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| Natura: | Preprint |
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2022
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| Accesso online: | https://arxiv.org/abs/2204.10900 |
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| _version_ | 1866916790937845760 |
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| author | Carvalho, Silas L. Vieira, Fabricio |
| author_facet | Carvalho, Silas L. Vieira, Fabricio |
| contents | We present in this work a proof of the exponential dichotomy for dynamically defined matrix-valued Jacobi operators in $(\mathbb{C}^{l})^{\mathbb{Z}}$, given for each $ω\in Ω$ by the law $[H_ω \textbf{u}]_{n} := D(T^{n - 1}ω) \textbf{u}_{n - 1} + D(T^{n}ω) \textbf{u}_{n + 1} + V(T^{n}ω) \textbf{u}_{n}$, where $Ω$ is a compact metric space, $T: Ω\rightarrow Ω$ is a minimal homeomorphism and $D, V: Ω\rightarrow M(l, \mathbb{R})$ are continuous maps with $D(ω)$ invertible for each $ω\inΩ$. Namely, we show that for each $ω\inΩ$,
\[ρ(H_ω)=\{z \in \mathbb{C}\mid (T, A_z)\;\mathrm{is\; uniformly\; hyperbolic}\}, \] where $ρ(H_ω)$ is the resolvent set of $H_ω$ and $(T, A_z)$ is the $SL(2l,\mathbb{C})$-cocycle induced by the eigenvalue equation $[H_ωu]_n=zu_n$ at $z\in\mathbb{C}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_10900 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Exponential dichotomy for dynamically defined matrix-valued Jacobi operators Carvalho, Silas L. Vieira, Fabricio Dynamical Systems 34L40, 37H15, 39A70, 47B36 We present in this work a proof of the exponential dichotomy for dynamically defined matrix-valued Jacobi operators in $(\mathbb{C}^{l})^{\mathbb{Z}}$, given for each $ω\in Ω$ by the law $[H_ω \textbf{u}]_{n} := D(T^{n - 1}ω) \textbf{u}_{n - 1} + D(T^{n}ω) \textbf{u}_{n + 1} + V(T^{n}ω) \textbf{u}_{n}$, where $Ω$ is a compact metric space, $T: Ω\rightarrow Ω$ is a minimal homeomorphism and $D, V: Ω\rightarrow M(l, \mathbb{R})$ are continuous maps with $D(ω)$ invertible for each $ω\inΩ$. Namely, we show that for each $ω\inΩ$, \[ρ(H_ω)=\{z \in \mathbb{C}\mid (T, A_z)\;\mathrm{is\; uniformly\; hyperbolic}\}, \] where $ρ(H_ω)$ is the resolvent set of $H_ω$ and $(T, A_z)$ is the $SL(2l,\mathbb{C})$-cocycle induced by the eigenvalue equation $[H_ωu]_n=zu_n$ at $z\in\mathbb{C}$. |
| title | Exponential dichotomy for dynamically defined matrix-valued Jacobi operators |
| topic | Dynamical Systems 34L40, 37H15, 39A70, 47B36 |
| url | https://arxiv.org/abs/2204.10900 |