Surjunctivity for cellular automata in Besicovitch spaces

TL;DR

This paper investigates cellular automata within Besicovitch spaces, demonstrating that under certain conditions, properties like injectivity imply surjectivity, similar to classical topologies.

Contribution

It introduces a new topological framework for cellular automata using Besicovitch pseudodistance and proves key properties are preserved in this setting.

Findings

Injectivity implies surjectivity under certain hypotheses

Cellular automata retain some classical properties in Besicovitch spaces

New topology provides a different perspective on cellular automata behavior

Abstract

The Besicovitch pseudodistance measures the relative size of the set of points where two functions take different values; the quotient space modulo the induced equivalence relation is endowed with a natural metric. We study the behavior of cellular automata in the new topology and show that, under suitable additional hypotheses, they retain certain properties possessed in the usual product topology; in particular, that injectivity still implies surjectivity.