---
title: Power-Free Shift Spaces
url: https://www.emergentmind.com/topics/power-free-shift-spaces
type: topic
---

# Power-Free Shift Spaces

Searching arXiv for recent papers on power-free shift spaces and related symbolic dynamics.
Power-free shift spaces are symbolic dynamical systems in which admissibility is defined by excluding prescribed powers. In the combinatorics-on-words setting, a finite alphabet $A$ and an exponent threshold $\beta>1$ determine the $d$-ary $\beta$-free shift $X_\beta^d$, consisting of bi-infinite sequences whose subwords avoid $\alpha$-powers with $\alpha \ge \beta$. In an arithmetic setting, a number field $K$ and an integer $k \ge 2$ determine the shift space $\mathbb D_{K,k}$ associated to the set of $k$th power-free integers in $\mathcal O_K$, realized as an orbit-closure under translations. These two constructions encode different notions of “power-free,” but both convert local exclusion laws into shift-invariant spaces with nontrivial entropy, rigidity, and ergodic structure [2507.18779] [2407.08438].

## 1. Symbolic-dynamical framework

For a finite alphabet $A$ of size $m \ge 2$ (written $d \ge 2$ in the repetition-avoidance literature), the full two-sided shift is $A^{\mathbb Z}$ with left shift $\sigma : A^{\mathbb Z} \to A^{\mathbb Z}$ given by $(\sigma x)_n = x_{n+1}$. A subshift $X \subset A^{\mathbb Z}$ is a closed, $\sigma$-invariant set. Its language is
$$
\mathcal L(X) = \{x_i \cdots x_j : x \in X,\ i \le j\},
$$
and $\mathcal L_n = \mathcal L \cap A^n$ denotes the set of words of length $n$ that appear in $X$. The topological entropy is
$$
h_{\mathrm{top}}(X)=h(X)=\lim_{n\to\infty}\frac{1}{n}\log|\mathcal L_n|
=\sup_{\mu\in\mathfrak M_\sigma(X)} h_\mu(\sigma),
$$
where $h_\mu$ is the Kolmogorov–Sinai entropy [2507.18779].

The arithmetic family uses a different acting group. If $K$ is a number field with ring of integers $\mathcal O_K$, then subsets of $\mathcal O_K$ are identified with $\{0,1\}^{\mathcal O_K}$, and $\mathcal O_K$ acts by translations: $(g \cdot X)=g+X$. A topological dynamical system $(X,G)$ in this setting consists of a compact space $X$ with continuous $G$-action. Morphisms, factor maps, topological conjugacies, and the extended symmetry group $\mathrm{ExSym}(X,G)$ are defined in the standard equivariant way; the Curtis–Hedlund–Lyndon principle is used in general form for these subshifts [2407.08438].

The shared formal structure is that both families are subshifts defined by infinite collections of forbidden local patterns. The divergence lies in the source of those patterns: repetition thresholds in words versus divisibility obstructions in number fields.

## 2. Repetition-avoidance shifts on finite alphabets

For $w \in A^n$ and rational $\alpha>1$ with $\alpha n \in \mathbb N$, the word $w^\alpha$ is the prefix of length $\alpha n$ of the infinite periodic word $w^\infty = wwww\cdots$. A word $u$ is an $\alpha$-power if $u=v^\alpha$ for some $v$. Given $d$ and $\beta>1$, the forbidden set is
$$
\mathcal F_\beta = \{v^\alpha : v\in A^+,\ \alpha\in[\beta,\infty)\cap\mathbb Q\},
$$
and the $d$-ary $\beta$-free shift is
$$
X_\beta^d = X(\mathcal F_\beta)
=\{x\in A^{\mathbb Z} : \text{no subword of } x \text{ lies in } \mathcal F_\beta\}.
$$
A variant $X_{\beta^+}^d$ is obtained by replacing $[\beta,\infty)$ with $(\beta,\infty)$ in $\mathcal F_\beta$; if $\beta \notin \mathbb Q$, then $X_{\beta^+}^d=X_\beta^d$ [2507.18779].

