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The dynamics of the Hénon map. (English) Zbl 0724.58042
This 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.
Reviewer: W.Sarlet (Gent)

##### MSC:
 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 37A05 Dynamical aspects of measure-preserving transformations
##### Keywords:
Hénon map; attractor
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