Nearly Simple Toeplitz Subshift

• Nearly simple Toeplitz subshift is the orbit closure of a simple Toeplitz sequence or its controlled morphic shift, ensuring the pattern Sturmian property.
• The structure exhibits uniform recurrence and minimal dynamics, with its maximal equicontinuous factor explicitly given by an odometer.
• It plays a pivotal role in classifying recurrent pattern Sturmian sequences, clearly distinguishing it from other Toeplitz system variants.

A nearly simple Toeplitz subshift is, in the precise sense used in the classification of recurrent pattern Sturmian sequences, the orbit closure of a sequence that is either a simple Toeplitz sequence or a shift of the image of a simple Toeplitz sequence under a single constant-length morphism of the form $0 \mapsto w0,\; 1 \mapsto w1$, subject to technical conditions ensuring the pattern Sturmian property (Le et al., 19 Aug 2025). Within the broader literature on symbolic dynamics, the expression “nearly simple” had earlier been used only informally for Toeplitz systems that are close to simple, regular, or odometer-based regimes; the explicit definition above places the notion inside the modern structure theory of low maximal pattern complexity (Downarowicz et al., 2015).

1. Terminology and scope

The term nearly simple Toeplitz subshift is not historically uniform. Several papers discuss “simple” Toeplitz systems, regular Toeplitz systems, Toeplitz systems with separated holes, Toeplitz systems with growing blocks, or strong Toeplitz systems of finite rank, and some summaries use “nearly simple” descriptively rather than as a formal class (Grigorchuk et al., 2019). By contrast, the paper on low maximal pattern complexity gives a concrete definition: a nearly simple Toeplitz sequence is one satisfying the conclusion of the one-hole pattern Sturmian classification, namely that a 1-hole Toeplitz sequence is pattern Sturmian if and only if it is either simple Toeplitz or a shift of the image of a simple Toeplitz sequence under a morphism $0 \mapsto w0,\; 1 \mapsto w1$; the corresponding subshift is the orbit closure of such a sequence (Le et al., 19 Aug 2025).

This terminological distinction matters. “Nearly simple” in this formal sense is not merely a synonym for “simple Toeplitz,” nor is it equivalent to other well-controlled Toeplitz subclasses such as regular Toeplitz systems, Toeplitz systems with separated holes, or finite-rank strong Toeplitz systems. Its defining role is instead tied to maximal pattern complexity and the classification of recurrent pattern Sturmian sequences (Le et al., 19 Aug 2025).

2. Construction from Toeplitz sequences

A Toeplitz sequence is built from an iterative filling-in of periodic data. In the formulation used for low maximal pattern complexity, a sequence $x$ is Toeplitz with period structure $(n_k)\subset \mathbb{N}$ if $n_k$ properly divides $n_{k+1}$ for all $k$, and there exists a partition

$$\mathbb{N}_0=\bigsqcup_{k,i}(a_{k,i}+n_k\mathbb{N}_0)$$

such that $x$ is constant on each infinite arithmetic progression $a_{k,i}+n_k\mathbb{N}_0$ (Le et al., 19 Aug 2025). The number of holes at step $0 \mapsto w0,\; 1 \mapsto w1$0 is the number of nonconstant residue classes modulo $0 \mapsto w0,\; 1 \mapsto w1$1. An $0 \mapsto w0,\; 1 \mapsto w1$2-hole Toeplitz sequence has exactly $0 \mapsto w0,\; 1 \mapsto w1$3 nonconstant residue classes at each step, while a simple Toeplitz sequence is a special 1-hole Toeplitz sequence in which, at each stage, all constant progressions modulo $0 \mapsto w0,\; 1 \mapsto w1$4 take the same value (Le et al., 19 Aug 2025).

