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The iteration of cubic polynomials. II: Patterns and parapatterns. (English) Zbl 0812.30008
For Part I see the authors in ibid. 160, No. 3/4, 143–206 (1988; Zbl 0668.30008).
This extensive paper (rather the monograph) consists of 12 chapters. In 1918 and 1920 G. Julia and P. Fatou proved that the Julia set for a polynomial \(P\) is connected if and only if none of the critical points escape to infinity under iteration and the Julia set for a polynomial \(P\) is a Cantor set if all the critical points do escape to infinity under iteration.
The authors give a complete solution of the problem when the Julia set of a cubic polynomial \(P\) is a Cantor set: The Julia set \(J_ P\) is a Cantor set if and only if the critical component \(K_ P(c)\) is not periodic. In the chapters 1-4 the theory of patterns is built and \(K_ P(c)\) is defined in terms of this theory. Very briefly: The pattern is a finite or infinite sequence of recursively determined Riemann surfaces. The notion of the end of patterns is defined in such a way, that the components of the filled-in Julia set \(K_ P\) for the polynomial \(P\) are in \(1-1\) correspondence with these ends. The component \(K_ P(c)\) corresponding to the critical end \(c\) is called the critical component. The chapters 7-11 are devoted to study so called parapatterns-parameter space for patterns. The last chapter deals with the polynomials of higher degree and this chapter says that most of the constructions of this paper go over to polynomials with two critical points of arbitrary degree.
This paper demonstrates the beauty of complex analysis.
Reviewer: A.Klíč (Praha)

MSC:
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
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