Singular function emerging from one-dimensional elementary cellular automaton Rule 150

Abstract

In this paper, we give a singular function on a unit interval derived from the dynamic of the one-dimensional elementary cellular automaton Rule 150. We describe properties of the resulting function, that is strictly increasing, uniformly continuous, and differentiable almost everywhere, and we show that it is not differentiable at dyadic rational points. We also give functional equations that the function satisfies, and show that the function is the only solution of the functional ones.

Discontinuous Riemann integrable functions emerging from cellular automata

“…A singular function is a function that is monotonically increasing (or decreasing) and continuous everywhere, with a zero derivative almost everywhere; for example, Salem's singular function [1,2,3,4]. We studied the relationship between Salem's singular function and elementary cellular automata, Rule 90 and two two-dimensional elementary cellular automata [5,6,7,8,9], and that between another new singular function and Rule 150 [10]. This paper presents new Riemann integrable functions with countable discontinuous points from two-dimensional elementary cellular automata.…”