Singular functions obtained via random function iteration

Abstract: In this paper we consider a discrete-time dynamical system on the real line by random iteration of two functions. These functions are assumed to satisfy appropriate monotonicity conditions; optionally, a symmetry condition may be imposed. Using Bernoulli measures on the space of binary sequences we show that sequences generated by the iteration process almost surely diverge to either plus or minus infinity. The function that assigns to each initial point the probability that the iterates diverge to plus infinity is shown to satisfy a functional equation that encodes self-similarity properties. In this way we obtain singular functions that are well-known from the literature: Cantor-like functions, Lebesgue singular functions, and the Minkowski question mark function.