Boundaries of Multiply Connected Fatou Components: A Unified Approach

Abstract We analyze the boundaries of multiply connected Fatou components of both rational and transcendental maps by means of universal covering maps and associated inner functions. A unified approach is presented, which includes invariant Fatou components (of any type) as well as wandering domains. We prove that any Fatou component admits a harmonic measure on its boundary whose support is the whole boundary. Consequently, we establish a precise connection between the geometric structure of such Fatou components (in terms of the limit sets of their universal covering maps), the dynamics induced on their boundary from an ergodic point of view, and analytic properties of the associated inner function. This approach leads to new results for both rational and transcendental maps.