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Remarks on iterated cubic maps. (English) Zbl 0762.58018
This article gives a survey on recent results concerning the dynamics of cubic polynomials viewed as self-mappings of the real line or the complex plane, especially on the work of Branner, Douady and Hubbard. In particular, certain phenomena are illustrated by computer created pictures and are described in detail.
First real cubic polynomials are studied as self-mappings of the real line. After introducing normal forms the author classifies the normalized polynomials in terms of the dynamics of critical points. Using simplified prototypical models he explains certain configurations in the parameter space like swallow configuration, arch configuration or product configuration. Relations to other mappings, e.g. Henon map, are pointed out in order to visualize phenomena like Arnold tongues.
The last chapters deal with complex cubic polynomials. The author discusses the geography of the parameter space, especially connectedness locus and hyperbolic components.
This article includes a comprehensive list of references.
Reviewer: H.Kriete (Aachen)

MSC:
37B99 Topological dynamics
26A18 Iteration of real functions in one variable
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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