Globality of bifurcation of a fixed point embedded into a simply connected topologically transitive set (Q2740278)

This paper deals with a two-parameter family of two-dimensional mappingsNEWLINE\[NEWLINEF:(x,y) \mapsto(-(1-2\varepsilon)xy,-(x^2+y^2)/2+a(a-2)/2).NEWLINE\]NEWLINE It is well known (see, e.g., the paper of \textit{M. Benedicks} and \textit{L. Carleson} [Ann. Math. (2) 122, 1-25 (1985; Zbl 0597.58016)]) that there exists a set \(\Delta \subset\mathbb R\) of positive Lebesgue measure such that for any \(a\in \Delta \) the mapping \(F_a:=F|_{x=0}\) has a simply connected invariant topologically transitive attractor \(\mathfrak A\) with dense set of periodic points. Under certain conditions (e.g. \(\varepsilon \in (3/8,1/2)\)) the set \(\{0\}\times \mathfrak A\) is the attractor for \(F\). The author shows that the bifurcation of the fixed point \(P_a=(0,a-2)\) has global character in the following sense: simultaneously with the bifurcation of the pair of 2-periodic points in a neighborhood of \(P_a\) there appears a nontrivial invariant topologically transitive set near \(\{0\}\times \mathfrak A\). This set lies outside of \(\{0\}\times \mathfrak A\) and contains a dense set of periodic points.