---
title: Pattern Sturmian Sequences
url: https://www.emergentmind.com/topics/pattern-sturmian-sequences
type: topic
---

# Pattern Sturmian Sequences

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 [2508.13420]. 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 $p(n)=n+1$ [2204.06413]. 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 $n$; accordingly, pattern Sturmian sequences are the nonperiodic sequences achieving the minimal value $p_x^*(n)=2n$ for all $n$ [2508.13420]. In the recurrent two-sided setting, the same threshold $2n$ is the minimal possible linear growth for aperiodic sequences, and recurrent aperiodic sequences with $p^*(n)=2n$ are called pattern Sturmian [1511.03834].

This notion strictly contains the class of classical Sturmian sequences. Classical Sturmian words are recurrent aperiodic binary sequences of minimal subword complexity,
$$
p(n)=n+1,
$$
where $p(n)$ counts contiguous factors of length $n$ [1511.03834]. All Sturmian sequences are pattern Sturmian, but the converse fails; in particular, the pattern Sturmian class includes Toeplitz sequences that are not classical Sturmian [1511.03834].

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 $p^*(n)=2n$. In the multidimensional work on $\mathbb{Z}^d$, the phrase is attached to symbolic codings characterized by exact pattern complexity on finite connected supports, and in dimension $d=1$ this reduces to the classical Sturmian factor-complexity law $p(n)=n+1$ rather than to maximal pattern complexity itself [2204.06413]. 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 $p(n)$ denotes the number of distinct length-$n$ factors, Sturmian means $p(n)=n+1$ for every $n$; 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 $1$’s differs by at most $1$ [1906.12103].

Their standard representation is by irrational circle rotation. For irrational slope $\alpha \in (0,1)$ and intercept $\rho \in [0,1)$, the standard mechanical word is
$$
s_n=\lfloor (n+1)\alpha+\rho\rfloor-\lfloor n\alpha+\rho\rfloor \in \{0,1\}.
$$
Equivalently, with $x_n=\{n\alpha+\rho\}$ and the partition $[0,1-\alpha)$, $[1-\alpha,1)$ of the circle, one sets
$$
s_n=
\begin{cases}
0,& x_n\in [0,1-\alpha),\\
1,& x_n\in [1-\alpha,1).
\end{cases}
$$
This is the classical coding by an irrational rotation [1906.12103].

Pattern Sturmian sequences generalize Sturmian minimality from contiguous blocks to arbitrary finite windows. The relationship is explicit: all Sturmian sequences satisfy both $p(n)=n+1$ and $p^*(n)=2n$ [1511.03834]. In the higher-dimensional framework of connected supports, the specialization to $d=1$ again yields the classical law. For an asymptotic pair satisfying the flip condition with difference set $F=\{0,-1\}$, the pattern-complexity statement gives
$$
\#L_{[0,n-1]}(x)=\#(F-[0,n-1])=n+1,
$$
recovering the Morse–Hedlund–Coven characterization [2204.06413].

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 $p^*(n)=2n$ 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 [1511.03834].

## 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 [2508.13420]. If $x \in \{0,1\}^{\mathbb{N}_0}$ is recurrent, then $x$ 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 [2508.13420]. This answers the question of classifying recurrent pattern Sturmian sequences posed by Kamae and Zamboni.

For nonrecurrent sequences, the classification is different. If $x \in \{0,1\}^{\mathbb{N}_0}$ is nonrecurrent and pattern Sturmian, then $x$ is either a nonrecurrent simple circle rotation coding sequence by two intervals, or almost constant in the sense that there exists an infinite set $S \subset \mathbb{N}_0$ with upper Banach density $d^*(S)=0$ such that $x$ is the characteristic function of $S$ or of $\mathbb{N}_0 \setminus S$ [2508.13420]. Kamae–Zamboni examples with $s_{k+1}>2s_k$ fall into this almost-constant regime.

