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We study self-adjoint bounded Jacobi operators of the form:  (J \psi)(n) = a_n \psi(n + 1) + b_n \psi(n) +a_{n-1} \psi(n - 1) on$\ell^2(\N)$. We assume that for some fixed q, the q-variation of $\{a_n\}$ and$\{b_n\}$ is square-summable and $\{a_n\}$ and $\{b_n\}$ converge to q-periodicsequences. Our main result is that under these assumptions the essentialsupport of the absolutely continuous part of the spectrum of J is equal to thatof the asymptotic periodic Jacobi operator. This work is an extension of arecent result of S.A.Denisov.

Preservation of absolutely continuous spectrum of periodic Jacobi operators under perturbations of square--summable variation