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Simultaneous continuation of infinitely many sinks at homoclinic bifurcations

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  • We prove that the $C^3$ diffeomorphisms on surfaces, exhibiting infinitely many sinks near the generic unfolding of a quadratic homoclinic tangency of a dissipative saddle, can be perturbed along an infinite dimensional manifold of $C^3$ diffeomorphisms such that infinitely many sinks persist simultaneously. On the other hand, if they are perturbed along one-parameter families that unfold generically the quadratic tangencies, then at most a finite number of those sinks have continuation.
    Mathematics Subject Classification: Primary: 37G25, 58F14; Secondary: 37D45, 37C29.

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