# American Institute of Mathematical Sciences

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

 Department of Mathematics, Kyoto University of Education, Kyoto, 612-8522, Japan

Received  October 2020 Revised  March 2021 Published  April 2021

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.

Citation: Akane Kawaharada. Singular function emerging from one-dimensional elementary cellular automaton Rule 150. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021125
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##### References:
Spatio-temporal pattern of Rule $150$, $\{T_{150} x_o\}_{n = 0}^{31}$
Limit set of Rule $150$ from the single site seed $x_o$
$\{cum_{150}(n)\}$ of Rule $150$ for $0 \leq n < 256$
$F(x)$ and the limit set of Rule $150$
Local rule of Rule $150$
 $x_{i-1} x_i x_{i+1}$ $111$ $110$ $101$ $100$ $011$ $010$ $001$ $000$ $(T_{150} x)_i$ $1$ $0$ $0$ $1$ $0$ $1$ $1$ $0$
 $x_{i-1} x_i x_{i+1}$ $111$ $110$ $101$ $100$ $011$ $010$ $001$ $000$ $(T_{150} x)_i$ $1$ $0$ $0$ $1$ $0$ $1$ $1$ $0$
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