of Claim 3.1.

The first inequality is clear; therefore, we only need to prove the second inequality. For $k\in {}[\![ {} 0,m_s-1{}]\!] {}$ , we denote $F_{n,s}^1(k):=\bigcup _{i=0}^k{}[\![ {} c_i,d_i{}]\!] {}$ .

Step 1. When $k=0$ , we are going to estimate $\lambda ^u(A_{n,s}^{\omega ^*}\cap P_{\omega ^*}^{F_{n,s}^1(0)}(x))$ . Note that ${}[\![ {} c_0,d_0{}]\!] {}={}[\![ {} 0, d_0{}]\!] {}$ is M- $\{(\omega ^*,x):\ x\in A_{n,s}^{\omega ^*}\}$ -admissible. Denote $\Delta _0=A_{n,s}^{\omega ^*}\cap \mathcal {P}_{\omega ^*}^{F_{n,s}^1(0)}(x)$ . By Lemma 2.3 and diam $(\mathcal {P})<\delta $ , and note that $x_s\in A_{n,s}^{\omega ^*}\cap P_{\omega ^*}^{F_n}(x)$ , we have

$$ \begin{align*} f_{\omega^*}^{d_0}(\Delta_0)\subset f_{\omega^*}^{d_0}(W_{4\delta}^u(\omega^*,x_s)\cap B_{d_0}(\omega^*,x_s,\delta))\subset W_{4\delta}^u(\theta^{d_0}\omega^*,f_{\omega^*}^{d_0}(x_s)). \end{align*} $$

Then, we have

$$ \begin{align*} \lambda^u(\Delta_0) & \leq e^{\gamma}\lambda^u(f_{\omega^*}^{d_0}\Delta_0)|\!\det (D_xf_{\omega^*}^{d_0}|_{E^u(\omega^*,x_s)})|^{-1}\quad \text{by Lemma }2.3 (2),\\ & \leq e^\gamma\lambda^u(W_{4\delta}^u(\theta^{d_0}\omega^*,f_{\omega^*}^{d_0}(x_s)))\cdot |\!\det (D_xf_{\omega^*}^{d_0}|_{E^u(\omega^*,x_s)})|^{-1}\\ & \leq e^{\gamma}\cdot e^{-\sum_{i\in[0,d_0]}J^u(f_{\omega^*}^i(x_s))}\cdot e^{\tilde{C}}\quad\text{by Corollary }2.1. \end{align*} $$

Step 2. We consider the case of $k=1$ . Denote $\Delta _1=A_{n,s}^{\omega ^*}\cap \mathcal {P}_{\omega ^*}^{F_{n,s}^1(1)}(x)$ , then

$$ \begin{align*} f_{\omega^*}^{d_0}(\Delta_1)=f_{\omega^*}^{d_0}(\Delta_0)\cap f_{\theta^{c_1}\omega^*}^{-(c_1-d_0)}\mathcal{P}_{\theta^{c_1}\omega^*}^{{}[\![{} 0,d_1-c_1{}]\!]{}}(f_{\omega^*}^{c_1}(x)). \end{align*} $$

In addition, note that ${}[\![ {} c_1,d_1{}]\!] {}$ is M- $\{(\omega ^*,x):\ x\in A_{n,s}^{\omega ^*}\}$ -admissible; therefore, we have

$$ \begin{align*} & f_{\omega^*}^{d_0}(\Delta_1)\subset W_{4\delta}^u(\theta^{d_0}\omega^*,f_{\omega^*}^{d_0}(x_s))\cap \Lambda_l(\theta^{d_0}\omega^*)\cap f_{\theta^{c_1}\omega^*}^{-(c_1-d_0)}\Lambda_l(\theta^{c_1}\omega^*)\cap f_{\theta^{c_1}\omega^*}^{-(c_1-d_0)}\\ &\quad\, \times\, B(f_{\omega^*}^{c_1}(x_s),\delta). \end{align*} $$

Then, there exists finite points $\{y_i^1\}_{i\in I_1}\subset f_{\omega ^*}^{d_0}(\Delta _1)$ such that:

1. (1) $W_{4\delta }^u(\theta ^{c_1}\omega ^* ,f_{\theta ^{d_0}\omega ^*}^{c_1-d_0}y_i^1)$ are mutually disjoint; and

2. (2) $f_{\omega ^*}^{d_0}(\Delta _1)\subset \bigcup _{i\in I_1}f_{\theta ^{c_1}\omega ^*}^{-(c_1-d_0)}(W_{4\delta }^u(\theta ^{c_1}\omega ^*,f_{\theta ^{d_0}\omega ^*}^{c_1-d_0}(y_i^1))\cap B(f_{\omega ^*}^{c_1}x_s,\delta ))\subset W_{8\delta }^u(\theta ^{d_0}\omega ^*, f_{\omega ^*}^{d_0}(x_s))$ .

