Characteristic measures of symbolic dynamical systems

Joshua Frisch; Omer Tamuz

REVIEW 3 major objections 4 minor 15 references

Characteristic measures of symbolic dynamical systems

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Zero-entropy shifts always admit a probability measure invariant under every automorphism of the system.

desk verdict A solid, mostly self-contained proof that zero entropy shifts admit characteristic measures, with two technical gaps in the write-up that are repairable rather than fatal. read the letter →

arxiv 1908.02930 v3 pith:JAHU5BPK submitted 2019-08-08 math.DS

Joshua Frisch , Omer Tamuz This is my paper · ORCID

classification math.DS MSC 37B1037B0537A35

keywords characteristicmeasurezeroentropyshiftautomorphismgroupsymbolicdynamicssoficsubexponentialgrowthminimalinvariant

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that every symbolic dynamical system with zero topological entropy carries a characteristic measure: a Borel probability measure fixed by all automorphisms of the system. This matters because automorphism groups of shifts are often large and non-amenable, so such a measure is not guaranteed by general fixed-point theorems. The proof uses the shift's growth function: when the number of allowed patterns on radius-$r$ balls grows subexponentially, uniform measures on restrictions of configurations to carefully chosen balls converge to a limit that every automorphism preserves. The same machinery also shows that automorphism groups of minimal zero-entropy shifts are sofic, a property weaker than amenability.

What carries the argument

The central object is the growth function $N_\Sigma(F)$, the number of distinct finite patterns the shift allows on a finite set $F$. The engine is a ratio condition: there must be an increasing exhaustive sequence of finite sets $F_n$ with $N_\Sigma(\bigcup_{g\in K} gF_n)/N_\Sigma(F_n)\to 1$ for every finite $K$, and zero entropy guarantees this for balls. Proposition 2.1 shows that under this condition, the uniform measures on sets of representatives of the restrictions to $F_n$ converge along a subsequence to a characteristic measure. Automorphisms enter through their memory sets—the finite sets on which the automorphism acts as a fixed block map—so that on each $F_n$ an automorphism is approximately a bijection between representative sets, forcing the limit measure to be invariant.

What would settle it

Exhibit a zero-entropy shift whose automorphism group has no invariant probability measure, or a minimal zero-entropy shift whose automorphism group is not sofic; either example would directly refute Theorems 1.2 and 1.7. A more local check is to compute ratios $N_\Sigma(B_{r+R})/N_\Sigma(B_r)$ for fixed $R$: a zero-entropy shift for which these ratios stay bounded away from 1 along every subsequence would break the subsequence step of Theorem 1.4.

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Extended reading notes

Core claim

The central theorem (Theorem 1.4) states that for every finitely generated group $G$ and every shift $(G,\Sigma)$ with $\liminf_{r\to\infty} \frac{1}{r}\log N_\Sigma(B_r)=0$, the shift admits a characteristic measure; specializing to $G=\mathbb{Z}$ gives Theorem 1.2, that every zero-entropy shift over the integers admits one. The proof constructs the measure as a weak limit of uniform measures on sets of representatives of restriction maps $\pi_n:\Sigma\to\Sigma^{F_n}$, and shows that any automorphism, being a block map with finite memory, pushes the limit back to itself. A parallel argument establishes that $\mathrm{Aut}(\mathbb{Z},\Sigma)$ is sofic whenever $(\mathbb{Z},\Sigma)$ is a minimal zero-entropy shift. The broader question of whether every shift, regardless of entropy, admits a characteristic measure is left open.

Load-bearing premise

The proof assumes that zero exponential growth lets you pick radii at which enlarging a ball by any fixed finite set adds only a negligible fraction of new allowed patterns; if this subsequence does not exist, the constructed limit measure need not be automorphism-invariant.

