Correspondence Between the Spectrum of Jacobi Operator and Dominated Splitting of its Cocycle Map
In this paper we show a version of Johnson’s theorem for $\M(2,\R)$-sequences. In particular, motivated by the work of \cite{zhang2} and \cite{marx}, we show that one can identify the spectrum of Jacobi operator $J_{a,b}:\ell^2(\Z, \R)\rightarrow\ell^2(\Z,\R)$ by those energies $E \in \R$, whose cocycle map $B^{E} \in \M(2,\R)$ does not admit dominated splitting, i.e.$$\sigma(J_{a,b})=\{E\in\R:B^{E}\notin\CD\CS\}.$$ This result extends a well-known Johnson's theorem for Schr\"odinger operators, which identifies the spectrum of Schr\"odinger operator by all energy values $E \in \R$, such that the cocycle map $A^E \in \SL(2,\R)$ is not uniformly hyperbolic. Jacobi operator is a natural extension to Schr\"odinger operator, and dominated splitting generalizes the notion of uniform hyperbolicity. The biggest difference lies in the fact that for Schr\"odinger operator, the cocycle map takes value in $\SL(2,\R)$, whereas for Jacobi operator, the cocycle map takes value in $\M(2,\R)$. In particular, it is allowed to be singular, i.e. have zero determinant. This is one of the main challenges of this paper that we had to overcome. In this paper we also discuss the importance of Jacobi operators, develop definition of dominated splitting for $\M(2,\R)$-sequences, show stability theorem and finish this paper with the discussion of possibility of extending this result to dynamically defined Jacobi operators.