arXiv:2609.07550v2

Unique ergodicity, and not, for primitive substitutions on compact alphabets containing isolated points

math.DS37B1037A2547B6552C23

Paper metadata

Original submission: September 7, 2026
Latest revision: October 19, 2026 · version 2
Claimed MSC: 37B10, 37A25, 47B65, 52C23
arXiv categories: math.DS
Paper license: CC BY 4.0 ↗
Comments: 29 pages

Abstract

We consider generalised subshifts generated by continuous substitution on compact Hausdorff alphabets. Although primitivity still implies minimality of the subshift generated by the substitution, Durand, Ormes and Petite showed, in contrast to the finite case, that primitivity no longer implies unique ergodicity, by constructing counterexamples with Cantor alphabet. Here we show that, even for the arguably simplest case of the one-point compactification of the natural numbers, primitivity is still insufficient for unique ergodicity, or even the existence of a natural length function (NLF). In previous work with Mañibo and Rust we showed that, for irreducible substitutions, unique ergodicity and existence of a NLF follow from strong power convergence of the renormalised substitution operator $T$ ; criteria were also developed that can sometimes confirm strong power convergence. Here we show the converse, so that strong power convergence of $T$ is equivalent to having both unique ergodicity and a NLF. We then consider the case of alphabets with only finitely many accumulation points, showing how upper bounds (and usually an exact formula) for the essential spectral radius of $T$ can be derived from associated finite substitutions, determined by the behaviour of substitution of the accumulation points. For many primitive substitutions this can be used to prove quasi-compactness of $T$ , and thus unique ergodicity. For primitive substitutions on alphabets with at least one isolated point, we show that strong power convergence and quasi-compactness of $T$ are equivalent and, in fact, that these properties are equivalent to iteration of substitution growing words in length uniformly across all seeds in the alphabet.

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Audited against arXiv v2

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated October 23, 2026

01Statements2 reported findingsNo statement issues found

The counterexample and the equivalences between operator convergence, length functions, and uniformly comparable word growth are supported. The essential-radius equality is consistent with the stated generality, but its printed proof incorrectly assumes countability and needs a replacement compactness construction.

Theorems 1.1, 4.1No issue found

The converse ergodicity implication and the counterexample have compatible growth estimates

PDF pp. 9–14 and 25–28 · arXiv:2609.07550v2

Unique ergodicity gives uniform averages over all legal superwords. A positive natural length function then converts normalized word averages to $T^nf$ , yielding the rank-one strong limit. Conversely the symbolic argument approximates block averages by continuous functions on the alphabet and uses long-superword decompositions to control boundaries. For the counterexample, the score recursion at $\alpha=39/20$ is an exact positive majorant: the displayed initial values and the $\beta=5/4$ induction yield scores at least one. Finite seeds grow at most like $\alpha^n$ , whereas the infinity seed has at least $2^n$ letters; Proposition 3.6 and the length-function comparability therefore exclude both unique ergodicity and a natural length function.

Theorem 1.2; Lemma 5.4; Corollary 5.9No issue found

Finite-rank deletion gives the strict essential-radius gap

PDF pp. 17–24 · arXiv:2609.07550v2

Input, output and constant deletions change the substitution operator by finite-rank operators, so preserve essential spectral radius. A weak coincidence allows one common output to be deleted from every substituted word. Under the distortion bound, its descendants occupy at least a fraction $D/L$ at every level, giving $\|M_\tau^n\|\leq(1-D/L)^n\|M^n\|$ . Gelfand's formula then gives the strict essential-radius gap required for quasi-compactness. The finite accumulated-letter matrix gives the upper bound, and the additional isolated-output hypothesis gives equality after the compactness repair described below. Taking powers gives the stated root in Corollary 5.9.

02Proofs2 reported findingsSuspected incomplete or incorrect proofs

The principal growth and operator arguments work, but Theorem 5.8's equality proof uses an invalid topological implication for its full compact-Hausdorff scope. A direct finite-set compactness construction repairs the equality without restricting the theorem. Some explicit error constants in the two ergodicity proofs also need correction.

Theorem 5.8Suspected incomplete as written

Finitely many accumulation points do not imply a countable alphabet

PDF p. 21 · essential-norm lower-bound proof · arXiv:2609.07550v2

The proof selects a sequence of clopen neighbourhoods whose intersection is exactly the accumulation set, claiming that a compact Hausdorff alphabet with finitely many accumulation points is countable. The one-point compactification of an uncountable discrete space is a counterexample; no such countable neighbourhood family exists there. The missing essential-norm lower bound can nevertheless be proved as follows. Fix the iterate and its maximal length $N$ . Construct increasing finite isolated sets $F_j$ . Choose a seed close to a maximizing accumulation point so its $N$ output letters are isolated and outside $F_j$ , and include those letters in $F_{j+1}$ . Such a seed exists by the theorem's extra hypothesis and continuity. Put $f_j=1_{A\setminus F_j}$ . Compactness of the perturbation $V$ supplies a norm-Cauchy subsequence of $Vf_j$ . For two successive subsequence indices, $f_j-f_k$ is supported on $F_k\setminus F_j$ , has norm at most one, and its image under the substitution iterate takes value $N$ at the chosen seed. Thus $\|M^n-V\|\geq N-o(1)$ . This supplies the omitted argument in the general, possibly uncountable, case.

Theorem 1.1 proof; Theorem A.4Suspected incorrect as written

The displayed error constants must retain their bounded constant terms

PDF pp. 12 and 27–28 · equations (3.7), (A.3) and following · arXiv:2609.07550v2

On p. 12, the error propagated from (3.7) is $\delta(c_2/c_1+c_2)$ , but the next line replaces $c_2/c_1$ by $c_1/c_2$ and chooses $\delta$ accordingly. In Appendix A, $mr^k/L=1\pm(2\delta+\delta^2)$ implies an error factor at most $2+c_2(1+2\delta+\delta^2)$ ; the printed $p_2$ drops the term $c_2$ . Use these larger constants throughout and take $\delta$ correspondingly smaller. Finally the asserted nesting of the intervals at the end does not follow from increasing their length thresholds, but the bounds $|c_\epsilon-c_{\epsilon'}|\leq\epsilon+\epsilon'$ give a common Cauchy limit directly. These repairs preserve both convergence conclusions; they do not furnish a counterexample to either theorem.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper: arXiv:2609.07550v2
Authors listed: James J. Walton
Audit date: October 23, 2026

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Unique ergodicity, and not, for primitive substitutions on compact alphabets containing isolated points

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The converse ergodicity implication and the counterexample have compatible growth estimates

Finite-rank deletion gives the strict essential-radius gap

Finitely many accumulation points do not imply a countable alphabet

The displayed error constants must retain their bounded constant terms

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