An introduction to multi-dimensional complex dynamics: Hénon mappings in ℂ2
✍ Scribed by Ralph W. Oberste-Vorth
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 759 KB
- Volume
- 30
- Category
- Article
- ISSN
- 0362-546X
No coin nor oath required. For personal study only.
✦ Synopsis
Dynamical systems as a subject has been around for some time, but this term represents quite different subjects to different mathematicians. For example, it may mean differential equations or iteration theory, that is, continuous or discrete dynamical systems. We are interested in the study of the iteration of functions. Two examples, well-known to undergraduate students, are Newton's method for finding zeros of a function and Euler's method for numerically solving differential equations.
Iterating functions of one complex variable was first studied extensively around 1918-20 by Fatou and Julia, facilitated by the work of Monte1 on normal families of meromorphic functions. The subject lay dormant until the 1980's when it was revisited by Douady, Hubbard, Milnor, Sullivan, Thurston, and others. New tools again led the charge: quasi-conformal mappings and computer experimentation. [See Blanchard [Bl] for a survey.]
Fatou first studied functions of several complex variables around 1922 [F], with clarifications by Bieberbach in 1933 [Bi]. They proved the existence of Fatou-Bieberbach domains: open subsets U c C" which are biholomorphically isomorphic to @" and whose complement, C" -U, has non-empty interior. These were constructed as basins of attractive fixed points.
Our work started with trying to understand Fatou-Bieberbach domains obtained from a cubic automorphism. This led us to look at simpler automorphisms, with quadratic polynomials, particularly the H&on family. There has since been an enormous amount of work on the dynamics of the H&on mappings in (c2 by Friedland and Milnor; Bedford, Lyubich, and Smillie; For-mess and Sibony; Hubbard and Oberste-Vorth; and others.
2. ITERATION OF MAPPINGS
Suppose we wish to iterate a mapping F : X + X. Define the iterates of x under F by F"" (x) = F( F""-' (x)) for all natural numbers n where F"'(x) = x. Moreover, if F is invertible, then define iterates F"-"(x) corresponding to negative integers n as iterates under F-' . The sequence of iterates of x under F--either {x, F(x), Fo2(x), . . . } or { . . . , Fop2(x), F-'(x), x, F(x), Fo2(x), . . . }-is called the orbit of x under F.
The solutions of F(x) = x are called fixed points of F. More generally, the solutions of F""(x) = x are called periodic points of F; the minimal such n is called the period of a periodic point.
Let us consider the case where X = @" and F is an analytic isomorphism. A periodic point p of period m is attracting if every eigenvalue of (c&, Fom is less than 1 in absolute value. Indeed, if p is an