Proof. Since $A(x)=A(y)$ for every $x,y\in X$ with $d(x,y)<\eta $ , we get $|A|_{\mathcal {C}^\alpha ,\eta }=0$ for every $\alpha>0$ . If $d_{\mathcal {C}^{\alpha }}(A,B)<\epsilon $ , we have $d_{\mathcal {C}^{\alpha },\eta }(A,B)<\epsilon $ . Therefore,

$$ \begin{align*}|B|_{\mathcal{C}^{\alpha},\eta}\leq |A-B|_{\mathcal{C}^{\alpha},\eta}+|A|_{\mathcal{C}^{\alpha},\eta}\leq \|A-B\|_{\mathcal{C}^{\alpha},\eta}<\epsilon.\end{align*} $$

This finishes the proof of Lemma.

Proof. Part (i): Since $A_{n-m}(f^m(x))=A_n(x)A_m(x)^{-1}$ ,

$$\begin{align*}\|A_{n-m}(f^m(x))\|.\|A_{n-m}(f^m(x))^{-1}\|\leq \|A_n(x)\|.\|A_m(x)^{-1}\|.\|A_n(x)^{-1}\|.\|A_m(x)\|\leq \kappa^2.\end{align*}$$

Part (ii) is a consequence of continuity of $A_n$ ; and Part (iii) follows immediately from part (i).

To prove part (iv), fix an element of $\omega (x)$ , say y and $n\in \mathbb {Z}$ . Then, there exists a sequence $n_1<n_2<\cdots $ of positive integers such that

(3.3) $$ \begin{align} \lim\limits_{j\to\infty}f^{n_j}(x)=y. \end{align} $$

Now, for sufficiently large j, $n_j+n>0$ and for such j’s, using Part (i), we get $\|A_n(f^{n_j}(x))\|.\|A_n(f^{n_j}(x))^{-1}\|\leq \kappa ^2$ . In view of (3.3), and since $A_n$ and $A_n^{-1}$ are both continuous, we conclude that $\|A_n(y)\|.\|A_n(y)^{-1}\|\leq \kappa ^2$ .

Proof. If $d>2$ and $x_0$ is a periodic point, then the conclusion follows from Proposition A.3 applied to $X=\mathcal {O}(x_0)$ , and Lemma A.4.

Now, let $x_0$ be a non-periodic point and let $d\geq 2$ . For every $k\geq 1$ , define

$$\begin{align*}\mathcal{B}_\kappa:= \big\{B:X \to \mathrm{SL}(d,\mathbb{R}): \sup\limits_{n\in \mathbb{N}} \|B_n(x_0)\| \leq \kappa\big\}.\end{align*}$$

It is clear that for every $k\geq 1$ , $\mathcal {B}_\kappa $ is closed. On the other hand, the set of continuous maps $B:X\to \mathrm {SL}(d,\mathbb {R})$ , for which $\{\|B_n(x_0)\|\}_{n\in \mathbb {N}}$ is unbounded, equals to $\bigcap _{\kappa \in \mathbb {N}} \mathcal {B}_\kappa ^c$ .

So, it suffices to prove that for every $\kappa \geq 1$ , the $\mathcal {C}^0$ -interior of $\mathcal {B}_\kappa $ is empty. This implies the lemma as the set of $\mathrm {SL}(d,\mathbb {R})$ cocycles over f is a Baire space.

Claim. The interior of $\mathcal {B}_\kappa $ is empty.

Let $\epsilon _0>0$ and $B \in \mathcal {B}_\kappa $ . For every $r>1$ , let $D_r:=\mathrm {Diag}(r,r^{-\frac {1}{d-1}},\ldots ,r^{-\frac {1}{d-1}})$ be a diagonal matrix and for every vector $v \in \mathbb {R}^d$ choose $R_v \in \mathrm {SO}(d,\mathbb {R})$ such that $R_v(e_1)={v}/{|v|}$ , where $(e_1,\ldots ,e_d)$ is the standard basis for $\mathbb {R}^d$ .

Consider an arbitrary unit vector $v_0 \in \mathbb {R}^d$ and $n \in \mathbb {N}$ . Now, for every $i=1,2,\ldots ,n$ , denote $v_i:= B_i(x_0)(B_n(x_0))^{-1}v_0$ and define

$$\begin{align*}\hat{B}(f^i(x_0))=R_{v_i}D_rR^{-1}_{v_i}B(f^i(x_0)).\end{align*}$$

Notice that as $x_0$ is not periodic, $\hat {B}$ is well-defined (for every n). Since $v_n=v_0$ , then $\hat {B}_n(x_0)(B_n(x_0))^{-1}v_0=r^nv_0$ . Consequently,

(3.4) $$ \begin{align} \| \hat{B}_n(x_0)\| \geq \frac{r^n}{\|B_n(x_0)^{-1}\|}\geq \frac{r^n}{\|B_n(x_0)\|^{d-1}}\geq r^n \kappa_0^{-(d-1)}. \end{align} $$

To complete the proof of claim, let $\delta $ be the positive number given by Lemma A.4 so that every $\delta $ -perturbation of cocycle B on a compact set can be extended to a cocycle on $X \epsilon _0$ -close to B.

Now, take $r>1$ close enough to $1$ , so that $\|D_r-\mathrm {Id}\|.\|B\|<\delta $ . Moreover, we can take n large enough to have $r^n \kappa _0^{-(d-1)}> k$ . We take K in Lemma A.4 equal to $\{x_0,\ldots , f^{n-1}(x_0)\}$ . According to Lemma A.4, there exist $\tilde {B}:X\to \mathrm {SL}(d,\mathbb {R})$ , with $d_{\mathcal {C}^0}(\tilde {B},B)<\epsilon _0$ and $\tilde {B}(f^i(x_0))=\hat {B}(f^i(x_0))$ . Moreover, (3.4) implies that $\tilde {B} \notin \mathcal {B}_\kappa $ and the proof of claim and lemma is complete.

Proof. Observe that for every $A:X \to \mathrm {SL}(d,\mathbb {R})$ , if $\mathrm {QC}(A,\kappa )\neq \emptyset $ for some $\kappa \geq 1$ , then, using Lemma 3.2, and minimality of f, $\mathrm {QC}(A,\kappa ^2)=X$ .

