Random iteration on hyperbolic Riemann surfaces

TL;DR

This paper investigates the long-term behavior of sequences of holomorphic self-maps on hyperbolic Riemann surfaces generated by iterative compositions, extending existing results from the unit disk to more general surfaces.

Contribution

It establishes new theorems for left iterated function systems on hyperbolic Riemann surfaces, generalizing prior results from the unit disk case.

Findings

Analogues of known theorems for right iterated systems are proved for left systems.

Results extend to general hyperbolic Riemann surfaces.

The paper generalizes previous work from the unit disk to broader settings.

Abstract

Let $\{f_\nu\}$ be a sequence of holomorphic self-maps of a hyperbolic Riemann surface $X$. In this paper we shall study the asymptotic behavior of the sequences obtained by iteratively left-composing or right-composing the maps $\{f_\nu\}$; the sequences of self-maps of $X$ so obtained are called left (respectively, right) iterated function systems. We shall prove the analogue for left iterated function systems of the theorems proved by Beardon, Carne, Minda and Ng for right iterated function systems with value in a Bloch domain; and we shall extend to the setting of general hyperbolic Riemann surfaces results obtained by Short and the second author in the unit disk for iterated function systems generated by maps close enough to a given self-map.