Appears in collection : Advancing Bridges in Complex Dynamics / Avancer les connections dans la dynamique complexe

Several important problems in complex dynamics are centered around the local connectivity of Julia sets of polynomials and of the Mandelbrot set. Importantly, when the Julia set of a polynomial is locally connected, the topological dynamics ofthe map can be completely described as a quotient of a power map on the circle.Local connectivity of the Julia set is less significant for transcendental entire functions. Nevertheless, by restricting to a class of transcendental entire functions, known as docile functions, we obtain a similar concept by describing the topological dynamics as a quotient of a simpler disjoint-type map. We will discuss the notion ofdocile functions, as well as some of their properties. This is joint work with Lasse Rempe.