Combinatorics of One-Dimensional Simple Toeplitz Subshifts
Abstract
This paper provides a systematic study of fundamental combinatorial properties of one-dimensional, two-sided infinite simple Toeplitz subshifts. Explicit formulas for the complexity function, the palindrome complexity function and the repetitivity function are proven. Moreover, a complete description of the de Bruijn graphs of the subshifts is given. Finally, the Boshernitzan condition is characterised in terms of combinatorial quantities, based on a recent result of Liu and Qu. Particular simple characterisations are provided for simple Toeplitz subshifts that correspond to the orbital Schreier graphs of the family of Grigorchuk's groups, a class of subshifts that serves as main example throughout the paper.
Cite
@article{arxiv.1801.08778,
title = {Combinatorics of One-Dimensional Simple Toeplitz Subshifts},
author = {Daniel Sell},
journal= {arXiv preprint arXiv:1801.08778},
year = {2020}
}
Comments
Minor corrections in the proofs of Proposition 3.4 and 5.4; results unchanged. In addition, several typos were corrected and some references were added