A classical special case forbids integer powers $w^k$ for all $k \ge r$; this is subsumed by the $\beta$-free model with $\beta=r$. Nonemptiness is governed by the repetition threshold $RT(d)$: $RT(2)=2$, $RT(3)=7/4$, $RT(4)=7/5$, and $RT(d)=d/(d-1)$ for $d \ge 5$. For $\beta \le RT(d)$ one has $X_\beta^d=\varnothing$, while for $\beta>RT(d)$ the shift is nonempty [2507.18779].

The entropy picture is sharply parameter-dependent. For $d=2$, $X_\beta^2$ is empty for $\beta \le 2$; for $2<\beta\le 7/3$, one has $h(X_\beta^2)=0$, described as the “polynomial plateau”; and for $\beta>7/3$, $h(X_\beta^2)>0$. These results are attributed to Restivo–Salemi and Karhumäki–Shallit. For $d=3,4$, every nonempty $X_\beta^d$ has positive entropy, due to Ochem. More generally, the “exponential conjecture” asserts that for $d \ge 3$, every nonempty $X_\beta^d$ has positive entropy; the cited state of the art proves this for all $d \ge 3$ except even $d$ from $12$ to $26$, with the equivalent Dejean-threshold formulation that $X_{RT(d)^+}^d$ has positive entropy for all $d \ge 3$ except possibly those even values [2507.18779].

Examples emphasize the dependence on alphabet size. For square-free words, $\beta=2$: over $d=2$, $X_2^2$ is empty, whereas over $d=3$, $X_2^3$ is nonempty and has positive entropy. For cube-free words, $\beta=3$: over $d=2$, $X_3^2$ has positive entropy. The uniqueness theory discussed below does not cover these small-$\beta$ cases; in particular, cube-free systems lie outside the current intrinsic-ergodicity range [2507.18779].

## 3. Entropy maximization, quasi-specification, and absence of periodic points

The central ergodic result for repetition-avoidance shifts is intrinsic ergodicity in a high-$\beta$ regime. A measure of maximal entropy (MME) is a measure $\mu \in \mathfrak M_\sigma(X)$ with $h_\mu(\sigma)=h_{\mathrm{top}}(X)$, and intrinsic ergodicity means uniqueness of the MME. For every $d \ge 2$ and $\beta>12$, both $X_\beta^d$ and $X_{\beta^+}^d$ have a unique measure of maximal entropy; for $d \ge 3$ the same holds at $\beta=12$ [2507.18779].

The proof does not use Bowen’s classical specification, because $X_\beta^d$ has no periodic orbits for any finite $\beta>1$. If $x \in X$ were periodic of period $p$ with period block $w$, then arbitrarily long windows of $x$ would equal $w^K$ for arbitrarily large $K$, and for $K \ge \lceil \beta \rceil$ the block $w^K$ would be an $\alpha$-power with $\alpha \ge \beta$, hence forbidden. This absence of periodic points is a basic structural difference from many specification-based systems, where periodic points are abundant and their growth rate is governed by entropy [2507.18779].

The replacement is a weak specification package on a core language. Define
$$
\mathfrak A := \{v^4 : v \in \mathcal L\},
$$
$$
\mathcal G := \{w \in \mathcal L : \text{no prefix or suffix of } w \text{ is a 4th power}\},
$$
and
$$
\mathcal C^p=\mathcal C^s:=\mathfrak A^* \cap \mathcal L.
$$
For $X_\beta^d$ with either $d \ge 3$, $\beta \ge 12$, or $d=2$, $\beta \ge 12^+$, the triplet $(\mathcal C^p,\mathcal G,\mathcal C^s)$ satisfies four properties: every word decomposes as $u^pvu^s$ with $u^p \in \mathcal C^p$, $v \in \mathcal G$, $u^s \in \mathcal C^s$; the boundary collections have an entropy gap, equivalently
$$
Q := \left(\sum_{i\ge0} |\mathcal C^p_i| e^{-ih}\right)
\left(\sum_{k\ge0} |\mathcal C^s_k| e^{-kh}\right) < \infty;
$$
there is variable-length 4-way specification on $\mathcal G$; and there is same-length specification on $\mathcal G$ [2507.18779].

