Power-Free Shift Spaces

• Power-free shift spaces are symbolic dynamical systems defined by excluding repeated blocks based on prescribed power thresholds, merging combinatorial and arithmetic frameworks.
• They exhibit sharp entropy transitions and support a unique measure of maximal entropy via quasi-specification methods despite the absence of periodic points.
• Their structure reveals rich rigidity and symmetry features, linking local exclusion rules to global dynamics and embedding arithmetic conditions from number fields.

Searching arXiv for 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 $\beta>1$0th power-free integers in $\beta>1$1, 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 (Climenhaga, 24 Jul 2025, Gundlach et al., 2024).

1. Symbolic-dynamical framework

For a finite alphabet $\beta>1$2 of size $\beta>1$3 (written $\beta>1$4 in the repetition-avoidance literature), the full two-sided shift is $\beta>1$5 with left shift $\beta>1$6 given by $\beta>1$7. A subshift $\beta>1$8 is a closed, $\beta>1$9-invariant set. Its language is

$d$0

and $d$1 denotes the set of words of length $d$2 that appear in $d$3. The topological entropy is

$d$4

where $d$5 is the Kolmogorov–Sinai entropy (Climenhaga, 24 Jul 2025).

The arithmetic family uses a different acting group. If $d$6 is a number field with ring of integers $d$7, then subsets of $d$8 are identified with $d$9, and $\beta$0 acts by translations: $\beta$1. A topological dynamical system $\beta$2 in this setting consists of a compact space $\beta$3 with continuous $\beta$4-action. Morphisms, factor maps, topological conjugacies, and the extended symmetry group $\beta$5 are defined in the standard equivariant way; the Curtis–Hedlund–Lyndon principle is used in general form for these subshifts (Gundlach et al., 2024).

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 $\beta$6 and rational $\beta$7 with $\beta$8, the word $\beta$9 is the prefix of length $X_\beta^d$0 of the infinite periodic word $X_\beta^d$1. A word $X_\beta^d$2 is an $X_\beta^d$3-power if $X_\beta^d$4 for some $X_\beta^d$5. Given $X_\beta^d$6 and $X_\beta^d$7, the forbidden set is

$X_\beta^d$8

and the $X_\beta^d$9-ary $\alpha$0-free shift is

$\alpha$1

A variant $\alpha$2 is obtained by replacing $\alpha$3 with $\alpha$4 in $\alpha$5; if $\alpha$6, then $\alpha$7 (Climenhaga, 24 Jul 2025).

A classical special case forbids integer powers $\alpha$8 for all $\alpha$9; this is subsumed by the $\alpha \ge \beta$0-free model with $\alpha \ge \beta$1. Nonemptiness is governed by the repetition threshold $\alpha \ge \beta$2: $\alpha \ge \beta$3, $\alpha \ge \beta$4, $\alpha \ge \beta$5, and $\alpha \ge \beta$6 for $\alpha \ge \beta$7. For $\alpha \ge \beta$8 one has $\alpha \ge \beta$9, while for $K$0 the shift is nonempty (Climenhaga, 24 Jul 2025).

The entropy picture is sharply parameter-dependent. For $K$1, $K$2 is empty for $K$3; for $K$4, one has $K$5, described as the “polynomial plateau”; and for $K$6, $K$7. These results are attributed to Restivo–Salemi and Karhumäki–Shallit. For $K$8, every nonempty $K$9 has positive entropy, due to Ochem. More generally, the “exponential conjecture” asserts that for $k \ge 2$0, every nonempty $k \ge 2$1 has positive entropy; the cited state of the art proves this for all $k \ge 2$2 except even $k \ge 2$3 from $k \ge 2$4 to $k \ge 2$5, with the equivalent Dejean-threshold formulation that $k \ge 2$6 has positive entropy for all $k \ge 2$7 except possibly those even values (Climenhaga, 24 Jul 2025).

Examples emphasize the dependence on alphabet size. For square-free words, $k \ge 2$8: over $k \ge 2$9, $\mathbb D_{K,k}$0 is empty, whereas over $\mathbb D_{K,k}$1, $\mathbb D_{K,k}$2 is nonempty and has positive entropy. For cube-free words, $\mathbb D_{K,k}$3: over $\mathbb D_{K,k}$4, $\mathbb D_{K,k}$5 has positive entropy. The uniqueness theory discussed below does not cover these small-$\mathbb D_{K,k}$6 cases; in particular, cube-free systems lie outside the current intrinsic-ergodicity range (Climenhaga, 24 Jul 2025).

