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

Abstract

<p style='text-indent:20px;'>This paper presents a singular function on the unit interval <inline-formula><tex-math id="M1">\begin{document}$ [0, 1] $\end{document}</tex-math></inline-formula> derived from the dynamics of one-dimensional elementary cellular automaton Rule <inline-formula><tex-math id="M2">\begin{document}$ 150 $\end{document}</tex-math></inline-formula>. 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.</p>