Random dynamics of transcendental functions

Abstract

This work concerns random dynamics of hyperbolic entire and meromor- phic functions of finite order and whose derivative satisfies some growth condition at infinity. This class contains most of the classical families of transcendental func- tions and goes much beyond. Based on uniform versions of Nevanlinna’s value distribution theory we first build a thermodynamical formalism which, in particular, produces unique geometric and fiberwise invariant Gibbs states. Moreover, spectral gap property for the associated transfer operator along with exponential decay of correlations and a central limit theorem are shown. This part relies on our construc- tion of new positive invariant cones that are adapted to the setting of unbounded phase spaces. This setting rules out the use of Hilbert’s metric along with the usual contraction principle. However these cones allow us to apply a contraction argument stemming from Bowen’s initial approach.