A proof of Rauzy's conjecture on abelian complexity

We resolve a conjecture posed by Rauzy in 1983 concerning the $d$-ary generalizations of Sturmian words.

A classical theorem by Coven and Hedlund from 1973 states that Sturmian words, which are classically defined by their subword complexity (a combinatorial refinement of topological entropy), or as the natural codings of irrational rotations on the circle, are also characterized by their abelian complexity (a combinatorial refinement of the notion of discrepancy).

More precisely, Sturmian words are exactly the binary infinite words with rationally independent letter frequencies and minimal abelian complexity, which is in this case constant and equal to $2$.

We prove that there exist no infinite ternary words with rationally independent letter frequencies and constant abelian complexity equal to $3$, thereby establishing Rauzy's conjecture.