Pattern Sturmian Sequences

• Pattern Sturmian sequences are symbolic sequences defined by a maximal pattern complexity of 2n, generalizing the classical Sturmian sequences.
• They are generated via circle rotation codings and Toeplitz systems, providing a framework for understanding recurrence and aperiodic order.
• Their study informs spectral theory, as seen in Schrödinger operators, and extends to multidimensional codings with applications in quasicrystals.

Pattern Sturmian sequences are symbolic sequences defined through a refinement of classical complexity that counts patterns along arbitrary finite sets of positions rather than only contiguous blocks. In the one-dimensional binary setting, a nonperiodic sequence is called pattern Sturmian when its maximal pattern complexity satisfies $p_x^*(n)=2n$ for all $n$, the minimal possible linear growth for nonperiodic sequences in the sense of Kamae–Zamboni (Le et al., 19 Aug 2025). The term also has a distinct higher-dimensional usage: in $\mathbb{Z}^d$, it refers to codings of codimension-one cut-and-project configurations whose combinatorics are captured by an exact connected-support complexity law generalizing the Sturmian formula $p(n)=n+1$ (Barbieri et al., 2022). Across these usages, the common theme is aperiodic order of minimal complexity relative to the ambient notion of pattern.

1. Terminology, complexity, and the scope of the concept

For a one-sided sequence $x \in A^{\mathbb{N}_0}$ and a finite pattern window $S \subset \mathbb{N}_0$ of size $n$, maximal pattern complexity is defined by

$$p_x(S):=\bigl|\{(x(m+s))_{s\in S}:m\in\mathbb{N}_0\}\bigr|, \qquad p_x^*(n):=\sup\{p_x(S):|S|=n\}.$$

Kamae–Zamboni’s Morse–Hedlund analogue states that $x$ is not eventually periodic if and only if $p_x^*(n)\ge 2n$ for every $n$0; accordingly, pattern Sturmian sequences are the nonperiodic sequences achieving the minimal value $n$1 for all $n$2 (Le et al., 19 Aug 2025). In the recurrent two-sided setting, the same threshold $n$3 is the minimal possible linear growth for aperiodic sequences, and recurrent aperiodic sequences with $n$4 are called pattern Sturmian (Damanik et al., 2015).

This notion strictly contains the class of classical Sturmian sequences. Classical Sturmian words are recurrent aperiodic binary sequences of minimal subword complexity,

$n$5

where $n$6 counts contiguous factors of length $n$7 (Damanik et al., 2015). All Sturmian sequences are pattern Sturmian, but the converse fails; in particular, the pattern Sturmian class includes Toeplitz sequences that are not classical Sturmian (Damanik et al., 2015).

A recurrent source of ambiguity is that the phrase “pattern Sturmian” is used in two related but nonidentical senses. In one dimension, the standard meaning is the maximal-pattern-complexity condition $n$8. In the multidimensional work on $n$9, the phrase is attached to symbolic codings characterized by exact pattern complexity on finite connected supports, and in dimension $\mathbb{Z}^d$0 this reduces to the classical Sturmian factor-complexity law $\mathbb{Z}^d$1 rather than to maximal pattern complexity itself (Barbieri et al., 2022). This suggests that the terminology is unified by minimal aperiodic complexity, but relative to different sampling geometries.

2. Relation to classical Sturmian sequences

Classical Sturmian sequences are infinite binary words characterized by minimal factor complexity and balancedness. If $\mathbb{Z}^d$2 denotes the number of distinct length-$\mathbb{Z}^d$3 factors, Sturmian means $\mathbb{Z}^d$4 for every $\mathbb{Z}^d$5; equivalently, a binary sequence is Sturmian and non-eventually periodic if and only if it is balanced, meaning that among any two equal-length factors the number of $\mathbb{Z}^d$6’s differs by at most $\mathbb{Z}^d$7 (Enter et al., 2019).

Their standard representation is by irrational circle rotation. For irrational slope $\mathbb{Z}^d$8 and intercept $\mathbb{Z}^d$9, the standard mechanical word is

$p(n)=n+1$0

Equivalently, with $p(n)=n+1$1 and the partition $p(n)=n+1$2, $p(n)=n+1$3 of the circle, one sets

$p(n)=n+1$4

This is the classical coding by an irrational rotation (Enter et al., 2019).

Pattern Sturmian sequences generalize Sturmian minimality from contiguous blocks to arbitrary finite windows. The relationship is explicit: all Sturmian sequences satisfy both $p(n)=n+1$5 and $p(n)=n+1$6 (Damanik et al., 2015). In the higher-dimensional framework of connected supports, the specialization to $p(n)=n+1$7 again yields the classical law. For an asymptotic pair satisfying the flip condition with difference set $p(n)=n+1$8, the pattern-complexity statement gives

$p(n)=n+1$9

recovering the Morse–Hedlund–Coven characterization (Barbieri et al., 2022).