We denote $\Delta _1^i(\theta ^{c_1}\omega ^*){\kern-1pt}={\kern-1pt}f_{\omega ^*}^{c_1}(\Delta _1){\kern-1pt}\cap{\kern-1pt} W_{4\delta }^u(\theta ^{c_1}\omega ^* ,f_{\theta ^{d_0}\omega ^*}^{c_1-d_0}(y_i^1)){\kern-1pt}\cap{\kern-1pt} B(f_{\omega ^*}^{c_1}(x_s),\delta )$ for ${i{\kern-1.2pt}={\kern-1.2pt}1{\kern-1.2pt}\in{\kern-1.2pt} I_1}$ , and by the choice of $\{y_i^1\}_{i\in I_1}$ ,

$$ \begin{align*} f_{\omega^*}^{c_1}(\Delta_1)=\bigcup_{i=1}^k\Delta_1^i(\theta^{c_1}\omega) \quad \text{and}\quad\Delta_1^i(\theta^{c_1}\omega^*) \quad \text{are mutually disjoint.} \end{align*} $$

We note that ${}[\![ {} 0,d_1-c_1{}]\!] {}$ is M- $\{(\theta ^{c_1}\omega ^*,x):\ x\in \Delta _1^i(\theta ^{c_1}\omega ^*)\}$ -admissible. Applying Lemma 2.3 on $\Delta _1^i(\theta ^{c_1}\omega ^*)$ , we have

$$ \begin{align*} f_{\theta^{c_1}\omega^*}^{d_1-c_1}(\Delta_1^i(\theta^{c_1}\omega^*))& \subset f_{\theta^{c_1}\omega^*}^{d_1-c_1}(W_{4\delta}^u(\theta^{c_1}\omega^*,f_{\theta^{d_0}\omega^*}^{c_1-d_0}(y_i^1))\cap B_{d_1-c_1}(f_{\theta^{d_0}\omega^*}^{c_1-d_0}(y_i^1),\theta^{c_1}\omega^*,2\delta))\\ & \subset W_{4\delta}^u(\theta^{d_1}\omega^*,f_{\theta^{c_1}\omega^*}^{d_1-c_1}f_{\theta^{d_0}\omega^*}^{c_1-d_0}(y_i^1)) \end{align*} $$

and

(3.13) $$ \begin{align} &\lambda^u(\Delta_1^i(\theta^{c_1}\omega^*))\nonumber\\&\quad \leq e^{\gamma}\lambda^u(f_{\theta^{c_1}\omega^*}^{d_1-c_1}\Delta_1^i(\theta^{c_1}\omega^*))\cdot |\!\det D_{f_{\theta^{d_0}\omega^*}^{c_1-d_0}(y_i^1)}f^{d_1-c_1}_{\theta^{c_1}\omega^*}|_{E^u(\theta^{c_1}\omega^*,f_{\theta^{d_0}\omega^*}^{c_1-d_0}(y_i^1))}|^{-1}\nonumber\\&\quad \leq e^{\gamma}\cdot\frac{|\!\det D_{f_{\theta^{d_0}\omega^*}^{c_1-d_0}(y_i^1)}f^{d_1-c_1}_{\theta^{c_1}\omega^*}|_{E^u(\theta^{c_1}\omega^*,f_{\theta^{d_0}\omega^*}^{c_1-d_0}(y_i^1))}|^{-1}}{ |\!\det D_{f_{\omega^*}^{c_1}(x_s)}f^{d_1-c_1}_{\theta^{c_1}\omega^*}|_{E^u(\theta^{c_1}\omega^*,f_{\omega^*}^{c_1}(x_s))}|^{-1}} \nonumber\\&\qquad \cdot |\!\det D_{f_{\omega^*}^{c_1}(x_s)}f^{d_1-c_1}_{\theta^{c_1}\omega^*}|_{E^u(\theta^{c_1}\omega^*,f_{\omega^*}^{c_1}(x_s))}|^{-1}\nonumber\\&\quad \leq e^\gamma \cdot e^{(d_1-c_1)\gamma} \cdot|\!\det D_{f_{\omega^*}^{c_1}(x_s)}f^{d_1-c_1}_{\theta^{c_1}\omega^*}|_{E^u(\theta^{c_1}\omega^*,f_{\omega^*}^{c_1}(x_s))}|^{-1}, \end{align} $$

where the second inequality is due to Corollary 2.1 and the third inequality is due to Lemma 2.4. Note that $\sum _i\lambda ^u(f_{\theta ^{c_1}\omega ^*}^{-(c_1-d_0)}W_{4\delta _1}^u(\theta ^{c_1}\omega ^*,f_{\theta ^{d_0}\omega ^*}^{c_1-d_0}(y_i^1)))\leq \lambda ^u(W_{8\delta _1}^u(\theta ^{d_0}\omega ^*, f_{\omega ^*}^{d_0}(x_s))) \leq ~1$ and

