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 ([LQ11]). 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.

Measure-theoretically mixing subshifts with low complexity

“…Our results already show that lim sup p(q)/q < 3/2 precludes X being Toeplitz; all Toeplitz shifts are minimal, and have no irrational continuous eigenvalues, so cannot have the structure of Theorem A. In the other direction, [Sel20] gives word complexity estimates for a subclass called simple Toeplitz subshifts, and those estimates show that there exist simple Toeplitz subshifts with lim sup p(q)/q = 3/2 (this happens whenever the parameter sequence (n k ) from that paper is unbounded). We now have the following.…”

Subshifts with leading sequences, uniformity of cocycles and spectra of Schreier graphs

“…If now ω ∈ A Z denotes the word that was obtained this way, we define the simple Toeplitz subshift as Ω := {T k ω : k ∈ N}. It is equal to the subshift that was defined above in terms of p (∞) (see for example [37], Proposition 2.6).…”

“…It was shown in [32], Corollary 2.1 that every simple Toeplitz subshift (Ω ω , T ) is minimal and uniquely ergodic. In addition, # A ≥ 2 implies that every simple Toeplitz word defined by (a k ) k is non-periodic (see for example [37], Proposition 2.2). Conversely, # A = 1 clearly gives periodicity of the subshift.…”