A RIGOROUS PARTIAL JUSTIFICATION OF GREENE CRITERION

We prove several theorems that lend support to Greene's criterion for the existence or not of invariant circles in twist maps. In particular, we show that some of the implications of the criterion are correct when the Aubry-Mather sets are smooth invariant circles or uniformly hyperbolic. We also suggest a simple modification that can work in the case that the Aubry-Mather sets have nonzero Lyapunov exponents. The latter is based on a closing lemma for sets with nonzero Lyapunov exponents, which may have several other applications.