Local connectedness of Julia sets for transcendental entire functions (Q2753325)

Let \(f\) be a transcendental entire function, and denote by \(f^n\) the \(n\)-th iterate of \(f\). The Fatou set \(\mathcal{F}(f)\) is the maximal open set in which all iterates \(f^n\) are defined and form a normal family in the sense of Montel, while the complement of \(\mathcal{F}(f)\) in \(\widehat{\mathbb C}\) is the Julia set \(\mathcal{J}(f)\). Note that \(\infty \in \mathcal{J}(f)\) and that \(\mathcal{J}(f)\) is unbounded in \(\mathbb C\). The residual Julia set of \(f\) is the subset of points in \(\mathcal{J}(f)\) that do not lie on the boundary of any Fatou component of \(f\). NEWLINENEWLINENEWLINEAn important role in studying the dynamics of \((f^n)\) is played by the set \(\text{sing }{f^{-1}}\) of singular values of \(f\) which is the set of points \(\zeta \in \mathbb C\) such that there is no neighborhood of \(\zeta\) where a holomorphic inverse function of \(f\) exists. Note that then \(\zeta\) is a critical value or an asymptotic value of \(f\). Let \(\mathcal{S}\) denote the set of transcendental entire functions \(f\) such that \(\text{sing }{f^{-1}}\) is a finite set. The author considers the following two conditions on singular values. NEWLINENEWLINENEWLINE(SV1) If \(\zeta \in \mathcal{F}(f) \cap \text{sing }{f^{-1}}\), then \(\zeta\) is a critical value and is absorbed by an attracting cycle. NEWLINENEWLINENEWLINE(SV2) If \(\zeta \in \mathcal{J}(f) \cap \text{sing }{f^{-1}}\), then \(\overline{\bigcup_{n \geq 0} f^n(\zeta)} \cap \partial D = \emptyset\) for any Fatou component \(D\). NEWLINENEWLINENEWLINENow, let \(\mathcal{SV}\) be the set of all \(f \in \mathcal{S}\) which satisfy (SV1) and (SV2). Then the author proves that if \(D\) is a bounded cyclic Fatou component of \(f \in \mathcal{SV}\), then \(\partial D\) is a Jordan curve. If all cyclic Fatou components of \(f\) are bounded, then \(\mathcal{J}(f)\) is locally connected. NEWLINENEWLINENEWLINEFurthermore, the author considers the family \(\mathcal{D}\) of functions \(f_\lambda(z)=\lambda ze^z\) with \(\lambda \in \mathbb C\). Then \(\mathcal{D} \subset \mathcal{S}\). Let \(\Lambda = \{ \lambda \in \mathbb C : |\text{Im}{\lambda}|\geq e\text{Arg}{\lambda} \}\). Then it is shown that if \(\lambda \in \Lambda\) and if \(f_\lambda\) has an attracting cycle of period greater than one, then each Fatou component of \(f_\lambda\) is bounded and \(\mathcal{J}(f_\lambda)\) is locally connected. Furthermore, he constructs a curve which is contained in the residual Julia set of \(f_\lambda\). NEWLINENEWLINENEWLINEFinally, for \(a>1\) let \(g_a(z)=ae^a(z-1+a)e^z\). Then the author shows that \(\mathcal{J}(g_a)\) is locally connected, and it is a Sierpinski carpet. Such a set is the complement of a countable dense family of open topological discs whose diameters tend to zero and whose closures are pairwise disjoint closed topological discs. Note that any two Sierpinski carpets are homeomorphic.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00060].