This implies that if there exists a $\mathcal {C}^0$ -open set of $\mathrm {SL}(d,\mathbb {R})$ cocycles over f which admit a quasi-conformal orbit, then for every $x\in X$ , the set of cocycles with bounded forward orbit at x has non-empty $\mathcal {C}^0$ -interior. But due to Lemma 3.5, this is impossible.

Proof. If such a $\delta $ did not exist, then for every positive integer k, we would have $\Lambda _k := \overline {\mathcal {U}} \setminus \big ( \mathcal {G}^{-1}\mathcal {U}_{(\frac {1}{k})}\big ) \neq \emptyset $ . Since $\{\Lambda _k\}_{k=1}^\infty $ is a nested sequence of non-empty compact sets, it follows that

$$\begin{align*}\bigcap\limits_{k=1}^\infty \Lambda_k=\overline{\mathcal{U}} \setminus\big( \mathcal{G}^{-1}\mathcal{U}\big)\neq \emptyset,\end{align*}$$

which contradicts the assumption $\overline {\mathcal {U}}\subset \mathcal {G}^{-1}\mathcal {U}$ .

Proof. Since $\overline {\mathcal {U}}$ is compact and it is covered by the union $\bigcup _{H\in \mathcal {H}} {H}^{-1}\mathcal {U}$ of open sets, it can be covered by the union of finitely many of them. So, $\bigcup _{H\in \mathcal {F}} {H}^{-1}\mathcal {U}$ , for some finite family $\mathcal {F}\subset \mathcal {H}$ . We may assume that $\mathcal {F}$ is the set of all words of length at most $N\in \mathbb {N}$ . By Lemma 4.4 there exists $\delta>0$ such that $\overline {\mathcal {U}}\subset \bigcup _{H\in \mathcal {F}} {H}^{-1}\mathcal {U}_{(\delta )}$ . Since linear maps are uniformly continuous on bounded sets, there exist positive real numbers $\epsilon _0 <\ldots <\epsilon _N<\delta /2$ such that for every $i\in \{1, \dots , m\}$ , $j\in \{1, \dots , N\}$ , and $X, Y \in \bigcup _{H\in \mathcal {F}} {H}^{-1}\mathcal {U}$ , if $\|X-Y\|< \epsilon _{j-1}$ then $\|H_iX-H_iY\|<\epsilon _j/2$ . Denote $\epsilon :=\frac {1}{2}\min \{\epsilon _0,\ldots ,\epsilon _N\}$ .

Fixing $u\in \overline {\mathcal {U}}$ , there exist positive integers $k\leq N$ , and $i_1,\ldots , i_k \leq m$ such that $\mathrm {B}_{\delta }(H_{i_k} \cdots H_{i_1} u) \subset \mathcal {U}$ . Let $u_0:=u$ , $\mathcal {V}_0:=\mathrm {B}_{\epsilon _0}(u_0)$ , and $u_j:=H_{i_j}u_{j-1}$ , $\mathcal {V}_j:=\mathrm {B}_{\epsilon _j}(u_j)$ for $1\leq j \leq k$ . It follows that $H_{i_j}\mathrm {B}_{\epsilon }(\mathcal {V}_{j-1})\subset \mathcal {V}_j$ and $\mathrm {B}_{\delta /2}(\mathcal {V}_k) \subset \mathcal {U}$ . Define $\mathcal {V}_u:= \bigcup _{j=0}^k \mathcal {V}_j$ .

Hence, the bounded open set $\mathcal {V}:= \bigcup _{u\in \overline {\mathcal {U}}}\mathcal {V}_u$ satisfies the covering property for $H_1,\ldots ,H_m$ .

Proof. For simplicity of notation, denote $A(\sigma ^i(x))$ and $B(\sigma ^i(x))$ by $A_i$ and $B_i$ , respectively. Then,

$$\begin{align*}A_k(x)-B_k(x)=\sum\limits_{i=1}^k A_k\cdots A_{i+1}(A_i-B_i)B_{i-1}\cdots B_1.\end{align*}$$

Employing the triangle inequality and using $\|A_i-B_i\|\leq d_{\mathcal {C}^0}(A,B)$ , we get

$$ \begin{align*} \|A_k(x)-B_k(x)\|& \leq \sum\limits_{i=1}^k \|A_k\|\cdots \|A_{i+1}\|.\|A_i-B_i\|.\|B_{i-1}\|\cdots \|B_1\| \\ & \leq d_{\mathcal{C}^0}(A,B) \sum\limits_{i=1}^k \|A\|^{k-i}\|B\|^{i-1} \end{align*} $$

Now, for $C:=\max \{\|A\|, \|B\|\}$ , we have $\sum \limits _{i=1}^k \|A\|^{k-i}\|B\|^{i-1}\leq k C^{k-1}$ , completing the proof.

Proof. To prove the Lemma, we first prove a prior upper bound for $\|B_m(x)-B_m(y)\|$ that works for every $x,y$ , and for every $\alpha $ -Hölder cocycle. Then, we use it as a part of the proof of the main statement.

Claim. Let $(\sigma ,B)$ be an $\alpha $ -Hölder cocycle over the shift map. Then, for every $x,y$

(5.3) $$ \begin{align} \|B_m(x)-B_m(y)\|\leq \|B\|^{m-1}|B|_{\mathcal{C}^{\alpha}}\sum\limits_{i=1}^m d(\sigma^i(x),\sigma^i(y))^\alpha. \end{align} $$

Denoting $x_i:=\sigma ^{i}(x)$ and $y_i=\sigma ^i(y)$ , we have

$$\begin{align*}B_m(x)-B_m(y)=\sum\limits_{i=0}^{m-1} B_{m-i-1}(x_i)\big(B(x_i)-B(y_i)\big) B_i(x).\end{align*}$$

Therefore, by the triangle inequality

(5.4) $$ \begin{align} \|B_m(x)-B_m(y)\| & \leq \sum\limits_{i=0}^{m-1} \|B_{m-i-1}(x_i)\|.\|B(x_i)-B(y_i)\|.\| B_i(x)\| \end{align} $$

(5.5) $$ \begin{align} & \leq \|B\|^{m-1}\sum\limits_{i=0}^{m-1} \|B(x_i)-B(y_i)\| \end{align} $$

By Hölder regularity of B, $\|B(x_i)-B(y_i)\| \leq |B|_{\mathcal {C}^{\alpha }}d(x_i,y_i)^\alpha $ . Combining this with (5.5) gives (5.3) and finishes the proof of the claim.