(3.14) $$ \begin{align} \begin{aligned} &\lambda^u(f_{\theta^{c_1}\omega^*}^{-(c_1-d_0)}W_{4\delta_1}^u(\theta^{c_1}\omega^*,f_{\theta^{d_0}\omega^*}^{c_1-d_0}(y_i^1))) \\ &\quad = \int_{W_{4\delta_1}^u(\theta^{c_1}\omega^*,f_{\theta^{d_0}\omega^*}^{c_1-d_0}(y_i^1))}\det (D_yf_{\theta^{c_1}\omega^*}^{-(c_1-d_0)}|_{E^u(\theta^{c_1}\omega^*,y)})\,d\lambda^u(y)\\ &\quad \geq e^{-(c_1-d_0)\gamma}\cdot (\det (D_{y_i^1}f_{\theta^{f_0}\omega^*}^{c_1-d_0}|_{E^u(\theta^{d_0}\omega^*,y_i^1)}))^{-1}\cdot \lambda^u(W_{4\delta_1}^u(\theta^{c_1}\omega^*,f_{\theta^{d_0}\omega^*}^{c_1-d_0}(y_i^1)))\\ &\quad \geq e^{-(c_1-d_0)\gamma}\cdot (\det (D_{y_i^1}f_{\theta^{f_0}\omega^*}^{c_1-d_0}|_{E^u(\theta^{d_0}\omega^*,y_i^1)}))^{-1}\cdot (C^*_{M})^{-1}, \end{aligned} \end{align} $$

where the first inequality is due to Lemma 2.4 and the second inequality is due to (3.7). The inequality (3.14) enables us to estimate the sum of determinants in the gaps of $F_{n,s}^1(1)$ , that is,

(3.15) $$ \begin{align} & \sum_i\bigg(\det (D_{y_i^1}f_{\theta^{f_0}\omega^*}^{c_1-d_0}|_{E^u(\theta^{d_0}\omega^*,y_i^1)})\bigg)^{-1} \nonumber\\&\quad \leq C^*_{M}e^{(c_1-d_0)\gamma} \sum_i\lambda^u(f_{\theta^{c_1}\omega^*}^{-(c_1-d_0)}W_{4\delta}^u(\theta^{c_1}\omega^*,f_{\theta^{d_0}\omega^*}^{c_1-d_0}(y_i^1)))\nonumber\\&\quad \leq C^*_{M}e^{(c_1-d_0)\gamma} \end{align} $$

Then, we have estimates

$$ \begin{align*} \lambda^u(f_{\omega^*}^{d_0}(\Delta_1))& = \sum_{i}\lambda^u(f_{\theta^{c_1}\omega^*}^{-(c_1-d_0)}\Delta_1^i(\theta^{c_1}\omega^*))\\ & \leq \sum_{i}e^{(c_1-d_0)\gamma}\cdot (\det (D_{y_i^1}f_{\theta^{d_0}\omega^*}^{c_1-d_0}|_{E^u(\theta^{d_0}\omega^*,y_i^1)}))^{-1}\cdot\lambda^u(\Delta_1^i(\theta^{c_1}\omega^*))\\\text{by }(3.13), & \leq e^{\gamma+(d_1-d_0)\gamma}|\!\det D_{f_{\omega^*}^{c_1}(x_s)}f^{d_1-c_1}_{\theta^{c_1}\omega^*}|_{E^u(\theta^{c_1}\omega^*,f_{\omega^*}^{c_1}(x_s))}|^{-1}\cdot \\ & \quad\times \sum_{i} (\det (D_{y_i^1}f_{\theta^{f_0}\omega^*}^{c_1-d_0}|_{E^u(\theta^{d_0}\omega^*,y_i^1)}))^{-1}\\\text{by }(3.15), & \leq e^{\gamma+(d_1-d_0)\gamma} |\!\det D_{f_{\omega^*}^{c_1}(x_s)}f^{d_1-c_1}_{\theta^{c_1}\omega^*}|_{E^u(\theta^{c_1}\omega^*,f_{\omega^*}^{c_1}(x_s))}|^{-1}C^*_{M}\cdot e^{(c_1-d_0)\gamma}. \end{align*} $$

Then, finally, we have

(3.16) $$ \begin{align} \lambda^u(\Delta_1) & \leq e^{\gamma}\lambda^u(f_{\omega^*}^{d_0}\Delta_1)|\!\det (D_xf_{\omega^*}^{d_0}|_{E^u(\omega^*,x_s)})|^{-1}\quad \text{ by Lemma }2.3(2)\nonumber\\& \leq e^{2\gamma+2(d_1-d_0)\gamma}\cdot C^*_{M}\cdot e^{-\sum_{i\in F_{n,s}^1(1)}J^u(f_{\omega^*}^i(x_s))}\cdot e^{2\tilde{C}}. \end{align} $$

