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Independence, sequence entropy and mean sensitivity for invariant measures

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Pith's one-line read For an ergodic measure on an amenable group action, a tuple is a sequence-entropy tuple exactly when it is mean-sensitive along a tempered Følner sequence.

desk verdict Solid extension of local entropy theory to amenable groups, with a real but repairable gap in the proof of the headline Theorem 1.6. read the letter →

arxiv 2501.08069 v2 pith:7IQJDPTE submitted 2025-01-14 math.DS

Chunlin Liu , Leiye Xu , Shuhao Zhang This is my paper · ORCID
classification math.DS MSC 37A1537A3537B05
keywords ITtuplesequenceentropymeansensitiveamenablegroupactionKroneckerfactortemperedFølnerergodicmeasureindependenceset
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 sets out to unify three local ways of measuring complexity in a topological dynamical system: sequence entropy tuples, mean-sensitive tuples, and sensitive-in-the-mean tuples. It proves that for an ergodic invariant measure on a system acted on by a countable infinite amenable group, all three notions coincide along any tempered Følner sequence. It also proves the half of the story that needs no amenability: every measure-theoretic sequence entropy tuple is an independence (IT) tuple. A reader should care because this turns sequence entropy, which is defined by a supremum over all time sequences, into a single orbit-averaging condition that can be checked along one fixed averaging schedule.

What carries the argument

The object that carries the argument is the diagonal measure $\lambda^X_K=\int_X \mu_x\times\cdots\times\mu_x\,d\mu(x)$, built from the disintegration of $\mu$ over its Kronecker factor (the largest factor with discrete spectrum); its support, minus the diagonal, is exactly the set of sequence-entropy tuples. The proof of Theorem 1.6 represents the Kronecker factor as a rotation on a compact homogeneous space $Z/H$, decomposes each product measure $\mu^{(K)}$ into ergodic components $\lambda^X_z$ indexed by $z\in Z^K$, and uses a level-set lemma to turn a positive value of $\lambda^X_K$ on a product of neighborhoods into a large set of fibers where every coordinate has uniformly positive mass. The converse direction uses the fact that a zero-sequence-entropy partition lies inside the Kronecker algebra, so its indicator functions are uniformly almost periodic along the tempered Følner sequence; an averaging contradiction then forces the tuple to be mean sensitive.

What would settle it

Compute, for an ergodic rotation on a compact abelian group (say the circle) and two small arcs, the diagonal measure of the product of the arcs and the Haar measure of the set of rotations for which both fibres have mass greater than $b$; the proof predicts the latter exceeds $b$ whenever the former exceeds $2b$. A concrete example violating this inequality would show that the step from diagonal measure to fibre level sets is false, breaking the first inclusion of Theorem 1.6.

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Extended reading notes

Core claim

The central claim, Theorem 1.6, is that for a topological dynamical system $(X,G)$ with $G$ amenable and $\mu$ ergodic, the set $\mathrm{SE}^\mu_K(X,G)$ of $\mu$-sequence-entropy $K$-tuples, the set $\mathrm{MS}^\mu_K(X,F)$ of $\mu$-mean-sensitive $K$-tuples along a tempered Følner sequence $F$, and the set $\mathrm{SM}^\mu_K(X,F)$ of $\mu$-sensitive-in-the-mean $K$-tuples along $F$ are the same for every $K\ge2$. The supporting result Theorem 1.1 shows that $\mathrm{SE}^\mu_K(X,G)\subset \mathrm{IT}_K(X,G)$, where IT tuples are those whose neighborhoods admit infinite independence sets, so that every sequence-entropy tuple is an IT tuple; together with the known inclusion $\mathrm{IT}_K\subset \mathrm{SE}_K$ this closes the chain of local complexity notions. Theorem 1.3 and Corollary 1.5 convert these local results into a global bound: if no essential IT $K$-tuples exist, then $h^*_\mu(G)\le \log(K-1)$ for every invariant measure $\mu$.

Load-bearing premise

The equality depends on two structural inputs: the almost-periodic factor of an ergodic system being a compact-group rotation (so fibre masses can be shifted and averaged), and its indicator functions being uniformly almost periodic along the averaging sequence; if either input fails for a concrete system, one of the two inclusions breaks.

Editorial extensions

If this is right

  • For ergodic measures on amenable groups, sequence-entropy tuples are computable by a fixed Følner averaging: a tuple is such a tuple exactly when every positive-measure set contains points that jointly visit the tuple's neighborhoods with positive limsup frequency.
  • Every measure-theoretic sequence entropy tuple is an IT tuple, so the measure-theoretic and combinatorial notions of independence are connected: $\bigcup_\mu \mathrm{SE}^\mu_K \subset \mathrm{IT}_K \subset \mathrm{SE}_K$.
  • Systems without essential IT K-tuples have maximal sequence entropy at most $\log(K-1)$ for every invariant measure, giving a multi-tuple version of Huang's tameness bound.
  • If an ergodic system has $h^*_\mu(G)>\log(K-1)$, it must contain essential mean-sensitive K-tuples, so high sequence entropy manifests as a local sensitivity phenomenon.
  • The mean-sensitive and sensitive-in-the-mean notions, previously known to be equivalent for pairs in abelian systems, now coincide with sequence entropy tuples for every K in all amenable group actions.

