arXiv:2608.09702v1

Entropies of compact subsets and supported measures

Qiang Huo, Xiangtong Wang

math.DS37B4037A3528A7860B0554H20

Paper metadata

Original submission: August 10, 2026
Claimed MSC: 37B40, 37A35, 28A78, 60B05, 54H20
arXiv categories: math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: Comments are welcome

Abstract

Let $(X,T)$ be a topological dynamical system and $(\mathcal M(X),T_*)$ be its induced system. For a non-empty compact subset $K\subset X$ , we define $\mathcal M(K)$ as the set of Borel probability measures supported on $K$ . In this paper, we systematically study the relationship between various entropies of $(T,K)$ and of $(T_*,\mathcal M(K))$ . We show that: $\begin{aligned} & h_{\mathrm{top}}^{\mathrm{UC}}(T,K)>0 \iff h_{\mathrm{top}}^{\mathrm{UC}}(T_*,\mathcal{M}(K))=\infty, \qquad &h_{\mathrm{top}}^{P}(T,K)>0\iff h_{\mathrm{top}}^{P}(T_*,\mathcal{M}(K))>0, \qquad &h_{\mathrm{top}}^{B}(T,K)>0 \implies h_{\mathrm{top}}^{B}(T_*,\mathcal{M}(K))>0 , \end{aligned}$ where $h_{\mathrm{top}}^{\mathrm{UC}}(T,K)$ , $h_{\mathrm{top}}^{P}(T,K)$ , and $h_{\mathrm{top}}^{B}(T,K)$ denote the upper capacity topological entropy, the packing topological entropy, and the Bowen topological entropy of $K$ , respectively. Additionally, we present a counterexample involving a non-invariant set, demonstrating that the converse of the third assertion is not valid in general.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections 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 August 18, 2026

01Statements4 reported findingsCorrect

For nonempty compact $K$ , the upper-capacity and packing entropy equivalences, the Bowen-entropy amplification, and the compact counterexample to the Bowen converse are correct. Theorem 1.3 omits compactness from its first displayed hypothesis; this is a local scope correction consistent with the abstract and the proof and does not lower the substantive status.

Theorem 1.1Correct

Upper-capacity entropy has the zero-or-infinite amplification dichotomy

Pages 2 and 6–9 · Theorem 1.1 and Section 3 · arXiv:2608.09702v1

Weighted atomic embeddings of $K^m$ into $\mathcal M(K)$ multiply upper-capacity entropy by at least $m$ , giving infinity whenever $K$ has positive entropy. Conversely, an exponentially large separated family of supported measures is discretized by a Bowen-spanning partition of $K$ and sent into a finite-dimensional $\ell_\infty$ space. The cited Glasner–Weiss combinatorial lemma forces that partition to have exponentially many atoms, proving positivity of the entropy of $K$ . The contrapositives yield the zero equivalence.

Theorem 1.2Correct

Packing entropy is positive for $K$ exactly when it is positive for $\mathcal M(K)$

Page 3 and pages 9–14 · Theorem 1.2 and Section 4 · arXiv:2608.09702v1

The Dirac embedding preserves pointwise local entropy and gives the forward inequality. For the converse, the compact-set packing variational principle supplies a measure on $\mathcal M(K)$ with positive upper local entropy. Its barycenter has zero upper local entropy if $K$ has zero packing entropy; after restricting most of its mass to a compact low-capacity subset, Lemma 4.2 transfers any remaining positive local entropy back to that subset, giving the required contradiction.

Theorem 1.3Correct

Bowen entropy amplification and failure of the converse

Page 3 and pages 14–17 · Theorem 1.3 and Section 5 · arXiv:2608.09702v1

For compact $K$ , product measures and the Bowen variational principle give $h_{\mathrm{top}}^B(T^{\times m},K^m)\geq m h_{\mathrm{top}}^B(T,K)$ , and the weighted atomic embedding then makes positive entropy on $K$ infinite on $\mathcal M(K)$ . In the binary-shift example, the two coordinate sets have alternating zero lower densities, so their union has zero Bowen entropy, while a family of mixtures of two Bernoulli product measures embeds full product alphabets of arbitrary finite size in $\mathcal M(K)$ and gives infinite Bowen entropy.