A common misconception is that “pattern Sturmian” is simply a synonym for “Sturmian.” The literature does not support this. The one-dimensional class defined by $x \in A^{\mathbb{N}_0}$0 is strictly larger than the Sturmian class, while the multidimensional usage shifts the emphasis from arbitrary windows to finite connected supports and from single sequences to asymptotic-pair and cut-and-project structure (Damanik et al., 2015).

3. Dynamical classification in one dimension

A recent structural classification separates recurrent and nonrecurrent pattern Sturmian sequences by the topology of the maximal equicontinuous factor (MEF) of the generated subshift (Le et al., 19 Aug 2025). If $x \in A^{\mathbb{N}_0}$1 is recurrent, then $x \in A^{\mathbb{N}_0}$2 is pattern Sturmian if and only if it is exactly one of two types: 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 answers the question of classifying recurrent pattern Sturmian sequences posed by Kamae and Zamboni.

For nonrecurrent sequences, the classification is different. If $x \in A^{\mathbb{N}_0}$3 is nonrecurrent and pattern Sturmian, then $x \in A^{\mathbb{N}_0}$4 is either a nonrecurrent simple circle rotation coding sequence by two intervals, or almost constant in the sense that there exists an infinite set $x \in A^{\mathbb{N}_0}$5 with upper Banach density $x \in A^{\mathbb{N}_0}$6 such that $x \in A^{\mathbb{N}_0}$7 is the characteristic function of $x \in A^{\mathbb{N}_0}$8 or of $x \in A^{\mathbb{N}_0}$9 (Le et al., 19 Aug 2025). Kamae–Zamboni examples with $S \subset \mathbb{N}_0$0 fall into this almost-constant regime.

The MEF viewpoint yields a broader theorem for sequences of non-superlinear maximal pattern complexity. If $S \subset \mathbb{N}_0$1 is recurrent, nonperiodic, and $S \subset \mathbb{N}_0$2, then $S \subset \mathbb{N}_0$3 is uniformly recurrent and its subshift is minimal with MEF either an odometer or $S \subset \mathbb{N}_0$4; correspondingly, $S \subset \mathbb{N}_0$5 is either a periodic interleaving of constant or circle-rotation-coding sequences for the same irrational $S \subset \mathbb{N}_0$6, or belongs to an $S \subset \mathbb{N}_0$7-hole Toeplitz subshift (Le et al., 19 Aug 2025). The proof mechanism is that finite boundary in an MEF coding partition forces linear lower bounds on $S \subset \mathbb{N}_0$8, while higher-dimensional connected compact abelian groups cannot be separated by finite boundaries. A plausible implication is that linear maximal pattern complexity is rigid enough to exclude most null-system geometries beyond circles and odometers.

The same work shows that recurrent but not uniformly recurrent sequences cannot stay in the linear regime: if $S \subset \mathbb{N}_0$9 is recurrent but not uniformly recurrent, then

$n$0

Thus pattern Sturmian behavior is inseparable from uniform recurrence in the recurrent case (Le et al., 19 Aug 2025).

4. Canonical constructions: rotation codings and Toeplitz systems

Circle rotation codings provide one canonical source of pattern Sturmian sequences. Let $n$1 with irrational $n$2, and partition $n$3 into two intervals $n$4 whose union is the circle. For letters $n$5, define

$n$6

For two-interval partitions, such codings satisfy $n$7 for all $n$8, and therefore nonperiodic examples are pattern Sturmian (Le et al., 19 Aug 2025). When the intervals are half-open and of lengths $n$9 and $$p_x(S):=\bigl|\{(x(m+s))_{s\in S}:m\in\mathbb{N}_0\}\bigr|, \qquad p_x^*(n):=\sup\{p_x(S):|S|=n\}.$$0, one recovers classical Sturmian words (Le et al., 19 Aug 2025).

Endpoint placement governs recurrence. A rotation coding is nonrecurrent if and only if one interval has endpoints at orbit points of $$p_x(S):=\bigl|\{(x(m+s))_{s\in S}:m\in\mathbb{N}_0\}\bigr|, \qquad p_x^*(n):=\sup\{p_x(S):|S|=n\}.$$1, namely

$$p_x(S):=\bigl|\{(x(m+s))_{s\in S}:m\in\mathbb{N}_0\}\bigr|, \qquad p_x^*(n):=\sup\{p_x(S):|S|=n\}.$$2

so certain local patterns occur exactly once (Le et al., 19 Aug 2025). This gives a precise mechanism by which a coding of an irrational rotation can fail recurrence without losing minimal maximal pattern complexity.