Step 3. We consider the case $k=2$ . Denote $\Delta _2=A_{n,s}^{\omega ^*}\cap \mathcal {P}_{\omega ^*}^{F_{n,s}^1(2)}(x)=\Delta _1\cap \mathcal {P}_{\omega ^*}^{{}[\![ {} c_2,d_2{}]\!] {}}(x)$ . Notice that ${}[\![ {} c_2,d_2{}]\!] {}$ is M- $\{(\omega ^*,x):\ x\in A_{n,s}^{\omega ^*}\}$ -admissible, and

$$ \begin{align*} f_{\omega^*}^{d_1}(\Delta_2)\subset\bigcup_{i\in I_1}( f_{\theta^{c_1}\omega^*}^{d_1-c_1}(\Delta_1^j(\theta^{c_1}\omega^*))\cap f_{\theta^{c_2}\omega}^{-(c_2-d_1)\mathcal{P}_{\theta^{c_2}\omega^*}^{{}[\![{} 0,d_2-c_2{}]\!]{}}}(f_{\omega^*}^{c_2}(x))), \end{align*} $$

then $\text {for all } j\in I_1$ , there exist finite points $\{z_i^{j,2}\}_{i\in I_{j,2}}\subset f_{\omega ^*}^{d_0}\Delta _2$ such that $\{W_{4\delta }^u(\theta ^{c_2}\omega ^*, f_{\theta ^{d_0}\omega ^*}^{c_2-d_0}(z_i^{j,2}))\}_{i\in I_{j,2}}$ are mutually disjoint, and

$$ \begin{align*} f_{\theta^{c_1}\omega^*}^{d_1-c_1}(\Delta_1^j(\theta^{c_1}\omega^*))&\subset \bigcup_{i\in I_{j,2}} f_{\theta^{c_2}\omega^*}^{-(c_2-d_1)}(W_{4\delta}^u(\theta^{c_2}\omega^*,f_{\theta^{d_0}\omega^*}^{c_2-d_0}(z_i^{j,2}))\cap B(f_{\omega^*}^{c_2}(x_s),\delta))\\ & \subset W_{8\delta}^{u}(\theta^{d_1}\omega^*,f_{\theta^{d_0}\omega^*}^{d_1-d_0}(y_j^1)). \end{align*} $$

For $j\in I_1$ and $i\in I_{j,2}$ , we denote

$$ \begin{align*} \Delta_1^{j,i}(\theta^{c_1}\omega^*)=f_{\omega^*}^{c_1}(\Delta_2)\cap f_{\theta^{c_2}\omega^*}^{-(c_2-d_1)}(W_{4\delta}^u(\theta^{c_2}\omega^*,f_{\theta^{d_0}\omega^*}^{c_2-d_0}(z_i^{j,2}))\cap B(f_{\omega^*}^{c_2}(x_s),\delta)). \end{align*} $$

Now, $\bigcup _{i\in I_{j,2}}\Delta _1^{j,i}(\theta ^{c_1}\omega ^*)$ is in the position of $\Delta _1$ in Step 2. By the similar estimates, we obtain

(3.17) $$ \begin{align} \lambda^u\bigg( \bigcup_{i\in I_{j,2}} \Delta_1^{j,i}(\theta^{c_1}\omega^*)\bigg)\leq e^{2\gamma+2(d_2-c_1)\gamma}\cdot C^*_{M}\cdot e^{-\sum_{i\in{}[\![{} c_1,d_1{}]\!]{}\cup{}[\![{} c_2,d_2{}]\!]{}}J^u(f_{\omega^*}^i(x_s))}\cdot e^{2\tilde{C}}. \end{align} $$

Then,

$$ \begin{align*} \begin{aligned} & \lambda^u(f_{\omega^*}^{d_0}(\Delta_2))\\ &\quad =\sum_{j\in I_1}\lambda^u\bigg(f_{\theta^{c_1}\omega^*}^{-(c_1-d_0)}\bigg(\bigcup_{i\in I_{j,2}}\Delta_{1}^{j,i}(\theta^{c_1}\omega^*)\bigg)\bigg)\\ &\quad \leq \sum_{j\in I_1}e^{(c_1-d_0)\gamma}\cdot (\det (D_{y_j^1}f_{\theta^{d_0}\omega^*}^{c_1-d_0}|_{E^u(\theta^{d_0}\omega^*,y_j^1)}))^{-1}\cdot\lambda^u\bigg(\bigcup_{i\in I_{j,2}}\Delta_1^{j,i}(\theta^{c_1}\omega^*)\bigg)\\ &\quad \leq (C^*_{M})^2\cdot e^{2\gamma+2(d_2-d_0)\gamma}\cdot e^{2\tilde{C}}\cdot e^{-\sum_{i\in{}[\![{} c_1,d_1{}]\!]{}\cup{}[\![{} c_2,d_2{}]\!]{}}J^u(f_{\omega^*}^i(x_s))}\quad \text{by }(3.15)\text{ and }(3.17). \end{aligned} \end{align*} $$

Similar as (3.16), we obtain

$$ \begin{align*} \begin{aligned} \lambda^u(\Delta_2) & \leq e^{\gamma}\lambda^u(f_{\omega^*}^{d_0}\Delta_2)|\!\det (D_xf_{\omega^*}^{d_0}|_{E^u(\omega^*,x_s)})|^{-1}\quad\text{by Lemma }2.3(2)\\ & \leq (C^*_{M})^2\cdot e^{3\gamma+2(d_2-d_0)\gamma}\cdot e^{3\tilde{C}}\cdot e^{-\sum_{i\in F_{n,s}^1(2)}J^u(f_{\omega^*}^i(x_s))}. \end{aligned} \end{align*} $$

Repeating the above procedure, we can obtain

$$ \begin{align*} \lambda^u(A_{n,s}^{\omega^*}\cap P_{\omega^*}^{F_{n,s}^1}(x))\leq (C^*_{M})^{m_s}\cdot e^{m_s\gamma+2(d_{m_s}-d_0)\gamma}\cdot e^{m_s\tilde{C}}\cdot e^{-\sum_{i\in F_{n,s}^1} J^u(f_{\omega^*}^i(x_s))} \end{align*} $$

The proof of Claim 3.1 is complete.

