%ABenedicks, Michael%APalmisano, Liviana%BJournal Name: Bulletin of the Brazilian Mathematical Society, New Series; Journal Volume: 54; Journal Issue: 3; Related Information: CHORUS Timestamp: 2023-08-30 09:14:47
%D2023%ISpringer Science + Business Media
%JJournal Name: Bulletin of the Brazilian Mathematical Society, New Series; Journal Volume: 54; Journal Issue: 3; Related Information: CHORUS Timestamp: 2023-08-30 09:14:47
%K
%MOSTI ID: 10432927
%PMedium: X
%TCoexistence Phenomena in the Hénon Family
%XAbstract
We study the classical Hénon family$$f_{a,b}:(x,y)\mapsto (1-ax^2+y,bx)$$${f}_{a,b}:(x,y)\mapsto (1-a{x}^{2}+y,bx)$,$$0$0<a<2$,$$0**$0<b<1$**, and prove that given an integer$$k\ge 1$$$k\ge 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)\in {E}_{k}$, has at leastkattractive 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.

%0Journal Article