Now, we turn to the proof of the main statement. We can write

$$ \begin{align*} B_{n_k}(x)-B_{n_k}(y)&=\sum\limits_{i=0}^{k-1} B_{n_k-n_{i+1}}(x_{n_{i+1}})\Big(B_{n_{i+1}-n_{i}}(x_{n_{i}})-B_{n_{i+1}-n_{i}}(y_{n_{i}})\Big)B_{n_{i}}(y). \end{align*} $$

Note that since $B_{n_k-n_{i+1}}(x_{n_{i+1}})=B_{n_k}(x)B_{n_{i+1}}(x)^{-1}$ , by assumption, $\|B_{n_i}(x)^{\pm 1}\|\leq K$ , we get $\|B_{n_k-n_{i+1}}(x_{n_{i+1}})\|\leq K^2$ . Therefore, applying the triangle inequality to the above equality, we get

(5.6) $$ \begin{align} \| B_{n_k}(x)-B_{n_k}(y)\| \leq K^{3}\sum\limits_{i=0}^{k-1} \|B_{n_{i+1}-n_{i}}(x_{n_i})-B_{n_{i+1}-n_{i}}(y_{n_i})\|. \end{align} $$

On the other hand, by (5.3) and $n_{i+1}-n_i\leq L$ , we have for every i

(5.7) $$ \begin{align} \|B_{n_{i+1}-n_{i}}(x_{n_i})-B_{n_{i+1}-n_{i}}(y_{n_i})\| & \leq \|B\|^{n_{i+1}-n_i-1}|B|_{\mathcal{C}^\alpha} \sum\limits_{j=n_{i-1}+1}^{n_i} d(x_j,y_j)^\alpha \end{align} $$

(5.8) $$ \begin{align} &~~~~~ \leq \max\{\|B\|,1\}^{L-1}|B|_{\mathcal{C}^\alpha} \sum\limits_{j=n_{i-1}+1}^{n_i} d(x_j,y_j)^\alpha. \end{align} $$

Therefore,

(5.9) $$ \begin{align} \| B_{n_k}(x)-B_{n_k}(y)\| \leq K^3 \max\{\|B\|,1\}^{L-1}|B|_{\mathcal{C}^\alpha}\sum\limits_{i=0}^{m-1}d(x_i,y_i)^\alpha. \end{align} $$

Now, since $y\in \sigma ^{-m}(W^u_{loc}\sigma ^m(x))$ , for every $i=0,\ldots ,m-1$ , $d(x_j,y_j)^\alpha < (\frac {1}{2^{m-j}})^\alpha |B|_{\mathcal {C}^\alpha }$ , which implies

$$ \begin{align*} \|B_{m}(x)-B_{m}(y)\| & \leq K^{3}\|B\|^{L-1}|B|_{\mathcal{C}^\alpha} \sum\limits_{i=0}^{m-1} \left(\frac{1}{2^{m-i}}\right)^\alpha \\ & \leq K^{3}\max\{\|B\|,1\}^{L-1}|B|_{\mathcal{C}^\alpha}\Big(\sum\limits_{i=0}^{\infty}\frac{1}{2^{i\alpha}}\Big) \\ & = ({1-2^{-\alpha}})^{-1}{K^{3}}\max\{\|B\|,1\}^{L-1}|B|_{\mathcal{C}^\alpha}. \end{align*} $$

This completes the proof.

Proof. The proof is by induction. For every $j\in \mathcal {A}$ , denote $D_j:=\phi (j)$ . First, in view of Lemma 4.4, we may assume that $\mathcal {U}$ contains the identity matrix, $\mathrm {Id}$ . Additionally, there is $\delta>0$ such that

(5.10) $$ \begin{align} \overline{\mathcal{U}}\subset \bigcup\limits_{i=1}^m {D_i}^{-1}\mathcal{U}_{(\delta)}. \end{align} $$

Since $\mathrm {Id}\in \mathcal {U}$ , it follows from (5.10) that there is $i_0\in \mathcal {A}$ such that $D_{i_0}\in \mathcal {U}_{(\delta )}$ . Clearly, $A|_{\mathsf {C}[i_0]}=D_{i_0}$ . Choose an element of ${\mathsf {C}[i_0]}$ and denote it by $x_0$ . Since $W_0=W^u_{loc}(x_0) \subset {\mathsf {C}[i_0]}$ , we conclude that for every $y\in W_0$ , $A(y)=D_{i_0}\in \mathcal {U}_{(\delta )}$ .

Now, if $\epsilon \in (0,\delta )$ and $d_{\mathcal {C}^{\alpha }}(A,B)<\epsilon $ , then $d_{\mathcal {C}^{0}}(A,B)<\epsilon $ . Consequently, for every $y\in W_0$ , $B(y)\in \mathcal {U}$ .

The above argument proves the proposition for $n=0$ . Now, suppose that the proposition holds for $n-1$ , that is, there exists $x_{n-1}\in \Sigma _m$ satisfying both items of the proposition. We aim to find $x_{n}\in \Sigma _m$ satisfying the desired properties. Note that by (5.10),

$$\begin{align*}B_{n}(x_{n-1})\in \overline{\mathcal{U}}\subset \bigcup\limits_{i=1}^m {D_i}^{-1}\mathcal{U}_{(\delta)}.\end{align*}$$

Therefore, there exists $i_{n}\in \mathcal {A}$ such that $D_{i_{n}}B_{n}(x_{n-1})\in \mathcal {U}_{(\delta )}$ and since A is locally constant depending on the zeroth position, $A|_{\mathsf {C}[i_n]}=D_{i_{n}}$ . Let $x_{n}$ be an arbitrary element in $\sigma ^{-(n-1)}(W_{n-1})\cap \sigma ^{-n}(\mathsf {C}[i_n])$ . We claim that this choice of $x_{n}$ satisfies the required properties (see Figure 2).

Figure 2

Proof of Proposition 5.4.

Figure 2 Long description

The diagram consists of three vertical lines representing mathematical spaces, with arrows indicating transformations between them.

* Leftmost Line: Labeled W sub loc super u (x sub 0). It contains a point x sub 0 and a point x sub n minus 1 above it. A red shaded interval surrounds x sub n minus 1, with a smaller dark brown segment nested within it. Brackets to the left identify the red interval as sigma super minus (n minus 1) (W sub n minus 1) and the dark brown segment as sigma super minus n (W sub n).

