Random Dynamics of Transcendental Functions

TL;DR

This paper develops a thermodynamical formalism for the random dynamics of a broad class of transcendental functions, establishing invariant measures, spectral properties, and statistical limit theorems in unbounded phase spaces.

Contribution

It introduces a novel thermodynamical framework for hyperbolic transcendental functions, including new invariant cones for unbounded phase spaces, extending classical dynamical systems theory.

Findings

Existence of unique geometric and fiberwise Gibbs states

Spectral gap and exponential decay of correlations

Central limit theorem for the dynamics

Abstract

This work concerns random dynamics of hyperbolic entire and meromorphic functions of finite order and whose derivative satisfies some growth condition at infinity. This class contains most of the classical families of transcendental functions 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 construction 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.