×

Quadratic polynomials and the Hénon attractor. (Polynômes quadratiques et attracteur de Hénon.) (French) Zbl 0754.58022

Séminaire Bourbaki, Vol. 1990/91, Exp. No. 734, Astérisque 201-203, 143-165 (1991).
[For the entire collection see Zbl 0742.00056.]
Several families of maps (i.e., quadratic polynomials, rational fractions and Hénon maps) are known to be characterized by an exponential expansion in spite of the presence of a “critical” compression region [M. Jakobson, Commun. Math. Phys. 81, 39-88 (1981; Zbl 0497.58017); M. Rees, Ann. Sci. Ec. Norm. Supér., IV. Sér. 19, 383-407 (1986; Zbl 0611.58038), and M. Benedicks and L. Carleson, Ann. Math., II. Ser. 133, No. 1, 73-169 (1991; Zbl 0724.58042)].
In the present paper, the author outlines the similar nature of these results, and suggests a common procedure for constructing the proof. He gives the complete proof for the case of quadratic polynomials, and indicates which are the difficulties involved in proving the other two results.
The basic idea is to chose the parameters localization such that to ensure a significant expansion for the first iterations, and to show that, with good probability, the passage through the “critical” region, which is inevitable, can not destroy the already achieved expansion. Although the recurrence on the number of iterations is complicated, the procedure might offer very interesting insights in understanding this behavior.
Reviewer: D.Savin (Montreal)

MSC:

37C70 Attractors and repellers of smooth dynamical systems and their topological structure
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30C10 Polynomials and rational functions of one complex variable
PDFBibTeX XMLCite
Full Text: Numdam EuDML