The standard topological object is the Toeplitz subshift, namely the shift orbit closure of a Toeplitz sequence. In the general symbolic-dynamical framework, Toeplitz subshifts are minimal when generated by a non-periodic Toeplitz sequence, and Toeplitz systems are equivalently minimal almost 1-1 extensions of an odometer (Kaya, 2016). The nearly simple class is obtained by starting from the 1-hole/simple Toeplitz mechanism and allowing one controlled constant-length morphic modification. This places it between the bare simple Toeplitz construction and more general Toeplitz systems with multiple holes or more elaborate $0 \mapsto w0,\; 1 \mapsto w1$5-adic presentations (Le et al., 19 Aug 2025).

A useful structural remark from the pattern-complexity classification is that if $0 \mapsto w0,\; 1 \mapsto w1$6 is nearly simple Toeplitz, meaning that $0 \mapsto w0,\; 1 \mapsto w1$7 is a shift of $0 \mapsto w0,\; 1 \mapsto w1$8 for simple Toeplitz $0 \mapsto w0,\; 1 \mapsto w1$9 and $x$0 of the form above, then for the $x$1-letter skeleton $x$2, all but one of these skeletons are constant, and the nonconstant one is simple Toeplitz (Le et al., 19 Aug 2025). This isolates the simple Toeplitz core inside the nearly simple object.

3. Recurrence, minimality, and maximal equicontinuous factor

Nearly simple Toeplitz sequences and their orbit closures inherit the basic recurrent features of classical Toeplitz systems. They are recurrent and, in fact, uniformly recurrent, and their subshifts are minimal (Le et al., 19 Aug 2025). In the general theory, Toeplitz systems are minimal almost 1-1 extensions of odometers, and their maximal equicontinuous factor is always an odometer (Downarowicz et al., 2015).

For nearly simple Toeplitz subshifts, the maximal equicontinuous factor is explicitly described as

$x$3

with addition by $x$4 as the transformation (Le et al., 19 Aug 2025). The subshift is an almost 1-1 extension of this odometer: away from a boundary set in the maximal equicontinuous factor partition, the coding map from the odometer back to the subshift is injective (Le et al., 19 Aug 2025). The pattern Sturmian constraint forces this boundary set to be a singleton in the nearly simple Toeplitz case (Le et al., 19 Aug 2025).

An important caveat is that not every point in a Toeplitz subshift is itself a Toeplitz sequence. The literature on low maximal pattern complexity emphasizes that borderline limit points may fail to be Toeplitz even when the generating sequence is nearly simple Toeplitz; the dynamical structure is carried by the subshift and its generator, not by every element individually (Le et al., 19 Aug 2025).

4. Role in pattern Sturmian classification

The principal mathematical significance of nearly simple Toeplitz subshifts is their appearance in the classification of recurrent pattern Sturmian sequences. For a sequence $x$5, maximal pattern complexity $x$6 generalizes ordinary word complexity, and pattern Sturmian means the nonperiodic minimal-growth condition

$x$7

(Le et al., 19 Aug 2025). The classification theorem states that every recurrent pattern Sturmian sequence is either a coding of an irrational circle rotation by two intervals or an element of a nearly simple Toeplitz subshift (Le et al., 19 Aug 2025).

This theorem identifies nearly simple Toeplitz subshifts as the non-rotation branch of the recurrent pattern Sturmian world. In particular, they exhaust the minimal uniformly recurrent pattern Sturmian subshifts that are not circle-rotation codings (Le et al., 19 Aug 2025). The structural dichotomy is reflected at the level of maximal equicontinuous factors: in the broader non-superlinear maximal pattern complexity regime, sequences are either nonrecurrent or minimal with maximal equicontinuous factor either an odometer or the product of a circle with a finite cyclic group (Le et al., 19 Aug 2025). The odometer branch is precisely where nearly simple Toeplitz subshifts live.

From this perspective, “nearly simple” is not a loose regularity adjective but a sharp classification label. It isolates the Toeplitz mechanisms that still attain the extremal pattern-complexity value $x$8 without collapsing to circle-rotation codings (Le et al., 19 Aug 2025).