Note that for $x\in A_n^{\omega ^*}$ , $I_M(\omega ^*,x)$ may not coincide with $I_M(s)$ . For the $x_s$ in the statement of Claim 3.1 and note that $x_s\in A_{n,s}^{\omega ^*}\cap P_{\omega ^*}^{F_n}(x)$ , moreover for those $i\in I_M(\omega ^*,x)\cap I_M(\omega ^*,x_s)$ , one has $(\theta ^i\omega ^*,f_{\omega ^*}^i(x))$ and $(\theta ^i\omega ^*,f_{\omega ^*}^i(x_s))$ both belong to $\Lambda _{le^{M\epsilon }}$ . Then, by Lemma 2.4, for those $i\in I_M(\omega ^*,x)\cap I_M(\omega ^*,x_s)$ , one has

$$ \begin{align*} \bigg|\log\frac{\det D_{f_{\omega^*}^i(x)}f_{\theta^i\omega^*}|_{E^u(\theta^i\omega^*,f_{\omega^*}^i(x))}}{\det D_{f_{\omega^*}^i(x_s)}f_{\theta^i\omega^*}|_{E^u(\theta^i\omega^*,f_{\omega^*}^i(x_s))}}\bigg|\leq \gamma. \end{align*} $$

For those i such that it may not satisfy the above estimate, we also have estimates

(3.18) $$ \begin{align} \#F_n\backslash(I_M(\omega^*,x)\cap I_M(\omega^*,x_s))\leq 2\sup_{y\in A_{n}^{\omega^*}}F_n\backslash I_M(\omega^*,y). \end{align} $$

Combining the above two cases, we obtain

(3.19) $$ \begin{align} \bigg|\log\prod_{i\in F_n}\frac{\det D_{f_{\omega^*}^i(x)}f_{\theta^i\omega^*}|_{E^u(\theta^i\omega^*,f_{\omega^*}^i(x))}}{\det D_{f_{\omega^*}^i(x_s)}f_{\theta^i\omega^*}|_{E^u(\theta^i\omega^*,f_{\omega^*}^i(x_s))}}\bigg|\leq n\gamma+2\tilde{C}\sup_{y\in A_{n}^{\omega^*} }F_n\backslash I_M(\omega^*,y). \end{align} $$

Therefore, for $x\in A_{n}^{\omega ^*}$ and $x_s\in A_{n,s}^{\omega ^*}\cap P_{\omega ^*}^{F_n}(x)$ , we have

$$ \begin{align*} & \lambda^u(A_{n,s}^{\omega^*}\cap P_{\omega^*}^{F_n}(x)) \\ &\quad \leq (C^*_{M}\cdot e^{\tilde{C}}\cdot e^{\gamma})^{m_s}\cdot e^{2n\gamma} \cdot e^{-\sum_{i\in F_{n,s}^1} J^u(f_{\omega^*}^i(x_s))}\quad\text{by Claim }3.1\\ &\quad \leq (C^*_{M}\cdot e^{\tilde{C}}\cdot e^{\gamma})^{m_s}\cdot e^{2n\gamma} \cdot e^{\tilde{C}\cdot \#(F_n\backslash I_M(s))}\cdot e^{-\sum_{i\in F_{n}} J^u(f_{\omega^*}^i(x_s))}\\ &\quad \leq (C^*_{M}\cdot e^{\tilde{C}}\cdot e^{\gamma})^{\# \partial F_n+\lceil({\#(F_n\backslash I_M(s))}/{M})\rceil}\cdot e^{2n\gamma} \cdot e^{\tilde{C}\cdot \#(F_n\backslash I_M(s))}\\ &\qquad \times e^{-\sum_{i\in F_{n}} J^u(f_{\omega^*}^i(x_s))}\quad\text{by }(3.11)\\&\quad \leq (C^*_{M}\cdot e^{\tilde{C}}\cdot e^{\gamma})^{\# \partial F_n+\lceil({\#(F_n\backslash I_M(s))}/{M})\rceil}\cdot e^{2n\gamma} \cdot e^{\tilde{C}\cdot \#(F_n\backslash I_M(s))}\cdot e^{-\sum_{i\in F_{n}} J^u(f_{\omega^*}^i(x))}\\&\qquad\times \cdot e^{n\gamma+2\tilde{C}\sup_{y\in A_n^{\omega^{*}}}F_n\backslash I_M(\omega^{*},y)}\quad\text{by }(3.19)\\&\quad \leq (C^*_{M}\cdot e^{\tilde{C}}\cdot e^{\gamma})^{\# \partial F_n+\sup_{y\in A_{n}^{\omega^*}}\lceil({\#(F_n\backslash I_M(\omega^*,y))}/{M})\rceil}\cdot e^{2n\gamma} \cdot e^{\tilde{C}\cdot \sup_{y\in A_{n}^{\omega^*}}\#(F_n\backslash I_M(\omega^*,y))}\\ &\qquad \times e^{-\sum_{i\in F_{n}} J^u(f_{\omega^*}^i(x))}\cdot e^{n\gamma+2\tilde{C}\sup_{y\in A_n^{\omega^*}}F_n\backslash I_M(\omega^*,y)}. \end{align*} $$

