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Numerical Study of Invariant Sets of a Volume-preserving Map

We study the behavior of invariant sets of a volume-preserving map, that is a quasi-periodic perturbation of a symplectic map, using approximation by periodic orbits. We present numerical results for analyticity domains of invariant surfaces, behavior after breakdown and a critical function describing breakdown of invariant surfaces as a function of their rotation vectors. We discuss implications of our results to the existence of a renormalization group operator describing breakdown of invariant surfaces.