Editorial extensions

If this is right

Every zero-entropy shift over $\mathbb{Z}$ has a probability measure invariant under all its automorphisms.
Every shift over a finitely generated group whose ball-pattern count grows subexponentially admits a characteristic measure; for amenable groups the measure can be chosen invariant under the group action as well.
Every minimal zero-entropy shift has a sofic automorphism group, placing these groups in a broad class that contains amenable and residually finite groups.
The construction is explicit enough that characteristic measures can be approached as limits of uniform cylinder measures for concrete shifts.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The ratio condition in Proposition 2.1 looks like the real sufficient hypothesis, and it does not require the group to be amenable; finding other sequences of sets with negligible relative boundary growth might yield characteristic measures outside the zero-entropy setting.
The soficity result is a step toward the open conjecture that automorphism groups of minimal zero-entropy shifts are amenable; showing that the constructed characteristic measure arises from an amenable action would close the remaining gap.
Lemma 2.4 suggests that in minimal shifts every nontrivial automorphism moves every configuration, and it is exactly this fixed-point-free behavior that converts the growth condition into soficity; non-minimal shifts would need a different mechanism.
A natural computational check on any proposed zero-entropy shift is whether the uniform representative measures converge in the sense of Proposition 2.1; shifts where the limits depend on the chosen subsequence would reveal extra structure in the automorphism group.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this is a real result. The paper shows every Z-shift of zero entropy has a characteristic measure, generalizes to finitely generated groups with sublinear log-growth of balls, and proves soficity of Aut for minimal zero entropy shifts. The Cyr-Kra conjecture is context, not an input. I read the construction carefully; the weak-limit argument in Prop 2.1 is sound and the total variation bound is right.

What is new: existence for zero entropy shifts answers a natural open question, the growth criterion is new, and the soficity theorem is a concrete step toward the Cyr-Kra amenability conjecture. No fitted parameters, no self-citation tricks, and the citation pattern is clean. The paper is honest about what remains open (Question 1.3).

Soft spots, in order:

1. The proof of Thm 1.4 skips a lemma. From liminf L(r)/r = 0 for a nondecreasing L, it claims there is a subsequence r_n with L(r_n+i) - L(r_n) -> 0 for each fixed i. That is not immediate from monotonicity, and it is exactly what feeds Prop 2.1. I believe the lemma is true -- choose r_n in a long flat stretch before a jump -- but the paper should prove it. Without that, Thm 1.4 has a gap, albeit a fillable one.

2. Prop 2.1 has a left/right multiplication mismatch. The statement assumes liminf N_Sigma(union_{g in K} g F_n)/N_Sigma(F_n) = 1, but the proof uses tilde F_n = union_{g in K} F_n g. In a nonabelian group those are different sets. For the ball application both are contained in B_{r_n + diam K}, so the proof of Thm 1.4 can be repaired; as stated, the proposition is not proved. This is a presentation bug, not a conceptual one.

Also minor: the phrase "and because L(r) is increasing" is doing too much work. Give the subsequence argument in a sentence or refer to a lemma.

Overall, the main theorem is true as far as I can tell, and the flaws are technical exposition issues. The paper deserves a serious referee; I would send it out. It will be useful to people working on automorphism groups of shifts, and to anyone following the Cyr-Kra conjecture. I would cite it if I worked in the area.

Referee Report

3 major / 4 minor

Summary. The paper studies characteristic measures for symbolic dynamical systems, i.e., probability measures invariant under the automorphism group of a shift. The main result, Theorem 1.4, states that for any finitely generated group G, every shift (G,Σ) with liminf_r (1/r) log NΣ(B_r) = 0 admits a characteristic measure; Theorem 1.2 is the Z specialization to zero entropy shifts. The paper also proves Theorem 1.7: the automorphism group of a minimal zero entropy Z-shift is sofic, via a more general criterion (Theorem 2.2). The proofs are based on a construction (Proposition 2.1) that produces a characteristic measure as a weak limit of uniform measures on sets of representatives of restrictions to a growing sequence of finite sets, provided that the growth is asymptotically unchanged by translating by finite sets.

