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In this note, we extend the renormalization horseshoe we have recentlyconstructed with N. Goncharuk for analytic diffeomorphisms of the circle totheir small two-dimensional perturbations. As one consequence, Herman ringswith rotation numbers of bounded type survive on a codimension one set ofparameters under small two-dimensional perturbations.

KAM-renormalization and Herman rings for 2D maps