If is a finite set (alphabet), the shift dynamical system consists of the space of sequences with entries in, 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 ergodic measures, where K is the limiting value of, and p is the complexity function of the language. As a consequence, we answer a question of Boshernitzan from 1984, providing a combinatorial proof for the bound on the number of ergodic measures for interval exchange transformations.
All Science Journal Classification (ASJC) codes
- Applied Mathematics
- classical ergodic theory
- formal languages
- symbolic dynamics