Spiders' webs and locally connected Julia sets of transcendental entire functions (Q2842231)

A connected subset \(E\) of the plane is called a spider's web if there exists a sequence \((G_k)\) of bounded, simply connected domains whose union is the plane such that \(\partial G_k\subset E\) and \(G_k\subset G_{k+1}\), for all \(k\). Spider's webs were introduced by Rippon and Stallard who showed that Julia sets of certain entire functions have this structure. The main result of the present paper says that if the Julia set of a transcendental entire function is locally connected, then it is a spider's web. In the opposite direction it is shown that if the Julia set is a spider's web, then it is locally connected at a dense subset consisting of buried points.