Reading between the lines

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

  • Because the inclusion SE⊂IT does not use amenability, the paper suggests that the independence character of sequence-entropy tuples is a non-amenable phenomenon; a natural test is whether the converse half of Theorem 1.6 can be formulated with invariant means in place of Følner sequences.
  • The fibre-shifting construction indicates an algorithmic way to certify high sequence entropy: sample fibers of the Kronecker factor and look for tuples whose neighborhoods have uniformly positive mass on a large level set; this could be implemented numerically for low-dimensional systems.
  • For non-ergodic measures the coincidence is known to fail, so the paper's use of ergodicity is not merely technical; extending the result would require an additional assumption controlling how Kronecker fibers vary across the ergodic decomposition.
  • The equality of the three tuple sets gives a new local analogue of the global statement that positive sequence entropy is equivalent to mean sensitivity, suggesting that further local-global bridge theorems for amenable group actions may hold for other complexity notions such as weak mixing or Devaney chaos.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 6 minor

Summary. The paper studies local entropy theory for actions of countable infinite discrete groups. Theorem 1.1 shows that every measure-theoretic sequence entropy K-tuple of an invariant measure is an IT K-tuple, yielding with known results the chain \cup_\mu SE^\mu_K \subset IT_K \subset SE_K. Theorem 1.3 gives an upper bound log(K−1) on maximal sequence entropy when no essential sequence entropy K-tuples exist, and Corollary 1.5 extends this to absence of essential IT tuples. Theorem 1.6 claims that for ergodic measures of amenable group actions, sequence entropy tuples, mean sensitive tuples, and sensitive-in-the-mean tuples coincide along any tempered Følner sequence. The proofs of Theorems 1.1 and 1.3 are essentially sound; the proof of Theorem 1.6 contains a gap in the SE⊂MS inclusion concerning the existence of an ergodic shift z satisfying both a positivity condition from Lemma 2.5 and a support condition from the given positive-measure set.

Significance. If Theorem 1.6 can be repaired, the paper settles an open question from the local entropy theory literature by giving a complete local characterization of sequence entropy via mean sensitivity for ergodic measures under amenable group actions, going beyond the abelian case treated in [6] and [25]. The novelty also includes the IT-tuple result (Theorem 1.1) and the entropy bound (Theorem 1.3), which are clean and appear correct. The proof technique, using the Kronecker factor and disintegration measures, is elegant and likely portable. The gap in Theorem 1.6 is localized and repairable, so the paper's contribution is substantial despite the current lack of full justification.

major comments (1)
  1. [Section 5.2] In the proof of SE^\mu_K ⊂ MS^\mu_K, the step 'By Lemma 5.6, ν_Z(∩ Z^b_{U_k}) > b, which together with Lemma 2.5 and Lemma 5.4, implies that there exists z=(z_1,…,z_K)∈Z^K such that …' is not justified as written. To apply Lemma 5.6 to λ^X_K one must first identify λ^X_K with λ^X_z for z=(e,…,e); this identification is true (via (2.5) and (5.5)) but is never stated. More importantly, the existence of an ergodic z satisfying both conditions (2) and (3) does not follow from Lemmas 2.5 and 5.4 alone: Lemma 2.5 gives an open neighborhood W of e such that (2) holds for every z∈W^K, and Lemma 5.4 gives a full-measure set E of z for which λ^X_z is ergodic, but the point z=(e,…,e) need not belong to E, and the diagonal has ν_Z^{(K)}-measure zero. A missing positive-measure argument is needed, for instance showing that S={z∈W^K: ν_Z(∩ Z_V z_k^{-1})>0} has positive ν_Z^{(K)}-measure by computing ∫_{W^K} ν_Z(∩ Z_V z_k^{-1}) dν_Z^{(K)}(z) = ∫_Z (ν_Z(z^{-1}Z_V∩W))^K dν_Z(z) > 0, then intersecting S with E to obtain the desired z. The manuscript omits this argument, so the inclusion SE^\mu_K ⊂ MS^\mu_K is not fully proved as written. The gap is fixable, but it is load-bearing for Theorem 1.6.
minor comments (6)
  1. [Section 4] In the proof of Theorem 1.3, the sentence 'We now suppose that (X,G) has essential sequence entropy K-tuples for µ' should read 'does not have essential sequence entropy K-tuples' to match the theorem statement and the subsequent use of Lemma 4.1.
  2. [Appendix A] In the proof of Proposition 2.7, the phrase 'an open neighborhood of (x_1,…,x_K)' refers to the set X\B_k, which is not necessarily open; it should be X\setminus \overline{B_k} throughout that argument.
  3. [Section 5.2] The final line of the proof of Theorem 1.6 writes SM^\mu_K(X,G) ⊂ SE^\mu_K(X,T); the last argument should be SE^\mu_K(X,G).
  4. [Theorem 1.6] The statement 'the µ-sequence entropy K-tuple, the µ-mean sensitive K-tuple along F and the µ-sensitive in the mean K-tuple along F coincide' should use the plural 'tuples'.
  5. [Section 5.2, SM⊂SE step] The application of (5.11) requires a measurable set A⊂L with µ(A)>0 and diam(A)<δ'; such a set exists because µ(L)>0 and µ is a finite Borel measure on a compact metric space, but this elementary fact is not stated.
  6. [Section 2.6 and Lemma 5.4] There are minor typos: 'supremun' should be 'supremum' in the definition of h^*_µ, and 'The later' should be 'The latter' in the proof of Lemma 5.4.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction: the main theorem's inputs are independently defined, Theorem 2.6 is backed by non-self references, and the only flagged issue is a missing existence step in Section 5.2, which is a proof gap rather than a circularity.

