Abstract
This overview paper is devoted to Sturmian words. The first part summarizes different characterizations of Sturmian words. Besides the well known theorem of Hedlund and Morse it also includes recent results on the characterization of Sturmian words using return words or palindromes. The second part deals with substitution invariant Sturmian words, where we present our recent results. We generalize one-sided Sturmian words using the cut-and-project scheme and give a full characterization of substitution invariant Sturmian words.
Highlights
In recent years, the combinatorial properties of finite and infinite words have become significantly important in fields of physics, biology, mathematics and computer science
In the last section we present some open problems related to generalizations of Sturmian words
One can see in Example 2.1 that there exist Sturmian words which are generated by certain maps
Summary
The combinatorial properties of finite and infinite words have become significantly important in fields of physics, biology, mathematics and computer science. Since many stable and unstable aperiodic structures have been discovered Sturmian words are infinite words over a binary alphabet with exactly n+1 factors of length n for each n 3 0 They represent the simplest family of quasi-crystals. Markoff was the first to prove the validity of Bernoulli’s description He did that in his work [7], where he described the terms of the sequence [(n + 1)a] - [na] -[a], n 3 1 (later known as a mechanical sequence). The first detailed investigation of Sturmian words is due to Hedlund and Morse [4], who studied such words from the point of view of symbolic dynamics and, introduced the term „Sturmian“; named after the mathematician Charles Francois Sturm. In the last section we present some open problems related to generalizations of Sturmian words
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