Lower dimensional invariant tori for quasiperiodically forced circle diffeomorphisms

For any analytic quasiperiodically forced circle diffeomorphisms , where f is fixed and ε is small, we show that if ω is Diophantine and the fibred rotation number of the diffeomorphism remains constant in a unilateral neighborhood of ε=0 (i.e., there is a unilateral phase-locking at ε=0), then the diffeomorphism has at least one analytic q-invariant torus, provided ε is small enough.