Greene (1979) proposed a relationship between the existence of an invariant circle for an area-preserving map and the 'residues' of periodic orbits with nearby rotation number. It remains without a doubt the most effective practical criterion for calculating the breakup of invariant circles. In this work four results are proved which go a long way towards placing the criterion on a rigorous foundation, and the issues remaining to be resolved in order to formulate a rigorous version of the criterion are discussed.
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Robert Sinclair MacKay (1992) studied this question.