  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.
