Random iteration on hyperbolic Riemann surfaces

Abstract

AbstractLet $$\{f_\nu \}\subset \mathop {\mathrm {Hol}}\nolimits (X,X)$$ { 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 behaviour of the sequences obtained by iteratively left-composing or right-composing the maps $$\{f_\nu \}$$ { 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 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 $$\mathbb {D}$$ D for iterated function systems generated by maps close enough to a given self-map.