Toeplitz sequences provide the other major family. In the framework of Schrödinger operators, a Toeplitz word is built by iteratively filling periodic “skeletons” with holes; simple Toeplitz words are 1-hole constructions over two letters, and every simple Toeplitz sequence over two letters is pattern Sturmian (Damanik et al., 2015). Gjini–Kamae–Tan–Xue showed that pattern Sturmian Toeplitz words are exactly compositions $$p_x(S):=\bigl|\{(x(m+s))_{s\in S}:m\in\mathbb{N}_0\}\bigr|, \qquad p_x^*(n):=\sup\{p_x(S):|S|=n\}.$$3 of a single partial skeleton with a simple Toeplitz word $$p_x(S):=\bigl|\{(x(m+s))_{s\in S}:m\in\mathbb{N}_0\}\bigr|, \qquad p_x^*(n):=\sup\{p_x(S):|S|=n\}.$$4; in particular, every pattern Sturmian Toeplitz word is 2-letter (Damanik et al., 2015).

The more recent dynamical classification refines this by identifying the relevant recurrent class as nearly simple Toeplitz subshifts. A nearly simple Toeplitz sequence is a 1-hole Toeplitz sequence that is either simple Toeplitz, or a shift of the image of a simple Toeplitz sequence under a single constant-length morphism $$p_x(S):=\bigl|\{(x(m+s))_{s\in S}:m\in\mathbb{N}_0\}\bigr|, \qquad p_x^*(n):=\sup\{p_x(S):|S|=n\}.$$5 of the form $$p_x(S):=\bigl|\{(x(m+s))_{s\in S}:m\in\mathbb{N}_0\}\bigr|, \qquad p_x^*(n):=\sup\{p_x(S):|S|=n\}.$$6, $$p_x(S):=\bigl|\{(x(m+s))_{s\in S}:m\in\mathbb{N}_0\}\bigr|, \qquad p_x^*(n):=\sup\{p_x(S):|S|=n\}.$$7, together with the technical constraint $$p_x(S):=\bigl|\{(x(m+s))_{s\in S}:m\in\mathbb{N}_0\}\bigr|, \qquad p_x^*(n):=\sup\{p_x(S):|S|=n\}.$$8 from the cited classification (Le et al., 19 Aug 2025). The orbit closure of such a sequence is a nearly simple Toeplitz subshift.

5. Multidimensional pattern Sturmian configurations

In the multidimensional setting, the central objects are configurations $$p_x(S):=\bigl|\{(x(m+s))_{s\in S}:m\in\mathbb{N}_0\}\bigr|, \qquad p_x^*(n):=\sup\{p_x(S):|S|=n\}.$$9 and asymptotic pairs $x$0 with finite difference set

$x$1

Such a pair is indistinguishable if, for every finite pattern $x$2, the occurrence sets in $x$3 and $x$4 differ by equally many points on each side; equivalently,

$x$5

The paper isolates a flip condition: $x$6, the restriction $x$7 is a bijection $x$8, and the values on $x$9 are cyclically permuted from $p_x^*(n)\ge 2n$0 to $p_x^*(n)\ge 2n$1 (Barbieri et al., 2022).

The main combinatorial theorem states that for an asymptotic pair satisfying the flip condition, the following are equivalent: indistinguishability; a one-occurrence condition for all patterns on finite connected supports; and the exact complexity formula

$p_x^*(n)\ge 2n$2

for every nonempty finite connected $p_x^*(n)\ge 2n$3 (Barbieri et al., 2022). For rectangles $p_x^*(n)\ge 2n$4 with side lengths $p_x^*(n)\ge 2n$5,

$p_x^*(n)\ge 2n$6

For $p_x^*(n)\ge 2n$7 this is $p_x^*(n)\ge 2n$8, and for $p_x^*(n)\ge 2n$9 it is $n$00 (Barbieri et al., 2022).

Under uniform recurrence, the structure becomes geometric. If $n$01 is uniformly recurrent, then $n$02 is an indistinguishable asymptotic pair satisfying the flip condition if and only if there exists a totally irrational $n$03 such that $n$04 and $n$05, the lower and upper characteristic $n$06-dimensional Sturmian configurations defined by a codimension-one cut-and-project scheme with internal space $n$07 (Barbieri et al., 2022). The symbolic coding is obtained by partitioning $n$08 into $n$09 consecutive intervals determined by $n$10, with star map

$n$11

The characteristic configurations admit the mechanical formulas

$n$12

This furnishes a multidimensional generalization of the Sturmian characterization by minimal complexity. The flip condition is essential: without it, higher-dimensional pathologies appear, recurrence need not imply uniform recurrence, and simple complexity bounds can fail (Barbieri et al., 2022).