The bridging constants are explicit. For same-length bridging, $\tau=1$ if $d \ge 3$ and $\beta \ge 8$, while $\tau=2$ if $d=2$ and $\beta>8$. For variable-length 4-way specification, $T=0$ if $d \ge 3$ and $\beta \ge 16$, $T=1$ if $d \ge 3$ and $\beta \ge 12$, and $T=2$ if $d=2$ and $\beta>12$. The combinatorial mechanism is control of short periods at boundaries via Fine–Wilf theory and counting of powers spanning concatenated core blocks. In particular, if a subword $t^\alpha$ spans across four concatenated core blocks $u,v,w,x \in \mathcal G$, then $|t^\alpha|<16|t|$ and therefore $\alpha<16$; with one-letter or two-letter separators the argument improves to $\alpha<12$ [2507.18779].

These structures yield quantitative counting bounds. If $h=h(X)$ and $\mathcal L_n$ is the extendable language at length $n$, then for all $n \in \mathbb N$,
$$
e^{nh} \le |\mathcal L_n| \le
\begin{cases}
(2.892)e^{(n+2)h}, & d=2,\\[4pt]
\left(1+\dfrac{d}{d^3-2d-1}\right)^2 e^{(n+1)h}, & d\ge3.
\end{cases}
$$
The boundary semigroup satisfies $h(\mathfrak A^*) \le \frac14 \log(2d)$, while the core has sufficiently large growth to force $h(\mathfrak A^*)<h$. Explicit lower bounds are
$$
e^{4h} \ge d^3-1 \quad (d\ge3), \qquad e^{8h} \ge 47 \quad (d=2).
$$
These are combined with a general uniqueness theorem for subshifts possessing decomposition, finite $Q$, variable-length 4-way specification, and multi-step same-length specification [2507.18779].

The proof of uniqueness proceeds by a Misiurewicz construction of an MME, Gibbs-type lower bounds on cylinders associated to $\mathcal G$, and a trimming/approximation lemma following Pacifico–Fan Yang–Jiagang Yang. The resulting method is summarized in the paper as a “quasi-specification + Gibbs lower bound + trimming” triad, which replaces periodic-orbit arguments entirely [2507.18779].

## 4. Arithmetic power-free shifts from number fields

A second family of power-free shift spaces arises from divisibility in number fields. Let $K$ be a number field of degree $n$ over $\mathbb Q$ with ring of integers $\mathcal O_K$. For a nonzero prime ideal $\mathfrak p$ of $\mathcal O_K$, the norm is $N(\mathfrak p)=|\mathcal O_K/\mathfrak p|$. Fix $k \ge 2$. An element $x \in \mathcal O_K$ is $k$th power-free if no prime ideal power $\mathfrak p^k$ divides the principal ideal $(x)$; equivalently, for every prime ideal $\mathfrak p$, one has $x \not\equiv 0 \bmod \mathfrak p^k$. The set of all such elements is denoted $V_{K,k}$ [2407.08438].

The associated shift space is defined on $\{0,1\}^{\mathcal O_K}$. Globally, $\mathbb D_{K,k}$ is the orbit-closure of $V_{K,k}$ under the $\mathcal O_K$-action by translations. Locally, a subset $S \subset \mathcal O_K$ is admissible if, for every prime ideal $\mathfrak p$, $S$ misses at least one residue class modulo $\mathfrak p^k$; equivalently, there exists a class $a \bmod \mathfrak p^k$ such that
$$
S \cap (a+\mathfrak p^k)=\varnothing.
$$
The paper proves that the admissible subshift equals the orbit-closure:
$$
\mathbb D_{K,k}=\overline{\mathcal O_K+V_{K,k}}.
$$
Thus admissibility gives the defining forbidden patterns: for each $\mathfrak p$, the system forbids the pattern “occupy at least one point in every residue class modulo $\mathfrak p^k$” [2407.08438].

