Abstract:

This is a joint work with M. Volker Mayer. Some background and historical outline will be provided. The talk will concern random dynamics of transcendental meromorphic functions $f: C \to \bar C$. We will discuss the existence of random conformal measures and their invariant versions.

An appropriately defined spectral gap property will be shown. In classical cases there is a natural and powerful proof of this property which stems from Garrett Birkhoff's contraction principle for operators preserving a positive cone.

This method however fails in our non-compact realm. We will nevertheless define appropriate invariant cones of positive functions and will revive an old approach of Rufus Bowen to overcome this difficulty.