Cascade of sinks

Abstract:

In this paper it is proved that if a one-parameter family $\{ {F_t}\}$ of ${C^1}$ dissipative maps in dimension two creates a new homoclinic intersection for a fixed point ${P_t}$ when the parameter $t = {t_0}$, then there is a cascade of quasi-sinks, i.e., there are parameter values ${t_n}$ converging to ${t_0}$ such that, for $t = {t_n}$, ${F_t}$ has a quasi-sink ${A_n}$ with each point $q$ in ${A_n}$ having period $n$. A quasi-sink ${A_n}$ for a map $F$ is a closed set such that each point $q$ in ${A_n}$ is a periodic point and ${A_n}$ is a quasi-attracting set (à la Conley), i.e., ${A_n}$ is the intersection of attracting sets $A_n^j, {A_n} = { \cap _j}A_n^j$, where each $A_n^j$ has a neighborhood $U_n^j$ such that $\cap \{ {F^k}(U_n^j):k \geqslant 0\} = A_n^j$. Thus, the quasi-sinks ${A_n}$ 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 ${A_n}$ are 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.