Coexistence phenomena in the H\'enon family

Abstract

We study the classical H\'enon family $f_{a,b}:(x,y)\mapsto(1-ax^2+y,bx)$, $0<a<2$, $0<b<1$, and prove that given an integer $k\geq 1$, there is a set of parameters $E_k$ of positive two-dimensional Lebesgue measure so that $f_{a,b}$, for $(a,b)\in E_k$, has at least $k$ attractive periodic orbits and one strange attractor. A corresponding statement also holds for the H\'enon-like families. The final main result of the paper is the existence, within the classical H\'enon family, of a positive Lebesgue measure set of parameters whose corresponding maps have two coexisting strange attractors.