The dynamics of the Hénon map.

*(English)*Zbl 0724.58042This gigantic paper studies the long time behaviour of the Hénon map \(T: (x,y)\to (1-ax^ 2+y,bx)\) for small positive \(b\) and \(0<a_ 0<a<2\). The starting model is the 1-dimensional map \(x\to 1-ax^ 2\) (-1\(\leq x\leq 1)\), for which the authors revise their earlier proof concerning chaotic behaviour in a way which exhibits that the derivatives grow exponentially. Next, a 2-dimensional model problem is studied, which is inspired by the idea that the attractor of the Hénon map (proved to be the closure of the unstable manifold \(W^ u)\) will roughly have the structure of the Cartesian product of a line segment and a Cantor set. Important notions for the detailed study of \(T\) are the concept of contractive directions and more predominantly still an inductive construction of a critical set. Other large sections deal with the geometry of \(W^ u\) and its relation to the choice of parameters.

The main theorem states that for all \(c<\log 2\), \(\exists b_ 0>0\) such that \(\forall b\in (0,b_ 0)\), \(\exists\) a set \(E(b)\) of positive 1-dimensional Lebesgue measure in such a way that \(\forall a\in E(b)\):

(i) for all \(z\) in some open \(U(a,b)\), \(T^ n(z)\) approaches the attractor;

(ii) \(\exists z_ 0(a,b)\in W^ u\) with dense orbit in \(W^ u\) and such that \(\| DT^ n(z_ 0)(0,1)\| \geq e^{cn}\).

Gettig there is a matter of an incredible series of entangled lemmas, assertions, claims, etcetera.

The paper is, despite its inevitably boring technicalities, quite well written.

The main theorem states that for all \(c<\log 2\), \(\exists b_ 0>0\) such that \(\forall b\in (0,b_ 0)\), \(\exists\) a set \(E(b)\) of positive 1-dimensional Lebesgue measure in such a way that \(\forall a\in E(b)\):

(i) for all \(z\) in some open \(U(a,b)\), \(T^ n(z)\) approaches the attractor;

(ii) \(\exists z_ 0(a,b)\in W^ u\) with dense orbit in \(W^ u\) and such that \(\| DT^ n(z_ 0)(0,1)\| \geq e^{cn}\).

Gettig there is a matter of an incredible series of entangled lemmas, assertions, claims, etcetera.

The paper is, despite its inevitably boring technicalities, quite well written.

Reviewer: W.Sarlet (Gent)