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Fix $m\in \mathbb{N}.$ For each $n\in \mathbb{N},$ let $A_n\in \mathbb{M}_{m}(\mathbb{C}),$ $X_n\in \mathbb{C}^m,$ and $B_n\in \mathbb{C}^m.$ Suppose that $$X_{n+1}=A_n X_n+B_n,$$ $$\lim_{n\rightarrow \infty} A_n=A_0,$$ $$|\det(A_0)|>1,$$ the moduli of all entries of $A_0$ greater than 1, and $$\limsup_{n\rightarrow \infty}\|B_n\|^{1/n}<1,$$ where $\|\cdot\|$ is the Euclidean norm.

Is it possible that $$\limsup_{n\rightarrow \infty}\|X_n\|^{1/n}=1?$$ I have the feeling that the answer should be "No." But I do not know how to prove it. For $m=1,$ it is not difficult to show that the answer is no. Could somebody suggest how to prove or disprove it? Any references (books or papers) are very welcome? Thank you.

Fix $m\in \mathbb{N}.$ For each $n\in \mathbb{N},$ let $A_n\in \mathbb{M}_{m}(\mathbb{C}),$ $X_n\in \mathbb{C}^m,$ and $B_n\in \mathbb{C}^m.$ Suppose that $$X_{n+1}=A_n X_n+B_n,$$ $$\lim_{n\rightarrow \infty} A_n=A_0,$$ $$|\det(A_0)|>1,$$ the moduli of all entries of $A_0$ are greater than 1, and $$\limsup_{n\rightarrow \infty}\|B_n\|^{1/n}<1,$$ where $\|\cdot\|$ is the Euclidean norm.

Is it possible that $$\limsup_{n\rightarrow \infty}\|X_n\|^{1/n}=1?$$ I have the feeling that the answer should be "No." But I do not know how to prove it. For $m=1,$ it is not difficult to show that the answer is no. Could somebody suggest how to prove or disprove it? Any references (books or papers) are very welcome? Thank you.

Fix $m\in \mathbb{N}.$ For each $n\in \mathbb{N},$ let $A_n\in \mathbb{M}_{m}(\mathbb{C}),$ $X_n\in \mathbb{C}^m,$ and $B_n\in \mathbb{C}^m.$ Suppose that $$X_{n+1}=A_n X_n+B_n,$$ $$\lim_{n\rightarrow \infty} A_n=A_0,$$ $$|\det(A_0)|>1,$$ and $$\limsup_{n\rightarrow \infty}\|B_n\|^{1/n}<1,$$ where $\|\cdot\|$ is the Euclidean norm.

Is it possible that $$\limsup_{n\rightarrow \infty}\|X_n\|^{1/n}=1?$$ I have the feeling that the answer should be "No." But I do not know how to prove it. For $m=1,$ it is not difficult to show that the answer is no. Could somebody suggest how to prove or disprove it? Any references (books or papers) are very welcome? Thank you.

Fix $m\in \mathbb{N}.$ For each $n\in \mathbb{N},$ let $A_n\in \mathbb{M}_{m}(\mathbb{C}),$ $X_n\in \mathbb{C}^m,$ and $B_n\in \mathbb{C}^m.$ Suppose that $$X_{n+1}=A_n X_n+B_n,$$ $$\lim_{n\rightarrow \infty} A_n=A_0,$$ $$|\det(A_0)|>1,$$ the moduli of all entries of $A_0$ greater than 1, and $$\limsup_{n\rightarrow \infty}\|B_n\|^{1/n}<1,$$ where $\|\cdot\|$ is the Euclidean norm.

Is it possible that $$\limsup_{n\rightarrow \infty}\|X_n\|^{1/n}=1?$$ I have the feeling that the answer should be "No." But I do not know how to prove it. For $m=1,$ it is not difficult to show that the answer is no. Could somebody suggest how to prove or disprove it? Any references (books or papers) are very welcome? Thank you.

Fix $m\in \mathbb{N}.$ For each $n\in \mathbb{N},$ let $A_n\in \mathbb{M}_{m}(\mathbb{C}),$ $X_n\in \mathbb{C}^m,$ and $B_n\in \mathbb{C}^m.$ Suppose that $$X_{n+1}=A_n X_n+B_n,$$ $$\lim_{n\rightarrow \infty} A_n=A_0,$$ $$|\det(A_0)|>1,$$ and $$\limsup_{n\rightarrow \infty}\|B_n\|^{1/n}<1,$$ where $\|\cdot\|$ is the Euclidean norm.

Is it possible that $$\limsup_{n\rightarrow n}\|X_n\|^{1/n}=1?$$ I have the feeling that the answer should be "No." But I do not know how to prove it. For $m=1,$ it is not difficult to show that the answer is no. Could somebody suggest how to prove or disprove it? Any references (books or papers) are very welcome? Thank you.

Fix $m\in \mathbb{N}.$ For each $n\in \mathbb{N},$ let $A_n\in \mathbb{M}_{m}(\mathbb{C}),$ $X_n\in \mathbb{C}^m,$ and $B_n\in \mathbb{C}^m.$ Suppose that $$X_{n+1}=A_n X_n+B_n,$$ $$\lim_{n\rightarrow \infty} A_n=A_0,$$ $$|\det(A_0)|>1,$$ and $$\limsup_{n\rightarrow \infty}\|B_n\|^{1/n}<1,$$ where $\|\cdot\|$ is the Euclidean norm.

Is it possible that $$\limsup_{n\rightarrow \infty}\|X_n\|^{1/n}=1?$$ I have the feeling that the answer should be "No." But I do not know how to prove it. For $m=1,$ it is not difficult to show that the answer is no. Could somebody suggest how to prove or disprove it? Any references (books or papers) are very welcome? Thank you.