full rationale

I found no step in which a claimed prediction or first-principles result is equivalent to its inputs by construction. The principal formula used as an input, h*_mu(G,alpha)=H_mu(alpha|K_mu(G)) (Theorem 2.6), is credited to the authors' [27] but also to Kerr-Li [20] and to Huang-Maass-Ye [16], and the text remarks that all proofs are essentially the same as in [16]; hence the self-citation is not load-bearing and does not force the conclusion. The proof of Theorem 1.6 derives the equalities from the independently defined notions of sequence entropy tuple, mean sensitive tuple and sensitive-in-the-mean tuple (Sections 2.6-2.7), using the support characterization Proposition 2.7 and the decomposition of ergodic measures via the Kronecker factor; no parameter is fitted to the target set, and no quantity called a prediction is defined in terms of the conclusion. The one local concern is in Section 5.2, where the text says: 'By Lemma 5.6, ν_Z(∩ Z^b_{U_k}) > b, which together with Lemma 2.5 and Lemma 5.4, implies that there exists z = (z1, . . . , zK) ∈ Z^K' satisfying (1)-(3). This needs a short additional positive-measure/continuity argument, because Lemma 5.4 guarantees ergodicity only for ν^{K}_Z-a.e. z and the diagonal point z=(e,...,e) may have measure zero. That omission is a proof gap, not a circular reduction: the existence of the desired z is not derived from the theorem's conclusion. Since all load-bearing ingredients are either proved in the paper or supported by independent references, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. It relies on standard and prior-cited results, primarily the sequence entropy formula and the structure of Kronecker factors. The main unproved input is Theorem 2.6, which is cited rather than proved here.

assumptions (5)
  • standard math Existence of tempered F\u00f8lner sequences and the pointwise ergodic theorem (Theorem 2.1, from Lindenstrauss).
    Used in Theorem 1.6 to pass from ergodic averages to measure values; standard for amenable group actions.
  • domain assumption Kushnirenko-type formula h*_mu(G,alpha) = H_mu(alpha|K_mu(G)) (Theorem 2.6, cited from [27] and [20]).
    Central tool connecting sequence entropy to the Kronecker algebra; assumed as a prior result.
  • standard math Mackey's structure theorem for ergodic discrete spectrum systems (Proposition 2.4).
    Used to represent the Kronecker factor as a quotient of a compact group, essential for the Section 5 decomposition.
  • standard math Regularity of disintegration and ergodic decomposition (Section 2.4).
    Standard measure theory used throughout the proofs.
  • standard math [31, Lemma 2.5] on almost periodic functions (used in the proof of SM subset SE).
    Provides the uniform continuity of averages of almost periodic indicators, needed for the contradiction argument.

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Pith. "Pith review of Independence, sequence entropy and mean sensitivity for invariant measures." pith.science (2026). https://pith.science/paper/7IQJDPTE

@misc{pith2026250108069,
  author       = {Pith},
  title        = {Pith review of: Independence, sequence entropy and mean sensitivity for invariant measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7IQJDPTE}},
  note         = {Machine review of arXiv:2501.08069}
}
read the original abstract

We investigate the connections between independence, sequence entropy, and mean sensitivity for a measure preserving system under the action of a countable infinite discrete group. We establish that every sequence entropy tuple for an invariant measure is an IT tuple. Furthermore, if the acting group is amenable, we show that for an ergodic measure, the sequence entropy tuples, the mean sensitive tuples along some tempered F{\o}lner sequence, and the sensitive in the mean tuples along some tempered F{\o}lner sequence coincide.

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Forward citations

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