Branner, Bodil; Hubbard, John H. The iteration of cubic polynomials. II: Patterns and parapatterns. (English) Zbl 0812.30008 Acta Math. 169, No. 3-4, 229-325 (1992). 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) Cited in 11 ReviewsCited in 106 Documents 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 Keywords:fundamental group; monodromy; critical points; Julia set; Cantor set; Riemann surfaces Citations:Zbl 0668.30009; Zbl 0668.30008 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Ahlfors, L.,Conformal Invariants. Topics in Geometric Function Theory. McGraw-Hill Book Company, 1973. · Zbl 0272.30012 [2] Blanchard, P., Disconnected Julia sets, inChaotic Dynamics and Fractals, ed. M. F. Barnsley and S. G. Demko. Academic Press (1986), pp. 181–201. [3] Blanchard, P., Symbols for cubics and other polynomials. To appear inTrans. Amer. Math. Soc. [4] Blanchard, P., Devaney, R. & Keen, L., Geometric realizations of automorphisms of the one-sidedd-shift. Preprint, 1989. [5] Branner, B. &Hubbard, J. H., The iteration of cubic polynomials. Part I: The global topology of parameter space.Acta Math., 160 (1988), 143–206. · Zbl 0668.30008 · doi:10.1007/BF02392275 [6] Brolin, H., Invariant sets under iteration of rational functions.Ark. Mat., 6 (1965), 103–144. · Zbl 0127.03401 · doi:10.1007/BF02591353 [7] Brown, M., A proof of the generalized Schoenflies theorem.Bull. Amer. Math. Soc., 66 (1960), 74–76. · Zbl 0132.20002 · doi:10.1090/S0002-9904-1960-10400-4 [8] Douady, A. & Hubbard, J. H., Étude dynamique des polynômes complexes. Première partie (1984) et deuxième partie (1985). Publications Mathématiques d’Orsay. [9] –, On the dynamics of polynomial-like mappings.Ann. Sci. École Norm. Sup. (4), 18 (1985), 287–343. · Zbl 0587.30028 [10] Fatou, P., Sur les équations fonctionnelles.Bull. Soc. Math. France, 47 (1919), 161–271; 48 (1920), 33–94 and 208–314. · JFM 47.0921.02 [11] Hubbard, J. H. & Oberste-Vorth, R., The dynamics of Henon mappings in the complex domain. Preprint, 1989. · Zbl 0839.54029 [12] Julia, G., Mémoires sur l’itération des fonctions rationelles.J. Math. Pures Appl. (7), 4 (1918), 47–245. · JFM 46.0520.06 [13] Lavaurs, P., Thesis, Université de Paris-Sud Centre d’Orsay, 1989. [14] Mañe, R., Sad, P. &Sullivan, D., On the dynamics of rational maps.Ann. Sci. École Norm. Sup. (4), 16 (1983), 193–217. · Zbl 0524.58025 [15] Milnor, J.,Morse Theory. Princeton University Press, 1963. [16] Milnor, J., Remarks on iterated cubic maps. Preprint, 1986. · Zbl 0762.58018 [17] Milnor, J., Hyperbolic components in spaces of polynomial maps. Preprint, 1987. · Zbl 0659.01014 [18] Sullivan, D., Quasi-conformal homeomorphisms and dynamics. I: Solutions of the Fatou-Julia problem on wandering domains.Ann. of Math., 122 (1985), 401–418. · Zbl 0589.30022 · doi:10.2307/1971308 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.