By (3.10) and (3.9), when $n\gg M$ , we have

$$ \begin{align*} \begin{aligned} & \lambda^u(A_{n}^{\omega^*}\cap P_{\omega^*}^{F_n}(x)) \leq (C^*_{M}\cdot e^{\tilde{C}}\cdot e^{\gamma})^{\# \partial F_n+\sup_{y\in A_{n}^{\omega^*}}\lceil({\#(F_n\backslash I_M(\omega^*,y))}/{M})\rceil}\cdot e^{2n\gamma} \\ &\quad \times e^{\tilde{C}\cdot \sup_{y\in A_{n}^{\omega^*}}\#(F_n\backslash I_M(\omega^*,y))}\\ &\quad\times e^{-\sum_{i\in F_{n}} J^u(f_{\omega^*}^i(x))}\cdot e^{n\gamma+2\tilde{C}\sup_{y\in A_n^{\omega^*}}F_n\backslash I_M(\omega^*,y)}\cdot e^{3n/M}. \end{aligned} \end{align*} $$

The proof of Proposition 3.2 is complete.

Proof of Lemma 4.1.

Denote $\Omega _0\subset \Omega $ to be the Oseledets full measure set relative to the system $(\Omega ,\theta ,\mathbb {P})$ . The following proof is processing in the set $\Omega _0\, \times \, \mathcal {M}$ . We first prove condition (1). We observe that

$$ \begin{align*} D_{(\omega,x)}F^n=\begin{pmatrix} D_\omega\theta ^n & 0 \\ C_n(\omega,x) & D_xf_\omega^n \end{pmatrix}, \end{align*} $$

where

$$ \begin{align*} C_n(\omega,x)=\sum_{j=0}^{n-1}D_{f_\omega^{j+1}(x)}f_{\theta^{j+1}\omega}^{n-1-j}C(F^j(\omega,x))D_\omega\theta^j. \end{align*} $$

By the right side of assumption (1.4), we have

(4.3) $$ \begin{align} \limsup_{n}\frac{1}{n}\log \|D_{(\omega,x)}F^n(u,0)^T\| & \geq \limsup_{n}\frac{1}{n}\log \|D_\omega\theta^n u\|\geq \lambda_{\theta,\min}^+ \nonumber\\& \geq \limsup_{n}\frac{1}{n}\log \|D_xf_\omega^nv\|\quad \text{by }(1.4)\nonumber\\&= \limsup_{n}\frac{1}{n}\log \|D_{(\omega,x)}F^n(0,v)^T\|, \end{align} $$

and, similarly,

(4.4) $$ \begin{align} & \limsup_{n}\frac{1}{n}\log \|D_{(\omega,x)}F^n(u,v)^T\| \nonumber\\ & \quad\geq \limsup_{n}\frac{1}{n}\log \|D_\omega\theta^n u\| \geq \limsup_{n}\frac{1}{n}\log \|D_{(\omega,x)}F^n(0,v)^T\|. \end{align} $$

Therefore, we have

$$ \begin{align*} \begin{aligned} & \limsup_{n}\frac{1}{n}\log \|D_{(\omega,x)}F^n(u,v)^T\| \\ &\quad \leq \limsup_{n}\frac{1}{n}\log (\|D_{(\omega,x)}F^n(u,0)^T\|+\|D_{(\omega,x)}F^n(0,v)^T\|)\\ &\quad =\max\bigg\{\!\limsup_{n}\frac{1}{n}\log \|D_{(\omega,x)}F^n(u,0)^T\|,\limsup_{n}\frac{1}{n}\log \|D_{(\omega,x)}F^n(0,v)^T\|\bigg\} \\ &\quad = \limsup_{n}\frac{1}{n}\log \|D_{(\omega,x)}F^n(u,0)^T\|\quad\text{by }(4.3), \end{aligned} \end{align*} $$

and

$$ \begin{align*} \begin{aligned} & \limsup_{n}\frac{1}{n}\log \|D_{(\omega,x)}F^n(u,0)^T\| \\ &\quad \leq \limsup_{n}\frac{1}{n}\log (\|D_{(\omega,x)}F^n(u,v)^T\|+\|D_{(\omega,x)}F^n(0,v)^T\|)\\ &\quad = \max\bigg\{\!\limsup_{n}\frac{1}{n}\log \|D_{(\omega,x)}F^n(u,v)^T\|,\limsup_{n}\frac{1}{n}\log (\|D_{(\omega,x)}F^n(0,v)^T\|)\bigg\}\\ &\quad =\limsup_{n}\frac{1}{n}\log \|D_{(\omega,x)}F^n(u,v)^T\|\quad\text{by }(4.4). \end{aligned} \end{align*} $$

