Parameter exclusions in H\'enon-like systems

M. Viana ; S. Luzzatto

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 58 (2003) no. 6, pp. 1053-1092

Résumé

This paper is devoted to the Hénon attractor and related systems. The Hénon attractor is observed when the diffeomorphism $H_{a,b}$ of the plane given by the rule $(x,y)\mapsto(1- ax^2+ by,x)$ is iterated. If the number $b$ is small, then this map is close to the one-dimensional map $x\mapsto1-ax^2$. The latter has a fold, which is also reflected by the properties of $H_{a,b}$ with $b\ne0$, namely, this map has some hyperbolicity, though the property is not uniform. Benedicks and Carleson found an approach to the study of such maps that gives fairly complete information about the attractor for “most” values of $a$. Subsequently Mora and Viana, together with many other authors, proved that similar attractors occur in much more general situations, and the theory of these systems has also been well developed. The present paper is an introduction to this area of investigation. While omitting detailed proofs, the authors have tried to single out the crucial points, referring the reader to the original sources for a thorough presentation.