Devaney, Robert L.; Fagella, Núria; Jarque, Xavier Hyperbolic components of the complex exponential family. (English) Zbl 1099.30011 Fundam. Math. 174, No. 3, 193-215 (2002). The authors consider parameter space of the exponential family \(E_{\lambda}(z) = \lambda \exp(z)\). A hyperbolic component is a maximal connected region in \(\lambda\)-space where the map \(E_{\lambda}\) has an attracting orbit. These hyperbolic components were studied by I. N. Baker and P. J. Rippon [Math. Proc. Camb. Philos. Soc. 105, No. 2, 357–375 (1989; Zbl 0705.33001)], who showed in particular that between any two components of period \(n\), there are infinitely many components of period \(n+1\). In the article under consideration, the authors assign a combinatorial object, called an “\(S\)-kneading sequence”, to each hyperbolic component. (Roughly speaking, this sequence encodes the relative positions of the periodic cycle of Fatou components of \(E_{\lambda}\).) The main result of the article is that every such sequence is indeed realized by a hyperbolic component. This provides a more precise version of Baker and Rippon’s result. The proof augments ideas which were already present in the earlier article with the new ingredient of labelling components combinatorially. The main result was obtained independently, and with a similar proof, by D. Schleicher [Ann. Acad. Sci. Fenn., Math. 28, No. 1, 3–34 (2003; Zbl 1088.30016)]. (Readers should note that the combinatorial language in both articles is slightly different; however, Schleicher’s intermediate external addresses are equivalent to the authors’ notion of \(S\)-kneading sequences.) Schleicher also established the uniqueness of the labeling; i.e., no two hyperbolic components correspond to the same combinatorial sequence. (These results were first presented in his habilitation thesis [On the dynamics of iterated exponential maps, TU München, May (1999)].) Reviewer: Lasse Rempe (Coventry) Cited in 6 Documents MSC: 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets Keywords:exponential family; hyperbolic component; kneading sequence; transcendental dynamics; parameter space Citations:Zbl 0705.33001; Zbl 1088.30016 PDFBibTeX XMLCite \textit{R. L. Devaney} et al., Fundam. Math. 174, No. 3, 193--215 (2002; Zbl 1099.30011) Full Text: DOI