Significance. If the technical gaps are repaired, the results are significant: they give the first general existence theorem for characteristic measures on zero entropy shifts, and the soficity conclusion is a concrete partial step toward the Cyr–Kra conjecture on amenability of automorphism groups of minimal zero entropy shifts. The central construction is elegant and self-contained, uses no fitted parameters, and does not rely on the conjectures it discusses. The paper is likely to be of interest to researchers in symbolic dynamics and topological dynamics.

major comments (3)

[Proof of Theorem 1.4] The step reading 'Thus, and because L(r) is increasing, there is another subsequence r_n such that for every i>0, lim_L(r_n+i)-L(r_n)=0' is not justified in the text. This local-vanishing property is exactly what is needed to verify the hypothesis of Proposition 2.1, so the proof of the main theorem is incomplete as written. The statement is, in fact, provable for monotone integer-valued L: if for some fixed i there were a bound R such that every interval of length i beyond R contained an increment of L, then L(r) ≥ (r-R)/i for large r, contradicting liminf L(r)/r = 0; hence arbitrarily large i-increment-free intervals exist, and a diagonal choice gives the desired subsequence. This argument should be included.
[Proposition 2.1] There is a mismatch between the hypothesis and the proof concerning left versus right multiplication. The hypothesis asserts liminf_n NΣ(∪_{g∈K} gF_n)/NΣ(F_n)=1, but the proof defines F~_n = ∪_{g∈K} F_n g and uses the inclusion gK ⊆ F_n K to show that φ' is well defined. In a nonabelian group these sets are generally different, and the proof as written therefore does not prove the proposition as stated. The same issue appears in Theorem 2.2. The applications to balls in Theorem 1.4 are unaffected because B_r K and K B_r are both contained in B_{r+diam K}, but the statements of Proposition 2.1 and Theorem 2.2 should be corrected (preferably to right multiplication) or the proof should be adapted accordingly.
[Proof of Theorem 2.2, property (4)] The claim that the fourth condition of Lemma 2.3 follows from 'the fact that K ⊆ F~' is not correct as written: K is not necessarily contained in F~ = ∪_{g∈K} gF_k. To make the argument valid, one should choose k large enough that F_k contains K (possible because the increasing sequence F_n exhausts G), so that σ_F = φ(σ)_F implies σ_K = φ(σ)_K and the defining property of K applies. This is a repairable gap, but it should be fixed explicitly.

minor comments (4)

[Abstract] There is a typo in the abstract: 'adm it' should be 'admit'.
[Lemma 2.3] In condition (4) of Lemma 2.3, the expression '~g(a) = ( a)' appears to be a typo for '~g(a) = a'.
[Theorem 1.7] Theorem 1.7 has a grammatical typo: 'Let (Z, Σ) a minimal shift' should be 'Let (Z, Σ) be a minimal shift'.
[Proof of Proposition 2.1] In the proof of Proposition 2.1, the phrase 'Since their union is contained in Σ_{F~_n}' is slightly imprecise: the union of R_n and S~_n is a subset of the set of projections Σ_{F~_n}, not of the full shift-invariant set Σ. The intended inequality is clear, but the wording could be tightened.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorems are proved from the definitions via Proposition 2.1 and standard external results; the zero-entropy hypothesis is an input, not a disguised version of the conclusion.

