Cascade of sinks

In this paper it is proved that if a one-parameter family (F₁) of C¹dissipative maps in dimension two creates a new homoclinic intersection for a fixed point P_twhen the parameter t = t₀, then there is a cascade of quasi-sinks, i.e., there are parameter values tn converging to t₀such that, for t = t_n, Ft has a quasi-sink An with each point q in An having period ti. A quasi-sink A_nfor a map F is a closed set such that each point q in A_nis a periodic point and A_nis a quasi-attracting set (å la Conley), i.e., A_nis the intersection of attracting sets Formula Present where each Formula Present has a neighborhood Formula Present. Thus, the quasi-sinks An are almost attracting sets made up entirely of points of period n. Gavrilov and Silnikov, and later Newhouse, proved this result when the new homoclinic intersection is created nondegenerately. In this case the sets A_nare single, isolated (differential) sinks. In an earlier paper we proved the degenerate case when the homoclinic intersections are of finite order tangency (or the family is real analytic), again getting a cascade of sinks, not just quasi-sinks.