Existence of invariant curves for area-preserving mappings under weaker non-degeneracy conditions

Abstract

We consider a class of analytic area-preserving mappings C^m-smoothly depending on a parameter. Without imposing on any non-degeneracy assumption, we prove a formal KAM theorem for the mappings, which implies many previous KAM-type results under some non-degeneracy conditions. Moreover, by this formal KAM theorem, we can also obtain some new interesting results under some weaker non-degeneracy conditions. Thus, the formal KAM theorem can be regarded as a general KAM theorem for areapreserving mappings.

Key words： Area-preserving mapping; invariant curve; KAM iteration; nondegeneracy condition

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