* Center Line: Connected to the first by an arrow labeled sigma super n minus 1. This line is labeled W sub loc super u (sigma super n (x sub n minus 1)) = W sub n minus 1. It shows a point sigma super n (x sub n minus 1) at the bottom. A dark brown segment is highlighted on this line, identified by a bracket as W sub n minus 1 intersection sigma super minus 1 (C [0, i sub n]).

* Rightmost Line: Connected to the center line by an arrow labeled sigma. This line is labeled sigma (W sub n minus 1) intersection C [0, i sub n] = W sub n. It shows a continuous dark brown vertical segment representing the final transformed set.

Claim. There exists $C>0$ depending on $\mathcal {U},A,\alpha $ such that for every $z\in \sigma ^{-(n+1)}(W_n)$ ,

$$\begin{align*}\|B_{n+1}(z)-D_{i_n}B_{n}(y)\|<C\epsilon.\end{align*}$$

To prove the claim, first using the triangle inequality, one has

(5.11) $$ \begin{align} \|B_{n+1}(z)-D_{i_n}B_{n}(y)\|\leq \|B_{n+1}(z)-D_{i_n}B_{n}(z)\|+ \|D_{i_n}B_{n}(z)-D_{i_n}B_{n}(y)\|. \end{align} $$

Since $D_{i_n}=A|_{\mathsf {C}[i_n]}$ and $\sigma ^{n}(z)\in \mathsf {C}[i_n]$ , and $d_{\mathcal {C}^{\alpha }}(A,B)<\epsilon $ , $\|B(\sigma ^{n}(z))-D_{i_n}\|<\epsilon $ . On the other hand, $z\in \sigma ^{-(n+1)}(W_{n+1}) \subset \sigma ^{-n}(W_n)$ . Therefore,

(5.12) $$ \begin{align} \|B_{n+1}(z)-D_{i_n}B_{n}(z)\| \leq \|B(\sigma^{n}(z))-D_{i_n}\|.\|B_{n}(z)\|\leq K_{\mathcal{U}}\epsilon, \end{align} $$

where $K_{\mathcal {U}}:=\max \{\|D^{\pm 1 }\|:D\in \overline {\mathcal {U}}\}$ .

To estimate the second term in (5.11), using Lemma 5.3, we get

(5.13) $$ \begin{align} \|D_{i_n}B_{n}(y)-D_{i_n}B_{n}(z)\| & \leq \|D_{i_n}\|. \|B_{n}(y)-B_{n}(z)\| \end{align} $$

(5.14) $$ \begin{align} & \leq (1-2^{-\alpha})^{-1}\|A\|K_{\mathcal{U}}^{3}|B|_{\mathcal{C}^\alpha}. \end{align} $$

Since all the points $y,z$ and $x_i$ ’s are in the same local unstable set, we can replace $|B|_{\mathcal {C}^\alpha }$ by $\sup _{x \in \Sigma }|B\big \vert _{W^u_{loc}(x)}|_{\mathcal {C}^\alpha }$ in (5.14). On the other hand, since A is a locally constant cocycle depending on zeroth position, it follows from Corollary 2.4, $\sup _{x \in \Sigma }|B\big \vert _{W^u_{loc}(x)}|_{\mathcal {C}^\alpha }<\epsilon $ . So

(5.15) $$ \begin{align} \|D_{i_n}B_{n}(y)-D_{i_n}B_{n}(z)\| \leq (1-2^{-\alpha})^{-1}\|A\|K_{\mathcal{U}}^{3}\epsilon. \end{align} $$

The estimates provided by (5.11), (5.12) and (5.15), complete the proof of Claim for $C=K_{\mathcal {U}}+(1-2^{-\alpha })^{-1}\|A\|K_{\mathcal {U}}^{3}$ .

To complete the proof of the proposition, it is enough to assume that $\epsilon <C^{-1}\delta $ , for C given by the claim. Then, according to the claim, for every $z\in \sigma ^{-(n+1)}(W_n)$ , $ \|B_{n+1}(z)-D_{i_n}B_{n}(y)\| \leq \delta $ and since $D_{i_n}B_{n}(y)\in \mathcal {U}_{(\delta )}$ , we get $B_{n+1}(z)\in \mathcal {U}$ .

Proof of Theorem 5.1

Let $\epsilon>0$ be the number provided by Proposition 5.4. Take cocycle B with $d_{\mathcal {C}^{\alpha }}(A,B)<\epsilon $ and $\{x_n\}_{n\in \mathbb {N}}$ be the sequence given in Proposition 5.4. According to this proposition, for $W_n:=W^u_{loc}(\sigma ^n(x_n))$ , $\{\sigma ^{-n}(W_n)\}_{n\in \mathbb {N}}$ forms a nested sequence of compact subsets of $\Sigma _m$ , ensuring that their intersection is non-empty. Now, in view of Proposition 5.4, for every point $x^*$ in this intersection $B_n(x^*)\in \mathcal {U}$ for every $n\geq 0$ .

Proof. Suppose that the cocycle $(\sigma ,A)$ satisfies the covering condition for $\mathcal {U}\subset G$ and . In view of Remark 4.7, we may assume that $\mathcal {U}$ contains the identity matrix and moreover, there exist $\delta>0$ and finite families $\Lambda _{\underline {a}}$ ( $\underline {a}\in \mathcal {A}'$ ) of transitions, such that for every $\underline {a}\in \mathcal {A}'$ , $\Lambda _{\underline {a}}\subset \bigcup _{\underline {b}\in \mathcal {A}'}T(\underline {a},\underline {b})$ and

(5.16) $$ \begin{align} \overline{\mathcal{U}} \subset \bigcup\limits_{\underline{w}\in \Lambda_{\underline{a}}}A_{[\underline{w}\rangle}^{-1}\mathcal{U}_{(\delta)}. \end{align} $$

Now, we take $L:=\max \{|\underline {w}| : \underline {w}\in \Lambda _{\underline {a}},\underline {a}\in \mathcal {A}'\}$ and construct the sequence $\{n_j\}_{j\geq 0}$ so that $n_0=0$ and for every $j\geq 0$ , $0<n_{j+1}-n_j\leq L$ . First, fix an arbitrary element $\underline {a}_0 \in \mathcal {A}'$ , and set $x_0$ as an arbitrary element in $\mathsf {C}[-k;\underline {a}_0]$ .

