Words with Factor Complexity $2n+1$ and Minimal Critical Exponent

Abstract

Let the word ${\mathbf G}$ be the fixed point of the morphism $\gamma$ sending $0$ to $01$, $1$ to $2$, and $2$ to $02$. In 2019, Shallit and Shur showed that ${\mathbf G}$ has factor complexity $2n+1$. They also showed that ${\mathbf G}$ has critical exponent $\mu=2+\frac{1}{\lambda^2-1}= 2.4808726\cdots$, where $\lambda=1.7548777$ is the real zero of $x^3-2x^2+x-1=0$. They conjectured that this was the least possible critical exponent among the words with factor complexity $2n+1$. We confirm their conjecture. We anticipate that our method, including an intricate case analysis by computer, will have wider application.