The MEF viewpoint yields a broader theorem for sequences of non-superlinear maximal pattern complexity. If $x$ is recurrent, nonperiodic, and $\liminf_n p_x^*(n)/n<\infty$, then $x$ is uniformly recurrent and its subshift is minimal with MEF either an odometer or $S^1 \times C_m$; correspondingly, $x$ is either a periodic interleaving of constant or circle-rotation-coding sequences for the same irrational $\alpha$, or belongs to an $m$-hole Toeplitz subshift [2508.13420]. The proof mechanism is that finite boundary in an MEF coding partition forces linear lower bounds on $p_x^*$, 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 $x$ is recurrent but not uniformly recurrent, then
$$
\liminf_{n\to\infty}\frac{p_x^*(n)}{n\ln n}>0.
$$
Thus pattern Sturmian behavior is inseparable from uniform recurrence in the recurrent case [2508.13420].

## 4. Canonical constructions: rotation codings and Toeplitz systems

Circle rotation codings provide one canonical source of pattern Sturmian sequences. Let $T(\theta)=\theta+\alpha \bmod 1$ with irrational $\alpha$, and partition $S^1$ into two intervals $I_0,I_1$ whose union is the circle. For letters $a_0,a_1$, define
$$
x_n=
\begin{cases}
a_0,&T^n(\theta)\in I_0,\\
a_1,&T^n(\theta)\in I_1.
\end{cases}
$$
For two-interval partitions, such codings satisfy $p_x^*(n)\le 2n$ for all $n$, and therefore nonperiodic examples are pattern Sturmian [2508.13420]. When the intervals are half-open and of lengths $\alpha$ and $1-\alpha$, one recovers classical Sturmian words [2508.13420].

Endpoint placement governs recurrence. A rotation coding is nonrecurrent if and only if one interval has endpoints at orbit points of $0$, namely
$$
I\in \big\{(k_1\alpha,k_2\alpha),\ [k_1\alpha,k_2\alpha]\big\}\pmod 1
\quad\text{for some }k_1\ne k_2\in\mathbb{N}_0,
$$
so certain local patterns occur exactly once [2508.13420]. 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 [1511.03834]. Gjini–Kamae–Tan–Xue showed that pattern Sturmian Toeplitz words are exactly compositions $B(w,n,\ell)\langle B\rangle$ of a single partial skeleton with a simple Toeplitz word $B$; in particular, every pattern Sturmian Toeplitz word is 2-letter [1511.03834].

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 $\theta$ of the form $0\mapsto w0$, $1\mapsto w1$, together with the technical constraint $D((w?)^\infty,L)\le 0$ from the cited classification [2508.13420]. 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 $x \in \Sigma^{\mathbb{Z}^d}$ and asymptotic pairs $(x,y)$ with finite difference set
$$
F=\{u\in \mathbb{Z}^d : x_u\ne y_u\}.
$$
Such a pair is indistinguishable if, for every finite pattern $p$, the occurrence sets in $x$ and $y$ differ by equally many points on each side; equivalently,
$$
\#(\operatorname{occ}_p(x)\setminus \operatorname{occ}_p(y))
=
\#(\operatorname{occ}_p(y)\setminus \operatorname{occ}_p(x)).
$$
The paper isolates a flip condition: $F=\{0,-e_1,\dots,-e_d\}$, the restriction $x|_F$ is a bijection $F\to \Sigma$, and the values on $F$ are cyclically permuted from $x$ to $y$ [2204.06413].

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
$$
\#L_S(x)=\#L_S(y)=\#(F-S)
$$
for every nonempty finite connected $S\subset \mathbb{Z}^d$ [2204.06413]. For rectangles $S(m)$ with side lengths $m=(m_1,\dots,m_d)\in \mathbb{N}^d$,
$$
\#L_m(x)=\#L_m(y)=m_1\cdots m_d\Bigl(1+\frac1{m_1}+\cdots+\frac1{m_d}\Bigr).
$$
For $d=1$ this is $n+1$, and for $d=2$ it is $mn+m+n$ [2204.06413].

