TL;DR
This survey explores the theory of infinite episturmian words, generalizing Sturmian words to larger alphabets, covering their properties, morphisms, and connections to other combinatorial sequences.
Contribution
It provides a comprehensive overview of the properties, morphisms, and related sequences of episturmian words, expanding the understanding of their combinatorial structure.
Findings
Characterization of factor complexity and palindromes in episturmian words
Analysis of lexicographical and balance properties
Connections to Arnoux-Rauzy sequences and episkew words
Abstract
In this paper, we survey the rich theory of infinite episturmian words which generalize to any finite alphabet, in a rather resembling way, the well-known family of Sturmian words on two letters. After recalling definitions and basic properties, we consider episturmian morphisms that allow for a deeper study of these words. Some properties of factors are described, including factor complexity, palindromes, fractional powers, frequencies, and return words. We also consider lexicographical properties of episturmian words, as well as their connection to the balance property, and related notions such as finite episturmian words, Arnoux-Rauzy sequences, and "episkew words" that generalize the skew words of Morse and Hedlund.
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