Bifurcation to infinitely many sinks

This paper considers one parameter families of diffeomorphisms {F _t } in two dimensions which have a curve of dissipative saddle periodic pointsP _t , i.e.F _t ⁿ (P _t )=P _t and |detDF _t ⁿ (P _t )|<1. The family is also assumed to create new homoclinic intersections of the stable and unstable manifolds ofP _t as the parameter varies throught ₀. Gavirlov and Silnikov proved that if the new homoclinic intersections are created nondegenerately att ₀, then there is an infinite cascade of periodic sinks, i.e. there are parameter valuest _n accumulating att ₀ for which there is a sink of periodn [GS2, Sect. 4]. We show that this result is true for real analytic diffeomorphisms even if the homoclinic intersection is created degenerately. We give computer evidence to show that this latter result is probably applicable to the Hénon map forA near 1.392 andB equal ?0.3. Newhouse proved a related result which showed the existence of infinitely many periodic sinks for a single diffeomorphism which is a perturbation of a diffeomorphism with a nondegenerate homoclinic tangency. We give the main geometric ideas of the proof of this theorem. We also give a variation of a key lemma to show that the result is true for a fixed one parameter family which creates a nondegenerate tangency. Thus under the nondegeneracy assumption, not only is there a cascade of sinks proved by Gavrilov and Silnikov, but also a single parameter valuet* with infinitely many sinks.