A local–global principle underlies the construction. For $k \ge 2$, reduction modulo $p^k$ is surjective on $k$-free sets: every $k$-free residue class in $\mathcal O_K/p^k\mathcal O_K$ has a $k$-free representative in $\mathcal O_K$. Consequently, global preservation statements can be reduced to local preservation statements for all rational primes $p$ [2407.08438].

The dynamical invariants are explicitly arithmetic. The density of $k$-free integers is
$$
\mathrm{dens}(V_{K,k})=\prod_{\mathfrak p}(1-N(\mathfrak p)^{-k})=\frac{1}{\zeta_K(k)},
$$
and the patch-counting entropy equals the topological entropy:
$$
h_{\mathrm{top}}(\mathbb D_{K,k})
=(\log 2)\prod_{\mathfrak p}(1-N(\mathfrak p)^{-k})
=\frac{\log 2}{\zeta_K(k)}.
$$
The Euler-product form reflects the prime-ideal decomposition of the local constraints [2407.08438].

The paper also places $\mathbb D_{K,k}$ in a broader sieve framework. A sieve $R$ on $K$ is a choice, for each prime ideal $\mathfrak p$, of a compact open subset $R_\mathfrak p \subseteq \mathcal O_{K,\mathfrak p}$, and the corresponding admissible shift space $\mathbb D(K,R)$ is hereditary. For the $k$-free sieve one takes $R_\mathfrak p=\mathfrak p^k$, recovering the power-free system as a special case [2407.08438].

## 5. Linear symmetries, extended symmetries, and rigidity

The arithmetic power-free shifts admit a sharp description of linear and dynamical symmetries. Let $A:\mathcal O_K \to \mathcal O_K$ be a $\mathbb Z$-linear bijection. The following are equivalent: $A(V_{K,k})=V_{K,k}$; for all rational primes $p$, the induced map on $\mathcal O_K/p^k\mathcal O_K$ preserves $V_{K,k,p}$; and there exist a field automorphism $\tau \in \mathrm{Aut}(K)$ and a unit $\varepsilon \in \mathcal O_K^\times$ such that
$$
A=M_\varepsilon \circ \tau, \qquad M_\varepsilon(x)=\varepsilon x.
$$
Equivalently, the group of $\mathbb Z$-linear bijections preserving $V_{K,k}$ is isomorphic to $\mathcal O_K^\times \rtimes \mathrm{Aut}(K)$ [2407.08438].

The local mechanism is especially transparent at rational primes that split completely in $K$. There one has $\mathcal O_K \otimes \mathbb Z_p \cong \mathbb Z_p^n$, and the local condition forces the matrix of $A$ modulo $p$ to preserve $(\mathbb F_p^\times)^n$. A geometric lemma then yields that the matrix must be a permutation times a diagonal unit, and Chebotarev compatibility across infinitely many split primes globalizes this to $A=M_\varepsilon \circ \tau$ [2407.08438].

At the dynamical level, the extended symmetry group of the $k$-free shift is
$$
\mathrm{ExSym}(\mathbb D_{K,k})
\cong \mathcal O_K \rtimes (\mathcal O_K^\times \rtimes \mathrm{Aut}(K)).
$$
Explicitly, an element is a pair $(f,A)$ with
$$
A=M_\varepsilon \circ \tau, \qquad f(S)=t+\varepsilon\cdot\tau(S),
$$
where $t \in \mathcal O_K$, $\tau \in \mathrm{Aut}(K)$, and $\varepsilon \in \mathcal O_K^\times$. The ordinary symmetry group, with $A=\mathrm{id}$, consists only of translations by $\mathcal O_K$ [2407.08438].