Consequently, we obtain

(4.5) $$ \begin{align} \limsup_{n}\frac{1}{n}\log \|D_{(\omega,x)}F^n(u,0)^T\| =\limsup_{n}\frac{1}{n}\log \|D_{(\omega,x)}F^n(u,v)^T\|. \end{align} $$

Now, we need to estimate $\limsup _{n}({1}/{n})\log \|C_n(\omega ,x)u\|.$ Note that for $P(H^u(\omega ))u\not =0$ , the following limit exists $\lambda _\theta (u):=\limsup _{n}({1}/{n})\log \|D_\omega \theta ^n u\|$ . Therefore, for any $\epsilon>0$ , there exists $b_\epsilon>0$ such that

(4.6) $$ \begin{align} \|D_\omega\theta^ju\|\leq b_\epsilon e^{j(\lambda_\theta(u)+\epsilon)}\quad\text{for all }j\in\mathbb{Z}^+. \end{align} $$

Note that F is a $C^2$ diffeomorphism; therefore, $\|C(F^j(\omega ,x))\|\leq C_*$ for some constant $C_*>0$ only depending on the system. By (4.1), one has $\|D_{f_\omega ^{j+1}(x)}f_{\theta ^{j+1}\omega }^{n-1-j}\|\leq e^{(n-1-j)M_f}.$ These estimates give

$$ \begin{align*} \begin{aligned} \|C_n(\omega,x)\| & \leq \sum_{j=0}^{n-1}e^{(n-1-j)M_f}\cdot C_*\cdot b_\epsilon e^{j(\lambda_\theta(u)+\epsilon)}\leq \sum_{j=0}^{n-1}e^{(n-1-j)\lambda_\theta(u)}\cdot C_*\cdot b_\epsilon e^{j(\lambda_\theta(u)+\epsilon)}\\ & \leq \frac{C_*b_\epsilon}{e^{\epsilon}-1}e^{(\lambda_{\theta}(u)+\epsilon)n}. \end{aligned} \end{align*} $$

Therefore, we have estimate $\limsup _{n}({1}/{n})\log \|C_n(\omega ,x)u\|\leq \lambda _\theta (u)+\epsilon $ . Note that

$$ \begin{align*}\|D_{(\omega,x)}F^n(u,0)^T\|^2=\|D_\omega\theta^nu\|^2+\|C_n(\omega,x)u\|^2,\end{align*} $$

and we obtain

$$ \begin{align*} \lambda_\theta(u):=\limsup_{n}\frac{1}{n}\log \|D_\omega\theta^n u\|\leq \limsup_n\frac{1}{n}\log\|D_{(\omega,x)}F^n(u,0)^T\|\leq \lambda_\theta(u)+\epsilon. \end{align*} $$

Since $\epsilon>0$ is arbitrary, we arrive at $\limsup _{n}({1}/{n})\log \|D_\omega \theta ^n u\|= \limsup _n({1}/{n})\log \|D_{(\omega ,x)}F^n(u,0)^T\|$ . Combining with (4.5), we complete the proof of condition (1).

The proof of condition (2) is similar to the proof of condition (1). Now, we assume $P(H^s(\omega ))u\not = 0$ . We observe that

$$ \begin{align*} D_{(\omega,x)}F^{-n}=\begin{pmatrix} D_\omega\theta ^{-n} & 0 \\ C_{-n}(\omega,x) & D_xf_\omega^{-n} \end{pmatrix}\!, \end{align*} $$

where

$$ \begin{align*} C_{-n}(\omega,x)=\sum_{j=0}^{n-1}D_{f_\omega^{-(j+1)}(x)}f_{\theta^{-(j+1)\omega}}^{-(n-1-j)}\tilde{C}(F^{-j}(\omega,x))D_\omega\theta^{-j}. \end{align*} $$

By the left side of assumption (1.4), we have

(4.7) $$ \begin{align} \limsup_n\frac{1}{n}\log\|D_{(\omega,x)}F^{-n}(u,v)^T\| & \geq \limsup_n\frac{1}{n}\log\|D_\omega\theta^{-n}u\|\geq -\lambda_{\theta,\max}^- \nonumber\\& \geq \limsup_n\frac{1}{n}\log \|D_xf_\omega^{-n}v\|\quad\text{by }(1.4)\nonumber\\& =\limsup_n\frac{1}{n}\log\|D_{(\omega,x)}F^{-n}(0,v)^T\|, \end{align} $$

and, similarly,

(4.8) $$ \begin{align} & \limsup_n\frac{1}{n}\log\|D_{(\omega,x)}F^{-n}(u,0)^T\| \nonumber\\ &\quad \geq \limsup_n\frac{1}{n}\log\|D_\omega\theta^{-n}u\| \geq \limsup_n\frac{1}{n}\log\|D_\omega\theta^{-n}u\|. \end{align} $$

