Orbifold expansion and bounded Fatou components

The notion of expansion on its various forms is fundamental in the study of dynamical systems. In holomorphic dynamics, expansion for a function f has frequently been understood in terms of a conformal metric defined on a neighbourhood of its Julia set J(f). In fact, hyperbolic transcendental entire functions, that is, those for which the postsingular set is a compact subset of the Fatou set, are equivalently characterized as being expanding with respect to a hyperbolic metric. In this talk I will generalize these ideas further and consider a class of maps for which the postsingular set is not even bounded. For each function f in this class, I will sketch the construction of associated orbifolds so that f is expands certain orbifold metric in a neighbourhood of J(f). In particular, this expansion will be used to generalize existing results for hyperbolic functions, giving criteria for the boundedness of Fatou components and local connectivity of Julia sets.