Claim. There exist $\epsilon>0$ and $\underline {a}_1,\underline {a}_2,\ldots $ in $\mathcal {A}'$ such that for every $\alpha $ -Hölder cocycle $(\sigma ,B)$ with $d_{\mathcal {C}^\alpha }(A,B)<\epsilon $ , every $j\in \mathbb {N}$ ,

$$\begin{align*}B_{n_j}(y)\in \mathcal{U}, \quad \text{for every~}y\in \mathsf{C}[-k;\underline{w}_1\vee\cdots \vee \underline{w}_j]\cap W^u_{loc}(x_0),\end{align*}$$

where $ \underline {w}_j \in T(\underline {a}_{j-1},\underline {a}_j)$ and $n_j:=-k+\sum _{i\leq j}|\underline {w}_i|$ .

We prove the claim inductively. Assume that $\underline {a}_0,\ldots ,\underline {a}_j\in \mathcal {A}'$ have been chosen satisfying the conditions. Let $x_j$ be an arbitrary point in $\mathsf {C}[-k;\underline {w}_1\vee \cdots \vee \underline {w}_j]\cap W^u_{loc}(x_0)$ . Then, by (5.16),

$$\begin{align*}B_{n_j}(x_j)\in \overline{\mathcal{U}} \subset \bigcup\limits_{\underline{w}\in \Lambda_{\underline{a}_j}}A_{[\underline{w}\rangle}^{-1}\mathcal{U}_{(\delta)}.\end{align*}$$

Therefore, there exists $\underline {a}_{j+1}\in \mathcal {A}'$ and $\underline {w}_{j+1}\in T(\underline {a}_j,\underline {a}_{j+1})$ such that

(5.17) $$ \begin{align} A_{[\underline{w}_{j+1}\rangle}B_{n_j}(x_j)\in \mathcal{U}_{(\delta)}. \end{align} $$

Next, we aim to show that for every point $z\in \mathsf {C}[-k; \underline {w}_1\vee \cdots \vee \underline {w}_{j+1}]\cap W^u_{loc}(x_0)$ , $B_{n_{j+1}}(z)\in \mathcal {U}$ , where $n_{j+1}=n_j+|\underline {w}_{j+1}|$ . To see this, first observe that

$$ \begin{align*} B_{n_{j+1}}(z)&-A_{[\underline{w}_{j+1}\rangle}B_{n_j}(x_j)\\ & = \big(B_{n_{j+1}-n_j+k}(\sigma^{n_j-k}(z))-A_{[\underline{w}_{j+1}\rangle}B_k(\sigma^{n_j-k}(x_j))\big)B_{n_j-k}(z)\\ & \quad +A_{[\underline{w}_{j+1}\rangle}B_k(\sigma^{n_j-k}(x_j))\big(B_{n_j-k}(z)-B_{n_j-k}(x_j)\big). \end{align*} $$

Therefore, by the triangle inequality,

(5.18) $$ \begin{align} \begin{split} \| B_{n_{j+1}}(z)-& A_{[\underline{w}_{j+1}\rangle}B_{n_j}(x_j)\| \\ & \leq \|B_{n_{j+1}-n_j+k}(\sigma^{n_j-k}(z))-A_{[\underline{w}_{j+1}\rangle}B_k(\sigma^{n_j-k}(x_j))\|.\|B_{n_j-k}(z)\| \\ &\quad + \|A_{[\underline{w}_{j+1}\rangle}B_k(\sigma^{n_j-k}(x_j))\|. \|B_{n_j-k}(z)-B_{n_j-k}(x_j)\|. \end{split} \end{align} $$

Note that since $\mathsf {C}[-k; \underline {w}_1\vee \cdots \vee \underline {w}_{j+1}]\subset \mathsf {C}[-k; \underline {w}_1\vee \cdots \vee \underline {w}_{j}]$ , it follows from induction hypothesis that $B_{n_j}(z)\in \mathcal {U}$ and so

$$ \begin{align*} \|B_{n_j}(z)\|\leq K_{\mathcal{U}}:=\sup\{\|D^{\pm 1}\| : D\in \overline{\mathcal{U}}\}. \end{align*} $$

Therefore,

$$ \begin{align*} \|B_{n_j-k}(z)\|\leq \|B_{n_j}(z)\|.\|B_k(\sigma^{n_j-k}(z))^{-1}\|\leq K_{\mathcal{U}}\|B^{-1}\|^k\leq K_{\mathcal{U}}\|B\|^{kd}. \end{align*} $$

Now, since $d_{\mathcal {C}^0}(A,B)< \epsilon $ , one has $\|B\|\leq \|A\|+\epsilon $ , and so

(5.19) $$ \begin{align} \|B_{n_j-k}(z)\|\leq K_{\mathcal{U}}(\|A\|+\epsilon)^{kd}. \end{align} $$

On the other hand, it follows from Lemma 5.2 that

(5.20) $$ \begin{align} \begin{split} \|B_{n_{j+1}-n_j+k}(\sigma^{n_j-k}(z))&-A_{[\underline{w}_{j+1}\rangle}B_k(\sigma^{n_j-k}(x_j))\|\\ &\leq (L+k)(\|A\|+\epsilon)^{L+k-1}\epsilon. \end{split} \end{align} $$

Meanwhile, by Lemma 5.3, we have

(5.21) $$ \begin{align} \| B_{n_{j}-k}(z)-B_{n_j-k}(x_j)\| &\leq (1-2^{-\alpha})^{-1}\|B\|^{L-1}{K_{\mathcal{U}}^{3}}|B|_{\mathcal{C}^\alpha}, \end{align} $$

Since all the points z and $x_j$ ’s and their relevant iterations are in the same local unstable set with distance less than $2^{-k}$ , we can replace $|B|_{\mathcal {C}^\alpha }$ in (5.21) by $\sup _{x \in \Sigma }|B\big \vert _{\sigma ^{-k}(W^u_{loc}(x))}|_{\mathcal {C}^\alpha }$ . On the other hand, since A is a locally constant cocycle depending on positions in , it follows from Corollary 2.4,

