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

Abstract

This paper presents a singular function on the unit interval [0, 1] derived from the dynamics of one-dimensional elementary cellular automaton Rule 150. We describe the properties of the resulting function, which is strictly increasing, uniformly continuous, and differentiable almost everywhere, and show that it is not differentiable at dyadic rational points. We also derive functional equations that this function satisfies and show that this function is the only solution of the functional equations.

Cellular automata that generate symmetrical patterns give singular functions

“…We normalized the dynamics of the number of nonzero states and obtained functions for them. (For Rule 150, we previously obtained the results in [12].) In Section 3.2, we provide a sufficient condition of singularity for a function and show that the resulting functions for the four CAs are singular functions, which are strictly increasing, continuous, and differentiable with the derivative zero almost everywhere.…”

“…For the one-dimensional elementary CA Rule 90, the resulting function equals Salem's singular function L 1/3 , a self-affine function [6,7,8,9], and for a two-dimensional elementary CA, the resulting function equals Salem's singular function L 1/5 (numerical results were obtained in [10,11] where we showed that the difference forms of the equations match Salem's in [5]). For the one-dimensional CA Rule 150, we previously demonstrated that the resulting function is a singular function that strictly increases, is continuous, and is differentiable almost everywhere [12]. In addition, we discuss two nonlinear elementary CAs, Rule 22 and Rule 126.…”

Discontinuous Riemann integrable functions emerging from cellular automata

“…In [6,7], we showed several examples, whose one-variable functions are Salem's singular function [10,1,8,14], that is a kind of strictly increasing singular functions. In [4], we showed the existence of D-dimensional symmetrical CAs that give one-variable functions such that the parameters of Salem's singular function are 1/(2D + 1) and 1/(2 D + 1), and in [5], we showed that the one-variable function obtained for one-dimensional elementary CA Rule 150 is a new singular function other than Salem's singular function. In [3], sufficient conditions are given for a function obtained from a CA to be a singular function.…”