full rationale

The paper's central result, Theorem 1.4, is derived by combining Proposition 2.1 with the growth hypothesis liminf_r (1/r) log N_Sigma(B_r) = 0. The proof of Proposition 2.1 constructs candidate measures as weak limits of uniform measures on representatives of restrictions, and then uses the growth condition to show that each automorphism preserves these measures. This is a genuine derivation from the definitions, not a restatement of the conclusion. The zero-entropy assumption appears as a hypothesis of the theorem, not as a consequence being asserted elsewhere. The paper does not fit any parameter to data, rename a known result, or rely on the authors' own prior theorems. The only reliance on prior work is standard, external material: Hedlund's theorem that automorphism groups of shifts are countable, the standard memory-set representation of cellular automata, and a standard reduction from the paper's four conditions to soficity. The Cyr-Kra conjecture is mentioned only as context and is not used as a premise. The proof of Theorem 1.4 does contain a compressed subsequence step ('Thus, and because L(r) is increasing'), and Proposition 2.1 as stated uses left multiplication while the proof defines F~_n using right multiplication; these are potential correctness or exposition issues, not circularity. There is no equation or definition that equates the theorem's output with its input, and no self-citation chain carries the argument. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. The arguments rest on standard theorems of symbolic dynamics and topology, and on the zero-entropy hypothesis. The only technical subtlety is a subsequence-selection step in Theorem 1.4 that is stated rather than proved.

assumptions (3)

standard math Every automorphism of a subshift is a block map with a finite memory set (Curtis-Hedlund-Lyndon theorem).
Used in Proposition 2.1 and Theorem 2.2 to write φ(σ)(g) = Φ((g^{-1}σ)_K) for a finite memory set K.
standard math The space of Borel probability measures on a compact metrizable space is compact, so the uniform measures ν_n have a weak limit.
Used in the proof of Proposition 2.1 to obtain the candidate characteristic measure ν.
standard math The growth function NΣ is monotone under inclusion and subadditive under concatenation of finite sets.
Used in the proof of Theorem 1.4 to infer that small average growth implies small local growth of balls, and to compare NΣ(F_n) and NΣ(∪ gF_n).

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Cite this review

Pith. "Pith review of Characteristic measures of symbolic dynamical systems." pith.science (2026). https://pith.science/paper/JAHU5BPK

@misc{pith2026190802930,
  author       = {Pith},
  title        = {Pith review of: Characteristic measures of symbolic dynamical systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JAHU5BPK}},
  note         = {Machine review of arXiv:1908.02930}
}

read the original abstract

A probability measure is a characteristic measure of a topological dynamical system if it is invariant to the automorphism group of the system. We show that zero entropy shifts always admit characteristic measures. We use similar techniques to show that automorphism groups of minimal zero entropy shifts are sofic.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

1, 71–114

Mike Boyle, Douglas Lind, and Daniel Rudolph, The automorphism group of a shift of ﬁnite type , Transactions of the American Mathematical Society 306 (1988), no. 1, 71–114

work page 1988

[2] [2]

Tullio Ceccherini-Silberstein and Michel Coornaert, Cellular automata and groups, Springer Science & Business Media, 2010

work page 2010

[3] [3]

Coven, Anthony Quas, and Reem Yassawi, Computing automorphism groups of shifts using atypical equivalence classes , Discrete Analysis (2016), Paper No

Ethan M. Coven, Anthony Quas, and Reem Yassawi, Computing automorphism groups of shifts using atypical equivalence classes , Discrete Analysis (2016), Paper No. 3, 28

work page 2016

[4] [4]

Sigma 3 (2015), Paper No

Van Cyr and Bryna Kra, The automorphism group of a shift of linear growth: beyond transitivity, Forum of Mathematics. Sigma 3 (2015), Paper No. e5, 27

work page 2015

[5] [5]

, The automorphism group of a minimal shift of stretched expon ential growth, Journal of Modern Dynamics 10 (2016), 483–495

work page 2016

[6] [6]

2, 613–621

, The automorphism group of a shift of subquadratic growth , Proceedings of the American Mathematical Society 144 (2016), no. 2, 613–621

work page 2016

[7] [7]

1, 64–95

Sebasti´ an Donoso, Fabien Durand, Alejandro Maass, and Samu el Petite, On automorphism groups of low complexity subshifts , Ergodic Theory and Dynam- ical Systems 36 (2016), no. 1, 64–95

work page 2016

[8] [8]

2, 335–386

Harry Furstenberg, A Poisson formula for semi-simple Lie groups , Annals of Mathematics 77 (1963), no. 2, 335–386

work page 1963