Equicontinuous factors of one dimensional cellular automata

TL;DR

This paper investigates the topological and ergodic properties of one-dimensional cellular automata, establishing limitations on eigenvalues, and demonstrating the existence of equicontinuous factors under certain conditions.

Contribution

It proves that ergodic cellular automata cannot have irrational eigenvalues and that cellular automata with equicontinuous factors also possess such factors, advancing understanding of their dynamical structure.

Findings

Ergodic cellular automata cannot have irrational eigenvalues.

Cellular automata with equicontinuous factors also have equicontinuous cellular automaton factors.

Cellular automata with almost equicontinuous points have equicontinuous measurable factors.

Abstract

We are interested in topological and ergodic properties of one dimensional cellular automata. We show that an ergodic cellular automaton cannot have irrational eigenvalues. We show that any cellular automaton with an equicontinuous factor has also as a factor an equicontinuous cellular automaton. We show also that a cellular automaton with almost equicontinuous points according to Gilman's classification has an equicontinuous measurable factor which is a cellular automaton. 2000 Mathematics Subject Classification.: 37B15, 54H20, 37A30. Key words and phrases. Cellular Automata, Dynamical systems, equicontinuous factor.