On the study of cellular automata on modulo-recurrent words

TL;DR

This paper investigates stable cellular automata acting on Sturmian words, analyzing their combinatorial properties, complexity functions, and balance characteristics to deepen understanding of their structural behavior.

Contribution

It introduces a class of stable cellular automata and explores their effects on Sturmian words, including complexity and balance properties, which is a novel analysis.

Findings

Determined combinatorial properties of Sturmian words under SCA

Calculated classical and palindromic complexity functions

Proved Sturmian words are 2-balanced and characterized their abelian complexity

Abstract

In this paper, we study a class of cellular automata (CA) called stable cellular automata (SCA) that preserve stability by reflection, modulo-recurrent, and richness. After applying these automata to Sturmian words, we determine some of their combinatorial properties. Next, we calculate the classical and palindromic complexity functions of these words. Finally, we demonstrate that these words are $2$-balanced and establish their abelian complexity function.