Abstract:

Alexandre Eremenko conjectured on 1989 that every escaping point of an entire function could be joined with infinity by a curve of escaping points. On 2011, G. Rottenfußer, J. Rückert, L. Rempe and D. Schleicher gave a counterexample to this in the Eremenko-Lyubich class B of entire functions with a bounded set of singular values. They constructed a function for which every path-connected component of its Julia set is bounded. Recently C. Bishop using a new technique called quasiconformal folding has been able to refine their arguments and construct a counterexample in class S, this is with a finite number of singular values (namely two).