The number of ergodic measures for transitive subshifts under the regular bispecial condition

If $\mathcal{A}$ is a finite set (alphabet), the shift dynamical system consists of the space ${\mathcal{A}}^{\mathbb{N}}$ of sequences with entries in $\mathcal{A}$, along with the left shift operator $S$. Closed $S$-invariant subsets are called subshifts and arise naturally as encodings of other systems. In this paper, we study the number of ergodic measures for transitive subshifts under a condition (“regular bispecial condition”) on the possible extensions of words in the associated language. Our main result shows that under this condition, the subshift can support at most $\frac{K+1}{2}$ ergodic measures, where $K$ is the limiting value of $p(n+1)-p(n)$, and $p$ is the complexity function of the language. As a consequence, we answer a question of Boshernitzan from `84, providing a combinatorial proof for the bound on the number of ergodic measures for interval exchange transformations.