Random iteration on hyperbolic Riemann surfaces

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