Combinatorics of One-Dimensional Simple Toeplitz Subshifts

TL;DR

This paper systematically analyzes the combinatorial properties of one-dimensional simple Toeplitz subshifts, providing explicit formulas, graph descriptions, and characterizations related to their complexity, palindromes, and repetitivity.

Contribution

It offers the first comprehensive formulas and descriptions for key combinatorial functions and structures of simple Toeplitz subshifts, including a characterization of the Boshernitzan condition.

Findings

Explicit formulas for complexity, palindrome complexity, and repetitivity functions.

Complete description of de Bruijn graphs for these subshifts.

Characterization of the Boshernitzan condition in combinatorial terms.

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.