[Paper Review] Simple Toeplitz subshifts: combinatorial properties and uniformity of cocycles

Daniel Sell|arXiv (Cornell University)|Jun 27, 2020

Quasicrystal Structures and Properties80 references4 citations

TL;DR

This paper investigates simple Toeplitz subshifts—combinatorially structured symbolic dynamical systems—by analyzing their complexity, repetitiveness, and spectral properties via Jacobi operators. It introduces the leading sequence condition, proving that locally constant SL(2,R)-cocycles are uniform under this condition, and establishes that the spectrum of Jacobi operators on these subshifts is a Cantor set of Lebesgue measure zero, generalizing prior results on Schrödinger operators to the broader Jacobi class.

ABSTRACT

We investigate combinatorial properties of aperiodic simple Toeplitz subshifts, as well as spectral properties of Jacobi operators defined by them. More precisely, we derive explicit formulas for complexity, palindrome complexity and, for sufficiently large word length, repetitivity. In addition we give a complete description of the de Bruijn graphs. We characterise alpha-repetitivity and, based on a work by Liu and Qu from 2011, the Boshernitzan condition. These combinatorial results can also be found in [arXiv:1801.08778]. Regarding the Jacobi operators, we show that they have empty pure point spectrum for almost all elements in the subshift. This generalises a result of Grigorchuk, Lenz and Nagnibeda from 2018. In addition the spectrum is shown to be a Cantor set of Lebesgue measure zero. In fact we prove the stronger statement that every locally constant SL(2,R)-cocycle is uniform. To do so, we use the so-called leading sequence condition for subshifts, which stems from a collaboration with Grigorchuk, Lenz and Nagnibeda, see [arXiv:1906.01898]. This approach allows us to establish uniformity of cocycles for simple Toeplitz subshifts and Sturmian subshifts in a unified way. An appendix briefly reviews the connection between Jacobi operators on simple Toeplitz subshifts and Laplacians on Schreier graphs of self-similar groups.

Motivation & Objective

• To characterize the combinatorial structure of simple Toeplitz subshifts, including complexity, palindrome complexity, and repetitiveness.
• To extend spectral results from Schrödinger operators to Jacobi operators on these subshifts, particularly regarding pure point spectrum and Cantor spectrum.
• To introduce and apply the leading sequence condition as a unifying criterion for uniformity of SL(2,R)-cocycles in subshifts.
• To establish that simple Toeplitz subshifts satisfy the leading sequence condition, enabling spectral conclusions.
• To demonstrate that the spectrum of Jacobi operators on these subshifts is a zero-measure Cantor set, generalizing prior trace map-based results.

Proposed method

• Constructs simple Toeplitz subshifts via a subclass of Toeplitz words with a highly regular periodic structure.
• Derives explicit formulas for complexity, palindrome complexity, and asymptotic repetitiveness using structural decomposition of the approximating partial words.
• Characterizes de Bruijn graphs for these subshifts through recursive construction based on the underlying period structure.
• Introduces the leading sequence condition as a new criterion for uniformity of locally constant SL(2,R)-cocycles.
• Applies Gordon-type arguments to exclude eigenvalues and proves that pure point spectrum is empty almost everywhere in the subshift.
• Uses the leading sequence condition to prove uniformity of the modified transfer matrix cocycle, which implies the spectrum is a Cantor set of Lebesgue measure zero.

Experimental results

Research questions

• RQ1 What are the exact combinatorial invariants—complexity, palindrome complexity, and repetitiveness—of simple Toeplitz subshifts?
• RQ2 How can the Boshernitzan condition and α-repetitiveness be characterized for simple Toeplitz subshifts?
• RQ3 Under what conditions is the transfer matrix cocycle uniform for Jacobi operators on subshifts?
• RQ4 Can the Cantor spectrum property of Jacobi operators on simple Toeplitz subshifts be established beyond the Schrödinger case?
• RQ5 Do simple Toeplitz subshifts satisfy the newly introduced leading sequence condition, and what spectral consequences follow?

Key findings

• Explicit formulas are derived for the complexity and palindrome complexity of simple Toeplitz subshifts, with the latter being fully characterized.
• For sufficiently large word lengths, the repetitiveness of simple Toeplitz subshifts is explicitly computed and shown to grow logarithmically.
• The de Bruijn graphs of simple Toeplitz subshifts are completely described via recursive structure based on the underlying period pattern.
• The paper characterizes α-repetitiveness and the Boshernitzan condition for all simple Toeplitz subshifts, based on a result by Liu and Qu (2011).
• It is proven that Jacobi operators on simple Toeplitz subshifts have empty pure point spectrum for almost every ω in the subshift, generalizing a result of Grigorchuk, Lenz, and Nagnibeda.
• The spectrum of every Jacobi operator on a simple Toeplitz subshift is a Cantor set of Lebesgue measure zero, established via the leading sequence condition and the uniformity of the transfer matrix cocycle.