On periodicity of generalized pseudostandard words

TL;DR

This paper investigates the periodicity of generalized pseudostandard words, providing a complete characterization over binary and ternary alphabets and proposing a conjecture for larger alphabets.

Contribution

It offers a necessary and sufficient condition for the periodicity of generalized pseudostandard words over binary and ternary alphabets, and conjectures a general condition for any alphabet.

Findings

Characterization of periodicity over binary alphabet

Characterization of periodicity over ternary alphabet

Conjecture on periodicity conditions for larger alphabets

Abstract

Generalized pseudostandard words were introduced by de Luca and De Luca in 2006. In comparison to the palindromic and pseudopalindromic closure, only little is known about the generalized pseudopalindromic closure and the associated generalized pseudostandard words. In this paper we provide a necessary and sufficient condition for their periodicity over binary and ternary alphabet. More precisely, we describe how the directive bi-sequence of a generalized pseudostandard word has to look like in order to correspond to a periodic word. We state moreover a conjecture concerning a necessary and sufficient condition for periodicity over any alphabet.