6. Spectral theory for Schrödinger operators with pattern Sturmian potentials

Pattern Sturmian sequences have been studied as potentials for discrete Schrödinger operators

$n$13

with bounded potential $n$14 generated by the symbolic sequence (Damanik et al., 2015). For Sturmian potentials, it is classical that the spectrum has zero Lebesgue measure and all spectral measures are purely singular continuous. Damanik, Liu, and Qu conjectured that the same picture extends to all pattern Sturmian potentials (Damanik et al., 2015).

Their main theorem confirms the conjecture for the Toeplitz subclass: if $n$15 has potential given by a pattern Sturmian Toeplitz sequence, then $n$16 and all spectral measures are purely singular continuous (Damanik et al., 2015). The proof combines strict ergodicity of Toeplitz subshifts, uniformity of the Schrödinger cocycle via the Boshernitzan condition, an explicit block-trace description of the spectrum through recursively defined words $n$17, and Gordon-type criteria excluding eigenvalues by exploiting repeated squares and cubes (Damanik et al., 2015).

The same paper records partial results for other subclasses. Circle map sequences

$n$18

with irrational $n$19, are pattern Sturmian. Zero-measure spectrum is proved under various parameter regimes, and absence of absolutely continuous spectrum holds in full generality, while absence of point spectrum is established for large parameter sets or arithmetic conditions (Damanik et al., 2015). By contrast, sparse one-sided pattern Sturmian sequences of the form $n$20 at $n$21 and $n$22 otherwise, with $n$23, are nonrecurrent; for the associated half-line operators, the absolutely continuous spectrum is empty, but zero-measure spectrum and absence of point spectrum fail in general (Damanik et al., 2015). This is a precise illustration that nonrecurrent pattern Sturmian behavior is spectrally less rigid than the recurrent Toeplitz case.

A useful conceptual distinction follows. Low pattern complexity and uniform recurrence support uniform transfer-matrix behavior and zero-measure Cantor spectrum, while rich repetition structure excludes eigenvalues by Gordon-type arguments (Damanik et al., 2015). This suggests that the recurrent/nonrecurrent divide in symbolic dynamics is mirrored by a divide between robust singular continuity and mixed spectral phenomena.

7. Forbidden-pattern characterizations, examples, and open directions

For classical Sturmian systems, low complexity can be recast in terms of forbidden local configurations. In the lattice-gas construction of Aubry-type ground states, a Sturmian sequence is characterized by forbidding two types of patterns: an infinite family of pairwise distances between $n$24’s, and one finite-range block of $n$25’s (Enter et al., 2019). If $n$26 is the position of the $n$27-th $n$28 and $n$29 is the associated sequence of average distances, then

$n$30

and the forbidden set consists of all distances not in $n$31 together with the block of $n$32 consecutive $n$33’s (Enter et al., 2019). The theorem states that elements in any given Sturmian system are uniquely determined by the absence of exactly these patterns.

The Fibonacci system provides the explicit example. With the reciprocal of the golden mean, the allowed distances between $n$34’s begin

$n$35

and the forbidden distances begin

$n$36

while $n$37 is forbidden (Enter et al., 2019). Positive energies assigned to these forbidden patterns yield non-frustrated, two-body, infinite-range lattice-gas Hamiltonians whose ground states are exactly the Sturmian configurations (Enter et al., 2019).

Several open problems remain. For general one-dimensional pattern Sturmian potentials, the conjecture that all two-sided operators have zero-measure spectrum and purely singular continuous spectral measures remains open outside the Toeplitz case (Damanik et al., 2015). In the multidimensional connected-support setting, the main open question is whether a single uniformly recurrent configuration with minimal connected-support complexity $n$38 must already be a multidimensional Sturmian configuration $n$39 or $n$40; only one implication is known (Barbieri et al., 2022). Other open questions concern étale limits of characteristic pairs, the structure of more general indistinguishable pairs in $n$41, and the relation between bispecial displacement vectors and simultaneous Diophantine approximation (Barbieri et al., 2022).

Taken together, these works locate pattern Sturmian sequences at the intersection of combinatorics on words, topological dynamics, quasicrystal codings, spectral theory, and statistical mechanics. What persists across the variants is not a single formal definition, but a common rigidity principle: minimal aperiodic pattern growth forces highly constrained symbolic, geometric, and dynamical structure (Le et al., 19 Aug 2025).