Powers in a class of A-strict standard episturmian words

TL;DR

This paper investigates the occurrence of integer powers in a specific class of standard episturmian words, extending known results from Sturmian words and utilizing canonical decompositions and generalized singular words.

Contribution

It explicitly characterizes all integer powers in a class of A-strict standard episturmian words, generalizing previous work on Sturmian and Fibonacci words.

Findings

Explicit determination of all integer powers in the class of words.

Extension of results from Sturmian to episturmian words.

Application to the k-bonacci word example.

Abstract

This paper concerns a specific class of strict standard episturmian words whose directive words resemble those of characteristic Sturmian words. In particular, we explicitly determine all integer powers occurring in such infinite words, extending recent results of Damanik and Lenz (2003), who studied powers in Sturmian words. The key tools in our analysis are canonical decompositions and a generalization of singular words, which were originally defined for the ubiquitous Fibonacci word. Our main results are demonstrated via some examples, including the $k$-bonacci word: a generalization of the Fibonacci word to a $k$-letter alphabet ($k\geq2$).