Singular functions obtained via random function iteration

TL;DR

This paper studies a random iterative process on the real line, showing that it almost surely diverges and produces well-known singular functions such as Cantor-like and Minkowski functions.

Contribution

It introduces a framework for generating singular functions via random iteration of monotonic functions with self-similarity properties.

Findings

Sequences diverge to infinity with probability one.

The probability function satisfies a self-similar functional equation.

Reproduces classical singular functions like Cantor and Minkowski functions.

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.