Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Kotani Theory for ergodic matrix-like Jacobi operators   We extend the so-called Kotani Theory for a particular class of ergodic
matrix-like Jacobi operators defined in $l^{2}(\mathbb{Z}; \mathbb{C}^{l})$ by
the law $[H_{\omega} \textbf{u}]_{n} := D^{*}(T^{n - 1}\omega) \textbf{u}_{n -
1} + D(T^{n}\omega) \textbf{u}_{n + 1} + V(T^{n}\omega) \textbf{u}_{n}$, where
$T: \Omega \rightarrow \Omega$ is an ergodic automorphism in the measure space
$(\Omega, \nu)$, the map $D: \Omega \rightarrow GL(l, \mathbb{R})$ is bounded,
and for each $\omega\in\Omega$, $D(\omega)$ is symmetric. Namely, it is shown
that for each $r\in\{1,\ldots,l\}$, the essential closure of $\mathcal{Z}_{r}
:= \{x \in \mathbb{R}\mid$ exactly $2r$ Lyapunov exponents of $A_z$ are
zero$\}$ coincides with $\sigma_{ac,2r}(H_{\omega})$, the absolutely continuous
spectrum of multiplicity $2r$, where $A_z$ is a Schr\"odinger-like cocycle
induced by $H_\omega$. Moreover, if $k\in\{1,\ldots,2l\}$ is odd, then
$\sigma_{ac,k}(H_{\omega})=\emptyset$ for $\nu$-a.e. $\omega\in\Omega$. We also
provide a Thouless Formula for such class of operators.

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The categories of this paper are: physics
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The categories of this paper are: comp-gas
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th
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The categories of this paper are: adap-org
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The categories of this paper are: q-fin
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The categories of this paper are: solv-int
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The categories of this paper are: nlin
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The categories of this paper are: q-bio
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The categories of this paper are: cmp-lg
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The categories of this paper are: patt-sol
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The categories of this paper are: ao-sci
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-th
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The categories of this paper are: stat
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The categories of this paper are: hep-lat