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

The central ergodic result for repetition-avoidance shifts is intrinsic ergodicity in a high-$\mathbb D_{K,k}$7 regime. A measure of maximal entropy (MME) is a measure $\mathbb D_{K,k}$8 with $\mathbb D_{K,k}$9, and intrinsic ergodicity means uniqueness of the MME. For every $\beta>1$00 and $\beta>1$01, both $\beta>1$02 and $\beta>1$03 have a unique measure of maximal entropy; for $\beta>1$04 the same holds at $\beta>1$05 (Climenhaga, 24 Jul 2025).

The proof does not use Bowen’s classical specification, because $\beta>1$06 has no periodic orbits for any finite $\beta>1$07. If $\beta>1$08 were periodic of period $\beta>1$09 with period block $\beta>1$10, then arbitrarily long windows of $\beta>1$11 would equal $\beta>1$12 for arbitrarily large $\beta>1$13, and for $\beta>1$14 the block $\beta>1$15 would be an $\beta>1$16-power with $\beta>1$17, 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 (Climenhaga, 24 Jul 2025).

The replacement is a weak specification package on a core language. Define

$\beta>1$18

$\beta>1$19

and

$\beta>1$20

For $\beta>1$21 with either $\beta>1$22, $\beta>1$23, or $\beta>1$24, $\beta>1$25, the triplet $\beta>1$26 satisfies four properties: every word decomposes as $\beta>1$27 with $\beta>1$28, $\beta>1$29, $\beta>1$30; the boundary collections have an entropy gap, equivalently

$\beta>1$31

there is variable-length 4-way specification on $\beta>1$32; and there is same-length specification on $\beta>1$33 (Climenhaga, 24 Jul 2025).

The bridging constants are explicit. For same-length bridging, $\beta>1$34 if $\beta>1$35 and $\beta>1$36, while $\beta>1$37 if $\beta>1$38 and $\beta>1$39. For variable-length 4-way specification, $\beta>1$40 if $\beta>1$41 and $\beta>1$42, $\beta>1$43 if $\beta>1$44 and $\beta>1$45, and $\beta>1$46 if $\beta>1$47 and $\beta>1$48. 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 $\beta>1$49 spans across four concatenated core blocks $\beta>1$50, then $\beta>1$51 and therefore $\beta>1$52; with one-letter or two-letter separators the argument improves to $\beta>1$53 (Climenhaga, 24 Jul 2025).

These structures yield quantitative counting bounds. If $\beta>1$54 and $\beta>1$55 is the extendable language at length $\beta>1$56, then for all $\beta>1$57,

$\beta>1$58

The boundary semigroup satisfies $\beta>1$59, while the core has sufficiently large growth to force $\beta>1$60. Explicit lower bounds are

$\beta>1$61

These are combined with a general uniqueness theorem for subshifts possessing decomposition, finite $\beta>1$62, variable-length 4-way specification, and multi-step same-length specification (Climenhaga, 24 Jul 2025).

The proof of uniqueness proceeds by a Misiurewicz construction of an MME, Gibbs-type lower bounds on cylinders associated to $\beta>1$63, 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 (Climenhaga, 24 Jul 2025).

4. Arithmetic power-free shifts from number fields

A second family of power-free shift spaces arises from divisibility in number fields. Let $\beta>1$64 be a number field of degree $\beta>1$65 over $\beta>1$66 with ring of integers $\beta>1$67. For a nonzero prime ideal $\beta>1$68 of $\beta>1$69, the norm is $\beta>1$70. Fix $\beta>1$71. An element $\beta>1$72 is $\beta>1$73th power-free if no prime ideal power $\beta>1$74 divides the principal ideal $\beta>1$75; equivalently, for every prime ideal $\beta>1$76, one has $\beta>1$77. The set of all such elements is denoted $\beta>1$78 (Gundlach et al., 2024).