Under uniform recurrence, the structure becomes geometric. If $x$ is uniformly recurrent, then $(x,y)$ is an indistinguishable asymptotic pair satisfying the flip condition if and only if there exists a totally irrational $\alpha\in [0,1)^d$ such that $x=c_\alpha$ and $y=c'_\alpha$, the lower and upper characteristic $d$-dimensional Sturmian configurations defined by a codimension-one cut-and-project scheme with internal space $\mathbb{R}/\mathbb{Z}$ [2204.06413]. The symbolic coding is obtained by partitioning $[0,1)$ into $d+1$ consecutive intervals determined by $\alpha$, with star map
$$
n^\star=\alpha\cdot n \bmod 1.
$$
The characteristic configurations admit the mechanical formulas
$$
c_\alpha(n)=\sum_{i=1}^d\bigl(\lfloor \alpha_i+n\cdot \alpha\rfloor-\lfloor n\cdot \alpha\rfloor\bigr),
\qquad
c'_\alpha(n)=\sum_{i=1}^d\bigl(\lceil \alpha_i+n\cdot \alpha\rceil-\lceil n\cdot \alpha\rceil\bigr).
$$

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 [2204.06413].

## 6. Spectral theory for Schrödinger operators with pattern Sturmian potentials

Pattern Sturmian sequences have been studied as potentials for discrete Schrödinger operators
$$
(H\psi)(n)=\psi(n+1)+\psi(n-1)+V(n)\psi(n),
\qquad n\in \mathbb{Z},
$$
with bounded potential $V:\mathbb{Z}\to \mathbb{R}$ generated by the symbolic sequence [1511.03834]. 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 [1511.03834].

Their main theorem confirms the conjecture for the Toeplitz subclass: if $H$ has potential given by a pattern Sturmian Toeplitz sequence, then $|\Sigma(H)|=0$ and all spectral measures are purely singular continuous [1511.03834]. 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 $s_k,t_k$, and Gordon-type criteria excluding eigenvalues by exploiting repeated squares and cubes [1511.03834].

The same paper records partial results for other subclasses. Circle map sequences
$$
V(n)=\chi_{[1-\beta,1)}(n\alpha+\theta \bmod 1),
$$
with irrational $\alpha$, 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 [1511.03834]. By contrast, sparse one-sided pattern Sturmian sequences of the form $V(n)=v$ at $n=n_k$ and $0$ otherwise, with $n_{k+1}>2n_k$, 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 [1511.03834]. 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 [1511.03834]. 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 $1$’s, and one finite-range block of $0$’s [1906.12103]. If $x_i$ is the position of the $i$-th $1$ and $d_j$ is the associated sequence of average distances, then
$$
x_{i+j}-x_i\in \{d_j,d_j+1\},
$$
and the forbidden set consists of all distances not in $\{d_j,d_j+1\}$ together with the block of $d_1+1$ consecutive $0$’s [1906.12103]. 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 $1$’s begin
$$
2,3,5,6,7,8,10,11,13,14,15,16,18,19,20,21,\dots
$$
and the forbidden distances begin
$$
1,4,9,12,17,22,25,\dots,
$$
while $000$ is forbidden [1906.12103]. 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 [1906.12103].

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 [1511.03834]. In the multidimensional connected-support setting, the main open question is whether a single uniformly recurrent configuration with minimal connected-support complexity $\#L_S(x)=\#(F-S)$ must already be a multidimensional Sturmian configuration $s_{\alpha,\rho}$ or $s'_{\alpha,\rho}$; only one implication is known [2204.06413]. Other open questions concern étale limits of characteristic pairs, the structure of more general indistinguishable pairs in $d>1$, and the relation between bispecial displacement vectors and simultaneous Diophantine approximation [2204.06413].

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 [2508.13420].

Source: https://www.emergentmind.com/topics/pattern-sturmian-sequences