5. Relation to simple Toeplitz subshifts and the leading-sequence framework

Simple Toeplitz subshifts form the immediate substrate for nearly simple Toeplitz systems. They have been studied in detail from both combinatorial and spectral viewpoints: explicit formulas are known for their complexity, palindrome complexity, and repetitivity, and their de Bruijn graphs are completely described (Sell, 2018). For aperiodic simple Toeplitz subshifts, every locally constant $x$9-cocycle is uniform, the associated Jacobi operators have empty pure point spectrum for almost all elements, and the spectrum is a Cantor set of Lebesgue measure zero (Sell, 2020).

A broader conceptual framework is provided by subshifts with leading sequences. This class covers all simple Toeplitz subshifts as well as all Sturmian subshifts, and for minimal uniquely ergodic subshifts satisfying the leading sequence condition, every locally constant cocycle is uniform (Grigorchuk et al., 2019). The same source states that the framework is stable under finite changes and certain morphisms, and therefore many “nearly simple Toeplitz subshifts,” in the sense of subshifts differing finitely from simple Toeplitz subshifts or obtained by suitable morphisms, will also satisfy the leading sequence condition (Grigorchuk et al., 2019).

That observation does not identify the formally defined nearly simple Toeplitz class with the full leading-sequence class. It does, however, place nearly simple Toeplitz subshifts in a setting where finite-word control, cocycle uniformity, and zero-measure Cantor spectrum are natural surrounding phenomena. In this sense, simple Toeplitz subshifts provide the explicit model case, while nearly simple Toeplitz subshifts inherit a controlled part of that structure through their morphic origin (Grigorchuk et al., 2019).

6. Distinction from other Toeplitz subclasses and invariants

The nearly simple Toeplitz class should be separated from several adjacent Toeplitz notions that organize different aspects of the theory. Toeplitz subshifts with separated holes and those with growing blocks are defined through the geometry of skeletons and filled blocks; for these subclasses, topological conjugacy is hyperfinite, and for growing blocks the conjugacy relation is Borel reducible to $(n_k)\subset \mathbb{N}$0 (Kaya, 2016). Strong Toeplitz subshifts of finite rank are defined through constant-length, primitive, proper, recognizable $(n_k)\subset \mathbb{N}$1-adic directive sequences or equivalently Bratteli–Vershik models with the equal path number property; they form a generic class among infinite minimal subshifts, but not every finite-rank Toeplitz subshift is strong (Gao et al., 8 Apr 2025).

Likewise, neither entropy nor automorphism-group size singles out nearly simple Toeplitz systems. Toeplitz automorphism groups can be as small as $(n_k)\subset \mathbb{N}$2 or much larger: in low-complexity settings they are often virtually generated by roots of the shift, yet there exist minimal Toeplitz subshifts with complexity bounded by $(n_k)\subset \mathbb{N}$3 and torsion-free, non-finitely generated automorphism groups (Donoso et al., 2017). There are also explicit Toeplitz examples whose automorphism group is the additive subgroup of $(n_k)\subset \mathbb{N}$4 generated by powers of $(n_k)\subset \mathbb{N}$5 (Salo, 2014), as well as regular Toeplitz $(n_k)\subset \mathbb{N}$6-free systems whose automorphism group consists solely of powers of the shift (Bartnicka, 2017). A closely related later study gives sufficient conditions for triviality of automorphism groups in regular Toeplitz and $(n_k)\subset \mathbb{N}$7-free settings, while also producing examples with elements of arbitrarily large finite order (Dymek et al., 2021).

This suggests that nearly simple Toeplitz subshifts are best understood through their combinatorial origin and maximal-pattern-complexity role rather than through a universal rigidity invariant. Their canonical place in current theory is as the odometer-based, morphically controlled branch of recurrent pattern Sturmian dynamics (Le et al., 19 Aug 2025).