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

## TL;DR

This paper establishes an upper bound on the number of ergodic measures for transitive subshifts satisfying a regular bispecial condition, linking it to the complexity function and answering a longstanding question in dynamical systems.

## Contribution

It provides a new combinatorial proof for the maximum number of ergodic measures supported by such subshifts, extending understanding of their measure-theoretic properties.

## Key findings

- Maximum of (K+1)/2 ergodic measures under the regular bispecial condition
- Connection between complexity function growth and ergodic measures
- Resolution of a question posed by Boshernitzan in 1984

## Abstract

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.