(5.22) $$ \begin{align} \sup_{x \in \Sigma}|B\big\vert_{\sigma^{-k}(W^u_{loc}(x))}|_{\mathcal{C}^\alpha}<\epsilon. \end{align} $$

Therefore, by (5.21),

(5.23) $$ \begin{align} \| B_{n_{j}-k}(z)-B_{n_j-k}(x_j)\| &\leq (1-2^{-\alpha})^{-1}\|B\|^{L-1}{K_{\mathcal{U}}^{3}}\epsilon. \end{align} $$

Now, by (5.18)–(5.23), we have

(5.24) $$ \begin{align} \| B_{n_{j+1}}(z)-A_{[\underline{w}_{j+1}\rangle}B_{n_j}(x_j)\|\leq C\epsilon. \end{align} $$

for some constant C depending on $\mathcal {U},L,d,\alpha ,\|A\|,k$ and independent of j. So, due to (5.17) and (5.24), if $\epsilon <C^{-1}\delta $ , we deduce that $B_{n_{j+1}}(z)\in \mathcal {U}$ and the proof of Claim is complete.

Proof of Theorem 5.5

Let $(\sigma ,A)$ be a cocycle that satisfies the given assumptions. According to Proposition 5.6, there exists $\epsilon> 0$ such that for every $\alpha $ -Hölder cocycle $(B, \sigma )$ with $d_{\mathcal {C}^{\alpha }}(A, B) < \epsilon $ , we can find a sequence $0 = n_0 < n_1 < n_2 < \cdots $ of integers with $L := \sup _j (n_{j+1} - n_j) < +\infty $ , a sequence $\{x_j\}_{j \geq 1}$ in $\Sigma _m$ , and a non-empty open set $\mathcal {U} \subset G$ with compact closure such that for $W_j = W^u_{loc} \big (\sigma ^{n_j}(x_j)\big )$ , $\sigma ^{-n_{j+1}}(W_{j+1}) \subset \sigma ^{-n_{j}}(W_j)$ , and for every $y \in W_j$ , $B_{n_j}(y) \in \mathcal {U}$ .

Since $\{\sigma ^{-n_{j}}(W_j)\}_{j \geq 0}$ forms a nested sequence of compact subsets of $\Sigma _m$ , their intersection must be non-empty. We claim that for every point $x^*$ in this intersection, the sequence $\{\|B_n(x^*)\|\}_{n \in \mathbb {N}}$ is bounded by some constant depending on A.

If $x^*$ belongs to this intersection, then for every $j \in \mathbb {N}$ , $B_{n_j}(x^*) \in \mathcal {U}$ and so

$$\begin{align*}\|B_{n_j}(x^*)^{\pm 1}\| \leq K_{\mathcal{U}},\end{align*}$$

where $K_{\mathcal {U}} := \sup \{\|D^{\pm 1}\| : D \in \overline {\mathcal {U}}\}$ . Given $n \in \mathbb {N}$ , there exists $j \geq 0$ such that $n_j \leq n < n_{j+1} \leq n_j + L$ . We use the identity $B_n(x^*)=B_{n-n_j}(\sigma ^{n_j}(x^*))B_{n_j}(x^*)$ to give upper bound for $\|B_n(x^*)^{\pm 1}\|$ . We have,

$$ \begin{align*} \|B_n(x^*)\| & \leq \|B_{n-n_j}(\sigma^{n_j}(x^*))\|.\|B_{n_j}(x^*)\| \leq \|B\|^{n-n_{j}} K_{\mathcal{U}}. \end{align*} $$

Similarly, $ \|B_n(x^*)^{-1}\|\leq \|B^{-1}\|^{n-n_{j}} K_{\mathcal {U}}$ . Now, since $\|A-B\|<\epsilon $ , we get that if $\epsilon $ is sufficiently small, there exists C depending on $\epsilon $ and on cocycle A such that $\|B^{-1}\|< \|A^{-1}\|+C\epsilon $ . Hence,

$$\begin{align*}\|B_n(x^*)^{\pm 1}\|\leq C'K_{\mathcal{U}},\end{align*}$$

for some constant $C'$ depending on $\epsilon ,L$ and the cocycle A. Finally, using Lemma 3.3 part (iv), we conclude that for every $y^*\in \omega (x^*)$ ,

$$\begin{align*}\sup_{n\in \mathbb{Z}}\|B_n(y^*)^{\pm 1}\| \leq K:=(C'K_{\mathcal{U}})^2.\end{align*}$$

This completes the proof.

Proof of Theorem A

Using Corollary 4.3, there exist $\{D_1,\ldots ,D_{2d^2}\}$ of matrices in $\mathrm {GL}(d,\mathbb {R})$ and open set $\mathcal {U} \subset \mathrm {GL}(d,\mathbb {R})$ which satisfy the covering condition. Let $\sigma :\Sigma _{2d^2} \to \Sigma _{2d^2}$ be the shift map. Define a locally constant cocycle $(\sigma ,A)$ , depending on zeroth position and associated to the IFS generated by $\langle D_1,\ldots ,D_{2d^2}\rangle ^+$ . Thus Theorem 5.1 proves that, there exists $\epsilon>0$ such that if $d_{\mathcal {C}^{\alpha }}(A,B)<\epsilon $ , for a $\mathrm {GL}(d,\mathbb {R})$ cocycle $(\sigma ,B)$ , then there exists $x^* \in \Sigma _{2d^2}$ , such that for every $n\in \mathbb {N}$

$$\begin{align*}\|B_n(x^*)^{\pm 1}\|\leq \kappa_{\mathcal{U}}.\end{align*}$$

To conclude the proof, notice that by Lemma 3.3 part (iv), for every point $y^*\in \omega (x^*)$ , $\omega $ -limit set of orbit of $x^*$ , we have $\|B_n(y^*)^{\pm 1}\| \leq \kappa ^2_{\mathcal {U}}$ and this completes the proof.

Proof. It is known that transitivity of $\sigma :\Sigma _H\to \Sigma _H$ implies that the set of periodic points is dense in $\Sigma _H$ . So, we need to verify the transition property.

