Discontinuous Riemann integrable functions emerging from cellular automata

Abstract

This paper presents discontinuous Riemann integrable functions on the unit interval [0, 1] derived from the dynamics of two-dimensional elementary cellular automata. Based on the self-similarities of their orbits, we write down the numbers of nonzero states in the spatial and spatio-temporal patterns and obtain discontinuous Riemann integrable functions by normalizing the values. We calculate the integrals of the two obtained functions over [0, 1] and demonstrate the relationship between them.