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.

arXiv comment18 pages

References in corpus (1)

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

Cited by in corpus (1)

• Cellular automata that generate symmetrical patterns give singular functions