Random iteration on hyperbolic Riemann surfaces

Let {f_ν} ⊂ Hol(X,X) 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_ν}; the sequences of self-maps of X so obtained are called left (respectively, right) iterated function systems. We shall obtain the analogue for left iterated function systems of the theorems proved by Bear-don, 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 D for iterated function systems generated by maps close enough to a given self-map.