Consider two periodic points $p,p'\in \Sigma _H$ of periods $n,n'$ , respectively. Let $\underline {a},\underline {a}'\in \mathcal {A}^{\mathrm {fin}}$ with $|\underline {a}|=n$ and $|\underline {a}'|=n'$ be such that $p=(\ldots ,\underline {a};\underline {a},\underline {a},\ldots )$ and $p'=(\ldots ,\underline {a}';\underline {a}',\underline {a}',\ldots )$ . Since $\sigma :\Sigma _H\to \Sigma _H$ is transitive, there exists $\underline {b},\underline {c}\in \mathcal {A}^{\mathrm {fin}}$ such that the concatenated words $\underline {a}'\underline {c},\underline {c} \,\underline {a},\underline {a} \, \underline {b},\underline {b} \, \underline {a}'$ are all admissible (with respect to H). This implies that for every $i,j\in \mathbb {N}$ , the periodic point for $\sigma $ associated with the following finite word is an element of $\Sigma _H$ :

$$\begin{align*}\underline{w}_{ij}:=\underline{c}\underbrace{\underline{a}\,\underline{a}\cdots \underline{a}}_{i}\underline{b}\underbrace{\underline{a}'\underline{a}'\cdots \underline{a}'}_{j}.\end{align*}$$

Now, fixing $\epsilon>0$ , since $A:\Sigma _H\to \mathrm {SL}(d,\mathbb {R})$ is uniformly continuous, there exists $k \in \mathbb {N}$ such that $\|A(x)-A(y)\|<\epsilon $ whenever $x,y\in \Sigma _H $ with $d(x,y)<2^{-k}$ .

Take $i,j>\max \{2k/n,2k/n'\}$ and denote the periodic point associated to $\underline {w}_{ij}$ by $p^*$ . Then, it is easy to check that the following is an $\epsilon $ -transition from $p'$ to p

$$ \begin{align*} \Big(A(\sigma^{-kj}(p^*)),A(\sigma^{-kj+1}(p^*)),\ldots,A(p^*),A(\sigma(p^*)),\ldots,A(\sigma^{|\underline{c}|+ki}(p^*))\Big), \end{align*} $$

as required.

Proof of Theorem 6.8

The assumption that the cocycle $(\boldsymbol {E},X,F,f)$ stably does not admit dominated splitting guarantees that there exists $\delta _0>0$ such that for every continuous volume-preserving cocycle $\hat {F}$ over f, if $d_{\mathcal {C}^0}(F,\hat {F})<\delta _0$ , then $(\boldsymbol {E},X,\hat {F},f)$ does not admit dominated splitting.

The proof is obtained by induction. The base case $N=1$ follows from Corollary 6.7. Now, suppose that the assertion holds for $N-1$ , that is, for every $\epsilon>0$ , there exists volume-preserving cocycle $F^{(N-1)}$ over f with $d_{\mathcal {C}^0}(F^{(N-1)},F)<\epsilon $ and $(\boldsymbol {E},X,F^{(N-1)},f)$ contains $N-1$ periodic points $p_1,\ldots ,p_{N-1}$ such that for every $i=1,2,\ldots ,N-1$ , $F^{(N-1)}_{n(p_i)} (p_i)=\mathrm {Id} $ . We aim to prove the statement for N. Fix $\epsilon>0$ and let $\delta _1:=\mathrm {min}\{\epsilon ,\delta _0\}/3$ . (Note that in this proof $F^{(i)}$ denotes the perturbations of cocycle F and should not be mistaken with $F_i$ that corresponds to iterations of the cocycle.)

For every $i=1,2,\ldots ,N-1$ , consider the map $\Theta _i :\mathrm {SL}(d,\mathbb {R})\to \mathrm {SL}(d,\mathbb {R}) $ defined by

$$\begin{align*}\Theta_i(H):=H. F^{(N-1)}(f^{n(p_i)-2}(p_i))\cdots F^{(N-1)}(f(p_i)) F^{(N-1)}(p_i).\end{align*}$$

It is easy to see that $\Theta _i$ is an open map. Since we can approximate $\mathrm {Id}$ matrix with hyperbolic matrices, for every $1\leq i \leq N-1$ , there exists $D_i\in \mathrm {SL}(d,\mathbb {R})$ , $\delta _1$ close to ${F}^{(N-1)} (f^{n(p_i)-1}(p_i))$ such that $\Theta _i(D_i)$ is a hyperbolic matrix. Thus, applying Lemma A.5 with K equal to the union of orbits of $p_i$ ’s ( $1\leq i \leq N-1$ ), we find a volume-preserving cocycle $\tilde {F}^{(N-1)}$ over f with $d_{\mathcal {C}^0}(\tilde {F}^{(N-1)},{F}^{(N-1)})<\delta _1$ such that for every $1\leq i\leq N-1$ ,

$$\begin{align*}\tilde{F}^{(N-1)}(f^j(p_i))=\begin{cases} {F}^{(N-1)}(f^j(p_i)) & 0\leq j <n(p_i)-1\\ D_i & j=n(p_i)-1 \end{cases} \end{align*}$$

By this way of extension, we conclude that $\widetilde {F}^{(N-1)}_{n(p_i)}(p_i)=\Theta _i(D_i)$ is hyperbolic for every $i=1,\ldots ,N-1$ . Then, for every $1\leq i\leq N-1$ consider the map $ \Phi _i :\mathrm {SL}(d,\mathbb {R})^{n(p_i)-1}\to \mathrm {SL}(d,\mathbb {R}) $ with

$$\begin{align*}\Phi_i(A_{n(p_i)-1},\ldots,A_{1}):= A_{n(p_i)-1}\cdots A_2 A_1. \end{align*}$$

Clearly, $\Phi _i$ is an open map. Now, since the set of hyperbolic matrices is open in $\mathrm {SL}(d,\mathbb {R})$ (see Remark A.2), there exists $\delta _2>0$ such that for every cocycle H with $d(H,\widetilde {F}^{(N-1)}) < \delta _2$ , all the linear maps $H_{n(p_i)}(p_i)$ for $i=1,\ldots ,N-1$ are hyperbolic.

Let $\delta _3<\mathrm {min}\{\delta _1,\delta _2\}$ be a positive number. Using Corollary 6.7, there is volume-preserving cocycle $\hat {F}^{(N)}$ over f with $d_{\mathcal {C}^0}(\hat {F}^{(N)},\widetilde {F}^{(N-1)})<\delta _3 $ , and a periodic point $p_N\in X$ for f such that $\hat {F}^{(N)}_{n(p_N)}(p_N)=\mathrm {Id} $ . Note that since $\delta _3<\delta _2$ , ${\widetilde {F}}^{(N)}_{n(p_i)}(p_i)$ is hyperbolic for every $1\leq i \leq N-1$ . This, in particular, implies that $p_N$ is not contained in the orbits of $p_i$ ’s for $i<N$ .

