Coexistence Phenomena in the Hénon Family

Abstract

AbstractWe study the classical Hénon family $$f_{a,b}:(x,y)\mapsto (1-ax^2+y,bx)$$ f a , b : ( x , y ) ↦ ( 1 - a x 2 + y , b x ) , $$0<a<2$$ 0 < a < 2 , $$0<b<1$$ 0 < b < 1 , and prove that given an integer $$k\ge 1$$ k ≥ 1 , there is a set of parameters $$E_k$$ E k of positive two-dimensional Lebesgue measure so that $$f_{a,b}$$ f a , b , for $$(a,b)\in E_k$$ ( a , b ) ∈ E k , has at least k attractive periodic orbits and one strange attractor of the type studied in Benedicks and Carleson (Ann Math (2) 133(1):73–169, 1991). A corresponding statement also holds for the Hénon-like families of Mora and Viana (Acta Math 171:1–71, 1993), and we use the techniques of Mora and Viana (1993) to study homoclinic unfoldings also in the case of the original Hénon maps. The final main result of the paper is the existence, within the classical Hénon family, of a positive Lebesgue measure set of parameters whose corresponding maps have two coexisting strange attractors.