Theorem 1.3 hypothesisMinor formal correction

Compactness should be retained in the first assertion

Page 3 · statement of Theorem 1.3 · arXiv:2608.09702v1

Insert 'compact' in 'If $K\subset X$ is non-empty.' Proposition 5.1 and its proof invoke the compact-set variational principle, and the abstract, the common setup, and the counterexample all concern compact subsets. No claim for arbitrary noncompact sets is established, while the one-word restriction makes the theorem match every use and leaves all advertised conclusions intact.

02Proofs4 reported findingsContains incorrect or incomplete proofs

The upper-capacity and Bowen arguments are correct and complete. The packing-entropy converse has one concrete support defect: a normalized restriction to a merely Borel set need not have support contained in that set. A verified repair is available by taking a compact subset before restricting, but the printed proof is incomplete as written.

Theorem 4.3Incomplete as written

Restriction to a Borel set need not produce a measure supported on that set

Pages 11–13 · proof of Theorem 4.3, definition of $R_A$ · arXiv:2608.09702v1

The paper defines $\mathcal M(A)$ by the condition $\operatorname{supp}\mu\subset A$ . Lemma 4.1 produces a Borel set $A$ , but the proof then asserts that $R_A(\nu)=\nu|_A/\nu(A)$ belongs to $\mathcal M(A)$ . For nonclosed $A$ , the support of this restriction can contain boundary points outside $A$ , so the pushed-forward measure $\lambda$ need not satisfy the support hypothesis of Lemma 4.2. Repair classification: Verified repair. By inner regularity on the compact metric space, replace $A$ by a compact subset $A'\subset A$ whose barycenter mass remains above the required threshold. Monotonicity preserves $h_{\mathrm{top}}^{UC}(T,A')<c$ , $R_{A'}(\nu)$ is supported on $A'$ , and all total-variation and local-entropy estimates continue with the same argument after an arbitrarily small adjustment of $\theta$ .

Propositions 3.2 and Theorem 3.4Correct and complete

Atomic embeddings and the finite-dimensional separation argument

Pages 6–9 · Section 3 · arXiv:2608.09702v1

The binary weights make the ordered atomic representation injective even when coordinates coincide, the map is equivariant, and product separated sets give the lower bound. In the converse direction, uniform continuity of finitely many test functions controls the discretization error, the tail of the metric contributes less than $\epsilon/16$ , and the Glasner–Weiss lemma applies with target dimension $nL$ , forcing exponential growth of the spanning number of $K$ .

Proposition 5.1 and Theorem 5.3Correct and complete

Product lower bound and binary-shift counterexample

Pages 14–17 · Section 5 · arXiv:2608.09702v1

For the product measure, the lower pointwise local entropy is superadditive under the max product metric, and integrating plus the compact Bowen variational principle yields the factor $m$ . In the example, each coordinate set has lower density zero along its alternating endpoint subsequence, which supplies arbitrarily long Bowen covers with subexponential weight. The supported mixture map records every coordinate parameter in the first test function, so an $n$ -ball fixes the first $n$ symbols of the product parameter and has mass at most $(r+1)^{-n}$ ; Lemma 5.2 then gives entropy at least $\log(r+1)$ for every $r$ .

Final invocation in Theorem 5.3Typo

The local-mass bound is Lemma 5.2

Page 17 · final paragraph of the proof of Theorem 5.3 · arXiv:2608.09702v1

Replace 'Theorem 5.2' by 'Lemma 5.2'. The immediately preceding result with that number is exactly the local Bowen-ball mass criterion applied here, so the correction is unique and harmless.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2608.09702v1
Authors listed: Qiang Huo, Xiangtong Wang
Audit date: August 18, 2026

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Entropies of compact subsets and supported measures

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Upper-capacity entropy has the zero-or-infinite amplification dichotomy

Packing entropy is positive for KKK exactly when it is positive for M(K)\mathcal M(K)M(K)

Bowen entropy amplification and failure of the converse

Compactness should be retained in the first assertion

Restriction to a Borel set need not produce a measure supported on that set

Atomic embeddings and the finite-dimensional separation argument

Product lower bound and binary-shift counterexample

The local-mass bound is Lemma 5.2

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match

Packing entropy is positive for $K$ exactly when it is positive for $\mathcal M(K)$