A Proof of Rauzy's Conjecture on Abelian Complexity
M\'elodie Andrieu, L\'eo Vivion
TL;DR
This paper proves Rauzy's conjecture, establishing that no infinite ternary words with rationally independent letter frequencies can have a constant abelian complexity of 3, extending the understanding of Sturmian word properties.
Contribution
It provides a proof that no infinite ternary words with rationally independent letter frequencies have constant abelian complexity of 3, confirming a long-standing conjecture.
Findings
Proved Rauzy's conjecture on ternary words
Established limitations on abelian complexity for certain infinite words
Extended the characterization of Sturmian words to ternary alphabets
Abstract
A celebrated theorem by Coven and Hedlund (1973) states that Sturmian words are characterized by their abelian complexity: they are precisely the infinite words with rationally independent letter frequencies and constant abelian complexity equal to 2. In this article, we prove a conjecture of Rauzy (1983), showing that there do not exist infinite ternary words with rationally independent letter frequencies and constant abelian complexity equal to 3.
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