Therefore,

$$ \begin{align*} \begin{aligned} &\limsup_n\frac{1}{n}\log\|D_{(\omega,x)}F^{-n}(u,v)^T\|\\ &\quad \leq \limsup_n\frac{1}{n}\log(\|D_{(\omega,x)}F^{-n}(u,0)^T\|+\|D_{(\omega,x)}F^{-n}(0,v)^T\|) \\ &\quad = \max\bigg\{\!\limsup_n\frac{1}{n}\log\|D_{(\omega,x)}F^{-n}(u,0)^T\|,\limsup_n\frac{1}{n}\log(\|D_{(\omega,x)}F^{-n}(0,v)^T\|)\bigg\}\\ &\quad =\limsup_n\frac{1}{n}\log\|D_{(\omega,x)}F^{-n}(u,0)^T\|\quad\text{by }(4.8) \end{aligned} \end{align*} $$

and

$$ \begin{align*} \begin{aligned} & \limsup_n\frac{1}{n}\log\|D_{(\omega,x)}F^{-n}(u,0)^T\|\\ &\quad \leq \limsup_n\frac{1}{n}\log(\|D_{(\omega,x)}F^{-n}(u,v)^T\|+\|D_{(\omega,x)}F^{-n}(0,v)^T\|)\\ &\quad =\max \bigg\{\!\limsup_n\frac{1}{n}\log\|D_{(\omega,x)}F^{-n}(u,v)^T\|,\limsup_n\frac{1}{n}\log\|D_{(\omega,x)}F^{-n}(0,v)^T\|\bigg\}\\ &\quad =\limsup_n\frac{1}{n}\log\|D_{(\omega,x)}F^{-n}(u,v)^T\|\quad\text{by }(4.7). \end{aligned} \end{align*} $$

Consequently, we have

(4.9) $$ \begin{align} \limsup_n\frac{1}{n}\log\|D_{(\omega,x)}F^{-n}(u,0)^T\|=\limsup_n\frac{1}{n}\log\|D_{(\omega,x)}F^{-n}(u,v)^T\|. \end{align} $$

Now, we are going to estimate $\limsup _n({1}/{n})\log \|\tilde {C}_{-n}(\omega ,x)u\|$ . Since $P(H^s(\omega ))u\not =0$ , the following limit exists:

$$ \begin{align*} \lambda_{\theta,-}(u):=\limsup_n\frac{1}{n}\log\|D_\omega\theta^{-n}u\|. \end{align*} $$

For any $\epsilon>0$ , there exists $b_\epsilon ^\prime>0$ such that $\|D_\omega \theta ^{-j}u\|\leq b_\epsilon ^\prime e^{(\lambda _{\theta ,-}(u)+\epsilon )j}$ for any $j\geq 0$ . Note that since F is a $C^2$ diffeomorphism, there exists $C_*^\prime>0$ only depending on the system such that $\|\tilde {C}(F^{-j}(\omega ,x))\|\leq C_*^\prime $ for any $j\geq 0$ . By (4.1), for any $0\leq j\leq n-1$ , we have

$$ \begin{align*} \|D_{f_\omega^{-(j+1)}(x)}f_{\theta^{-(j+1)\omega}}^{-(n-1-j)}\|\leq e^{-m_f(n-1-j)}\leq e^{\lambda_{\theta,-}(u)(n-1-j)}. \end{align*} $$

These estimates give

$$ \begin{align*} \|C_{-n}(\omega,x)u\|\leq \sum_{j=0}^{n-1}e^{\lambda_{\theta,-}(u)(n-1-j)}\cdot C_*^\prime \cdot b_\epsilon^\prime e^{(\lambda_{\theta,-}(u)+\epsilon)j}\leq \frac{C_*^\prime b_\epsilon^\prime}{e^\epsilon-1}e^{n(\lambda_{\theta,-}(u)+\epsilon)}. \end{align*} $$

Therefore, we have estimate $\limsup _n({1}/{n})\log \|C_{-n}(\omega ,x)u\|\leq \lambda _{\theta ,-}(u)+\epsilon $ . Note that

$$ \begin{align*}\|D_{(\omega,x)}F^{-n}(u,0)^T\|^2=\|D_\omega\theta^{-n}u\|^2+\|C_{-n}(\omega,x)u\|^2,\end{align*} $$

so we obtain

$$ \begin{align*} \lambda_{\theta,-}(u)& :=\limsup_{n}\frac{1}{n}\log \|D_\omega\theta^{-n} u\|\\ & \leq \limsup_n\frac{1}{n}\log\|D_{(\omega,x)}F^{-n}(u,0)^T\|\leq \lambda_{\theta,-}(u)+\epsilon. \end{align*} $$

Since $\epsilon>0$ is arbitrary, we arrive at $\limsup _{n}({1}/{n})\log \|D_\omega \theta ^{-n} u\|= \limsup _n({1}/{n})\log \|D_{(\omega ,x)}F^{-n}(u,0)^T\|$ . Combining with (4.9), we complete the proof of condition (2).