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Complex analytic dynamics on the Riemann sphere. (English) Zbl 0558.58017

There has been a great revival of interest in the dynamics of complex analytic functions. This revival has been spurred on by the alluring computer graphics of Mandelbrot and the important recent results of Douady and Hubbard, Sullivan, Thurston, and many others. This survey of both the classical and modern work in this area will serve to heighten interest in this rapidly expanding field.

The subject of complex dynamics blossomed in the early twentieth century under the influence of such mathematicians as Fatou and Julia. This area, however, lay completely dormant until just recently, with the notable exceptions of the Siegel linearization theorem in the 1940’s and the work of I. N. Baker.

This survey article is a good introduction to much of the classical literature in the field, as well as to the modern contributions. The author begins with a rapid review of the properties of the Julia set, the subset of the Riemann sphere which carries the interesting dynamics of the system. Many of the easier proofs are included. Also, there are many of the now standard but still alluring computer pictures of specific Julia sets.

Other classical topics include a discussion of the local dynamics near a neutral periodic point. Both the rationally indifferent and the Siegel disk case are described.

Modern topics include the important Sullivan No Wandering Domain Theorem and the work of Manẽ, Sad, and Sullivan on structural stability. Douady and Hubbard’s results concerning the dynamics of quadratic polynomials and the structure of the Mandelbrot set are also described. Finally, the author includes a long description of the applications of the Measurable Riemann Mapping Theorem in complex dynamics.

Since many of the classical papers in this subject are relatively inaccessible, and since many of the modern papers have not yet appeared, this survey is a most welcome and timely addition to the literature.

Reviewer: R.Devaney

37C75Stability theory
30D05Functional equations in the complex domain, iteration and composition of analytic functions
30D35Distribution of values (one complex variable); Nevanlinna theory