# Some topological properties of one dimensional cellular automata

## TL;DR

This paper investigates the topological properties of one-dimensional cellular automata, focusing on their periodic factors and classifying these factors based on their periods, revealing complex behaviors in certain automata.

## Contribution

It introduces a classification of periodic factors of cellular automata based on their periods and analyzes the existence of multiple classes of periodic factors in specific automata.

## Key findings

- Classification of periodic factors by their periods
- Existence of infinitely many classes of periodic factors in certain automata
- Surjective automata with equicontinuity points can have complex periodic factor structures

## Abstract

Cellular automata, CA for short are continuous maps defined on the set of configurations over a finite alphabet A that commutes with the shift. They are characterized by the existence of local function which determine by local behavior the image of an element of the configurations space. We will study periodic factors of cellular automata. We classify periodic factors according to their periods. We show that for a surjective cellular automaton with equicontinuity points but without being equicontinuous there is an infinity of classes of equivalence of periodic factors