Finally, Lemma A.5, for $K:= \mathcal {O}(p_1)\cup \mathcal {O}(p_2)\cup \cdots \cup \mathcal {O}(p_N) $ , implies that there exists a volume-preserving cocycle ${F}^{(N)}$ over f with $d_{\mathcal {C}^0}({F}^{(N)},\hat {F}^{(N)})<\epsilon $ , such that

• • ${F}^{(N)}|_{\mathcal {O}(p_N)}=\hat {F}^{(N)}|_{\mathcal {O}(p_N)}$ ,

• • and for every $i=1,\ldots ,N-1$ , ${F}^{(N)}|_{\mathcal {O}(p_i)}={F}^{(N-1)}|_{\mathcal {O}(p_i)}$ .

This completes the proof of Theorem 6.8.

Proof of Proposition 7.3

Let $ l \in \mathbb {N} $ be such that the cocycle $ (\sigma ,A) $ depends on the positions in $ \mathbb {I}_l $ . We aim to provide $\tilde {A}$ as a locally constant cocycle that depends on positions in $ \mathbb {I}_n $ for a sufficiently large $ n> l $ , which will be specified at the end of the proof.

Throughout the proof by $A_{[{\underline {w}_{i,j}}\rangle }$ we mean $A_{[{2n+1;\underline {w}_{i,j}}\rangle }$ and remove $2n+1$ in the notation for simplicity.

For every $1\leq i \leq m$ , let $p_i:=(\ldots ,i;i,i,\ldots )$ be the fixed point associated with $\underline {p}_i$ . For every distinct $i,j\in \{1,\ldots ,m\}$ define

$$ \begin{align*} \underline{w}_{i,j}:= \big(\underbrace{i,i,\ldots,i}_{2n+1},\underbrace{j,j,\ldots,j}_{2n+1}\big). \end{align*} $$

Clearly, $\underline {w}_{i,j}$ gives a transition from $\underline {p_i}$ to $\underline {p_j}$ . For every $1\leq i, j \leq m$ and $1\leq k \leq 2n$ , let $\phi ^{(k)}_{i,j}\in \mathrm {SL}(d,\mathbb {R})$ be such that

$$\begin{align*}\phi^{(k)}_{i,j}:=A|_{\mathsf{C}[-n;(\underbrace{i,\ldots,i}_{2n+1-k},\underbrace{j,\ldots,j}_{k})]}.\end{align*}$$

Figure 3

A transition following a homoclinic point. On grey regions, the cocycle is identity. The dashed rectangle is zoomed in the next figure.

Figure 3 Long description

A line drawing on a white background features two large grey squares representing regions where the cocycle is identity.

* The first grey square is in the bottom-left. It is centered on a coordinate cross with arrows pointing inward along the vertical axis and outward along the horizontal axis. A small vertical rectangle sits on the right edge of this square.

* A series of curved arrows points from this rectangle to the right, hopping through two smaller vertical rectangles.

* An ellipsis indicates a continuation of this horizontal path.

* The path continues through three small square boxes on the horizontal axis, leading to a junction where the horizontal axis meets a vertical axis.

* At this junction, curved arrows transition the flow from the horizontal axis to a vertical upward direction.

* The upward path moves through two small square boxes, separated by another vertical ellipsis.

* The final curved arrow points into a dashed rectangle located at the bottom of the second large grey square in the top-right.

* This second grey square also features a coordinate cross with arrows pointing inward vertically and outward horizontally, mirroring the first square's structure.

Since $A(p_i)=A(p_j)=\mathrm {Id} $ , and A depends on positions in , for every k with $k\leq n-l$ or $k\geq n+1+l$ , $\phi ^{(k)}_{i,j}=\mathrm {Id} $ . Hence,

$$ \begin{align*}A_{[{\underline{w}_{i,j}}\rangle}=\underbrace{\phi^{(2n)}_{i,j}\cdots \phi^{(n+l+1)}_{i,j}}_{\mathrm{Id} } \phi^{(n+l)}_{i,j}\cdots \phi^{(n-l+1)}_{i,j} \underbrace{\phi^{(n-l)}_{i,j} \cdots \phi^{(1)}_{i,j}}_{\mathrm{Id} }=\phi^{(n+l)}_{i,j}\cdots \phi^{(n-l+1)}_{i,j}, \end{align*} $$

For every $1\leq i \leq m$ , let $\{D_{i,j}:1\leq j \leq m, j\neq i\}$ be an arbitrary permutation of $S_1,\ldots ,S_{m-1}$ . One can apply Lemma 7.5 to each $(A_{[{\underline {w}_{i,j}}\rangle })^{-1}D_{i,j}$ ( $1\leq i ,j\leq m$ , $i\neq j$ ), and obtain $n_{i,j}\in \mathbb {N}$ and an $n_{i,j}$ -tuple $(D^{(1)}_{i,j},\ldots ,D^{(n_{i,j})}_{i,j})$ of elements of $\mathrm {SL}(d,\mathbb {R})$ , $\epsilon $ -close to $\mathrm {Id}$ such that for every $1\leq i,j\leq m$ with $i\neq j$ , and $1\leq s \leq n_{i,j}$

(7.2) $$ \begin{align} (A_{[{\underline{w}_{i,j}}\rangle})^{-1}D_{i,j}=D^{(n_{i,j})}_{i,j}D^{(n_{i,j}-1)}_{i,j}\cdots D^{(1)}_{i,j}, \quad \|D_{i,j}^{(s)}-\mathrm{Id}\|<\epsilon. \end{align} $$

Note that without loss of generality, one may assume that all the numbers $n_{i,j}$ have equal value, say $n_0$ . Indeed, for given $n_0>\max \{n_{i,j}\}$ , one can define $D^{(s)}_{i,j}=\mathrm {Id}$ for $n_{i,j}<s\leq n_0$ . By this definition, all the properties listed in (7.2) hold. Now, we take $n:=n_0+l$ and define $\tilde {A}:\Sigma _m\to \mathrm {SL}(d,\mathbb {R})$ depending on positions in such that for every $1\leq i,j\leq m$ and $1\leq k \leq n-l$ ,