The resulting rigidity is strong. For number fields $K,L$ and integers $k,l \ge 1$, the following are equivalent: $\mathbb D_{K,k}$ and $\mathbb D_{L,l}$ are topologically conjugate; $\mathbb D_{L,l}$ is a factor of $\mathbb D_{K,k}$; and $K \cong L$ with $k=l$. Thus no two such dynamical systems with different fields or different exponents are topologically conjugate, and no one is a factor system of another [2407.08438].

Examples illustrate how arithmetic data enters the symmetry group. For $K=\mathbb Q$, one has $\mathcal O_K=\mathbb Z$, $\mathrm{Aut}(K)$ trivial, and $\mathcal O_K^\times=\{\pm1\}$, so
$$
\mathrm{ExSym}(\mathbb D_{\mathbb Q,k}) \cong \mathbb Z \rtimes \{\pm1\}.
$$
For real quadratic fields, $\mathcal O_K^\times$ is infinite and $\mathrm{Aut}(K)$ has order $2$, so the quotient by translations is infinite. For imaginary quadratic fields, $\mathcal O_K^\times$ is finite. For cyclotomic fields $K=\mathbb Q(\zeta_m)$, both $\mathrm{Gal}(K/\mathbb Q)\cong (\mathbb Z/m\mathbb Z)^\times$ and the roots of unity enlarge the extended symmetry group [2407.08438].

## 6. Conceptual contrasts, misconceptions, and open directions

The phrase “power-free shift space” can refer to two distinct constructions. In the repetition-avoidance literature, “power” means a repeated block $v^\alpha$ in a word; in the arithmetic literature, “power-free” refers to avoidance of divisibility by $\mathfrak p^k$ in $\mathcal O_K$. The common terminology reflects a shared exclusion principle, but the acting groups, alphabets, entropy formulas, and rigidity phenomena differ substantially [2507.18779] [2407.08438].

A second common misconception is that specification-type arguments in symbolic dynamics require periodic points. The repetition-avoidance systems $X_\beta^d$ show otherwise: they have no periodic orbits for any finite $\beta>1$, yet for all $d \ge 2$ with $\beta>12$, and for $d \ge 3$ with $\beta=12$, they admit a unique MME via a nonuniform specification framework on a core set $\mathcal G$ [2507.18779]. By contrast, the arithmetic systems are not analyzed through specification in the cited work; their main structural results concern local–global admissibility, exact entropy, symmetry groups, and topological rigidity [2407.08438].

Open problems in the repetition-avoidance setting are explicit. The paper asks whether every $X_\beta^d$ with $h>0$ has a unique MME, whether the unique MME is fully supported, mixing, $K$, and Bernoulli, whether a thermodynamic formalism for Hölder potentials can be developed using the present quasi-specification toolkit, whether ergodic measures are entropy-dense and $\mathfrak M_\sigma(X)$ is Poulsen, and whether the current $\beta$ thresholds and counting constants can be improved by refining the core/boundary decomposition [2507.18779].

Open directions in the arithmetic and sieve setting are of a different kind. For general sieves, the symmetry group can be larger than translations, and full classification of symmetries or factors is open in many cases. The existence of factor maps with exceptional local behavior at finitely many places is described as subtle and combinatorial; necessary conditions from sieve morphism theorems and entropy are known, but sufficiency is not settled [2407.08438].

Taken together, these results place power-free shift spaces at the intersection of symbolic dynamics, combinatorics on words, and arithmetic dynamics. One branch emphasizes entropy maximization without periodic-orbit methods; the other emphasizes local–global admissibility, arithmetic entropy formulas, and rigidity under conjugacy and factor maps.

Source: https://www.emergentmind.com/topics/power-free-shift-spaces