Structural instability of the mapping z\(\to \lambda \,\exp (z)(\lambda >e^{-1})\)
For an entire function f denote by J(f) the Julia set of f (i.e. the set of all points for which the family \(\{f^ n\}\) of iterates of f fails to be normal in the sense of Montel). \textit{P. Fatou} [Acta Math. 47, 337-370 (1926)] asked for an example of a transcendental function f such that \(J(f)={\mathbb{C}}\) and he conjectured that \(J(\exp)={\mathbb{C}}\). \textit{I. N. Baker} [Ann. Acad. Sci. Fenn., Ser. A I 467, 3-11 (1970; Zbl 0197.053)] proved that there exists a constant \(k>e^ 2\) such that for \(f(z)=kze^ z\), \(z\in {\mathbb{C}}\), we have \(J(f)={\mathbb{C}}\). The next step in this direction has been done by \textit{W. Ogińska} and the reviewer [Ann. Univ. Marie Curie-Skłodowska, Sect. A 33, 111-118 (1979; Zbl 0463.30023)]. Making use of the Baker's idea we proved that J(2k\(\pi\) i exp)\(={\mathbb{C}}\) for every integer \(k\neq 0\). (Unfortunately, this paper is not cited by later researchers). Two years later \textit{M. Misiurewicz} [Ergodic Theory Dyn. Syst. 1, 103-106 (1981; Zbl 0466.30019)] proved the Fatou's conjecture. \textit{R. L. Devaney} [Bull. Am. Math. Soc., New Ser. 11, 167-171 (1984; Zbl 0542.58021)] generalized this result showing among others that J(\(\lambda\) exp)\(={\mathbb{C}}\) for every \(\lambda >e^{-1}\). The main result of the present paper says that for any given \(\lambda >e^{- 1}\) and \(\epsilon >0\) there exists a \(\lambda^*\in {\mathbb{C}}\) such that \(| \lambda^*-\Lambda | <\epsilon\) and \(J(\lambda^* \exp)\neq {\mathbb{C}}\).
scientific article; zbMATH DE number 4154826