Basins for general nonlinear Hénon attracting sets

Abstract:

When a new homoclinic intersection is created for a dissipative diffeomorphism in dimension two, there results a cascade of sinks. We show that immediately after one of these sinks ${q_n}$ is formed, its basin boundary is made up of the stable manifold of the saddle periodic point ${p_n}$ formed at the same time. After this sink undergoes a period doubling, there still remains a trapping region with an attracting set inside. In fact, we show that until this saddle periodic point ${p_n}$ has its own homoclinic bifurcation, there is an attracting set whose boundary is made up of the stable manifold of ${p_n}$. By picking a rectangle ${B_n}$ carefully, the one-parameter family of maps ${f_{{t^n}}}$ creates these sinks and attracting sets by pulling the image ${f_{{t^n}}}({B_n})$ across ${B_n}$ and eventually forming a horseshoe in ${B_n}$. The maps, ${f_{{t^n}}}$ on ${B_n}$, are well approximated for large $n$ by quadratic maps equivalent to the Hénon map. We prove our results for general nonlinear Hénon maps which include not only the quadratic maps but also other nonlinear maps which also create horseshoes, including those arising from homoclinic tangencies.