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DOMINATED SPLITTINGS AND SPECTRA FOR SINGULAR QUASI-PERIODIC JACOBI OPERATORS

We prove that the resolvent set of a (possbly singular) quasi-periodic Jacobi operator is characterized dynamically as the set of all energies whose associated Jacobi cocycles enjoy a dominated splitting. Consequently, the notion of a dominated splitting is seen to be the appropriate analog of uniform hyperbolicity when passing from Schrödinger operators to Jacobi operators.