The associated shift space is defined on $\beta>1$79. Globally, $\beta>1$80 is the orbit-closure of $\beta>1$81 under the $\beta>1$82-action by translations. Locally, a subset $\beta>1$83 is admissible if, for every prime ideal $\beta>1$84, $\beta>1$85 misses at least one residue class modulo $\beta>1$86; equivalently, there exists a class $\beta>1$87 such that

$\beta>1$88

The paper proves that the admissible subshift equals the orbit-closure:

$\beta>1$89

Thus admissibility gives the defining forbidden patterns: for each $\beta>1$90, the system forbids the pattern “occupy at least one point in every residue class modulo $\beta>1$91” (Gundlach et al., 2024).

A local–global principle underlies the construction. For $\beta>1$92, reduction modulo $\beta>1$93 is surjective on $\beta>1$94-free sets: every $\beta>1$95-free residue class in $\beta>1$96 has a $\beta>1$97-free representative in $\beta>1$98. Consequently, global preservation statements can be reduced to local preservation statements for all rational primes $\beta>1$99 (Gundlach et al., 2024).

The dynamical invariants are explicitly arithmetic. The density of $d$00-free integers is

$d$01

and the patch-counting entropy equals the topological entropy:

$d$02

The Euler-product form reflects the prime-ideal decomposition of the local constraints (Gundlach et al., 2024).

The paper also places $d$03 in a broader sieve framework. A sieve $d$04 on $d$05 is a choice, for each prime ideal $d$06, of a compact open subset $d$07, and the corresponding admissible shift space $d$08 is hereditary. For the $d$09-free sieve one takes $d$10, recovering the power-free system as a special case (Gundlach et al., 2024).

5. Linear symmetries, extended symmetries, and rigidity

The arithmetic power-free shifts admit a sharp description of linear and dynamical symmetries. Let $d$11 be a $d$12-linear bijection. The following are equivalent: $d$13; for all rational primes $d$14, the induced map on $d$15 preserves $d$16; and there exist a field automorphism $d$17 and a unit $d$18 such that

$d$19

Equivalently, the group of $d$20-linear bijections preserving $d$21 is isomorphic to $d$22 (Gundlach et al., 2024).

The local mechanism is especially transparent at rational primes that split completely in $d$23. There one has $d$24, and the local condition forces the matrix of $d$25 modulo $d$26 to preserve $d$27. 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 $d$28 (Gundlach et al., 2024).

At the dynamical level, the extended symmetry group of the $d$29-free shift is

$d$30

Explicitly, an element is a pair $d$31 with

$d$32

where $d$33, $d$34, and $d$35. The ordinary symmetry group, with $d$36, consists only of translations by $d$37 (Gundlach et al., 2024).

The resulting rigidity is strong. For number fields $d$38 and integers $d$39, the following are equivalent: $d$40 and $d$41 are topologically conjugate; $d$42 is a factor of $d$43; and $d$44 with $d$45. Thus no two such dynamical systems with different fields or different exponents are topologically conjugate, and no one is a factor system of another (Gundlach et al., 2024).

Examples illustrate how arithmetic data enters the symmetry group. For $d$46, one has $d$47, $d$48 trivial, and $d$49, so

$d$50

For real quadratic fields, $d$51 is infinite and $d$52 has order $d$53, so the quotient by translations is infinite. For imaginary quadratic fields, $d$54 is finite. For cyclotomic fields $d$55, both $d$56 and the roots of unity enlarge the extended symmetry group (Gundlach et al., 2024).

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 $d$57 in a word; in the arithmetic literature, “power-free” refers to avoidance of divisibility by $d$58 in $d$59. The common terminology reflects a shared exclusion principle, but the acting groups, alphabets, entropy formulas, and rigidity phenomena differ substantially (Climenhaga, 24 Jul 2025, Gundlach et al., 2024).

A second common misconception is that specification-type arguments in symbolic dynamics require periodic points. The repetition-avoidance systems $d$60 show otherwise: they have no periodic orbits for any finite $d$61, yet for all $d$62 with $d$63, and for $d$64 with $d$65, they admit a unique MME via a nonuniform specification framework on a core set $d$66 (Climenhaga, 24 Jul 2025). 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 (Gundlach et al., 2024).

Open problems in the repetition-avoidance setting are explicit. The paper asks whether every $d$67 with $d$68 has a unique MME, whether the unique MME is fully supported, mixing, $d$69, 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 $d$70 is Poulsen, and whether the current $d$71 thresholds and counting constants can be improved by refining the core/boundary decomposition (Climenhaga, 24 Jul 2025).

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 (Gundlach et al., 2024).

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.