The authors present a new method to prove twist theorems. The method applies to C(sup 3 + gamma), measure-preserving twist maps which are not necessarily perturbations of the classical Twist, but it assumes 'good behavior' for some finite orbit (which is guaranteed in particular if the map is a perturbation or if F has a good rotation number on a boundary circle) and guarantees a set of positive measure of C(sup 2 + gamma - epsilon) invariant curves.
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