Entropies of compact subsets and supported measures

Qiang Huo; Xiangtong Wang

REVIEW 1 major objections 5 minor 18 references

Entropies of compact subsets and supported measures

T0 review · 1 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that passing from a compact set K to the probability measures it supports sends upper capacity and Bowen entropy to infinity whenever they are positive, while packing entropy stays positive exactly when it is positive for…

desk verdict Solid, genuinely new subset-level entropy dichotomy with a striking Bowen counterexample; one repairable gap in the packing-entropy proof. read the letter →

arxiv 2608.09702 v1 pith:QYE6IYWK submitted 2026-08-10 math.DS

Qiang Huo , Xiangtong Wang This is my paper · ORCID

classification math.DS MSC 37B4037A3528A7860B0554H20

keywords uppercapacityentropyBowenpackinglocalinducedsystemonmeasuressupportedvariationalprinciplecompactsubsets

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

Every compact set carries several notions of topological entropy that measure how complex its orbits are. This paper asks what happens to those entropies when the set $K$ is replaced by $M(K)$, the space of Borel probability measures supported on $K$, with the dynamics pushed forward. The answer is a precise trichotomy: the upper capacity entropy of $K$ is zero exactly when it is zero for $M(K)$, and positive exactly when the entropy of $M(K)$ is infinite; the packing entropy of $K$ is positive exactly when it is positive for $M(K)$; and positive Bowen entropy of $K$ forces the Bowen entropy of $M(K)$ to be infinite, although the paper constructs a compact non-invariant $K$ with zero Bowen entropy whose supported-measure space has infinite Bowen entropy. A reader should care because this determines which local entropy notions survive the passage from points to measures and which ones explode, completing the picture started by the classical zero-or-infinite dichotomy for induced systems.

What carries the argument

The load-bearing object is the local variational principle recalled as Theorem 2.2, which identifies the Bowen and packing entropies of a compact set $K$ with the suprema of the measure-theoretic upper and lower local entropies over measures $\mu\in M(K)$. The paper applies this principle twice, once to $(X,T)$ and once to the induced system $(M(X),T_*)$ with compact set $M(K)$, thereby turning topological statements about entropies into statements about local entropies of measures. Two further devices carry the quantitative estimates: the embedding $\Phi_m(x_1,\dots,x_m)=\sum_{i=1}^m a_i\delta_{x_i}$ with weights $a_i=2^{i-1}/(2^m-1)$, which embeds the $m$-fold product system into $(M(X),T_*)$ and yields lower bounds of $m$ times the original entropy; and a combinatorial lemma on separated sets in $\ell^1$ balls that transfers upper bounds from $M(K)$ back to $K$. For the Bowen counterexample, the mechanism is a pair of zero-density coordinate blocks in the full shift whose product measures have coordinates at scale $1/(r+1)^n$, which forces exponentially many separated measures in $M(K)$ for every $r$.

What would settle it

A compact set $K$ with $h^{\mathrm{B}}_{\mathrm{top}}(T,K)>0$ but finite $h^{\mathrm{B}}_{\mathrm{top}}(T_*,M(K))$ would overturn Theorem 1.3; the decisive check is to estimate Bowen covers of $M(K)$ for such a $K$, and the paper predicts they never stabilize at a finite exponential rate.

Watch

Extended reading notes

Core claim

The central claim is that the three subset entropies of a non-empty compact set $K$ and of $M(K)$ relate in exactly the following way. Writing $h^{\mathrm{UC}}_{\mathrm{top}}$, $h^{\mathrm{P}}_{\mathrm{top}}$, and $h^{\mathrm{B}}_{\mathrm{top}}$ for upper capacity, packing, and Bowen topological entropies, and $T_*$ for the pushforward map on probability measures, the paper proves: $h^{\mathrm{UC}}_{\mathrm{top}}(T,K)=0$ if and only if $h^{\mathrm{UC}}_{\mathrm{top}}(T_*,M(K))=0$, and $h^{\mathrm{UC}}_{\mathrm{top}}(T,K)>0$ if and only if $h^{\mathrm{UC}}_{\mathrm{top}}(T_*,M(K))=+\infty$; $h^{\mathrm{P}}_{\mathrm{top}}(T,K)>0$ if and only if $h^{\mathrm{P}}_{\mathrm{top}}(T_*,M(K))>0$; and $h^{\mathrm{B}}_{\mathrm{top}}(T,K)>0$ implies $h^{\mathrm{B}}_{\mathrm{top}}(T_*,M(K))=+\infty$. It then shows the last implication cannot be reversed: in the one-sided full shift on $\{0,1\}$, the compact non-invariant set $K=K_0\cup K_1$ built from two interleaved zero-density coordinate sets has zero Bowen entropy, while the supported-measure space $M(K)$ has infinite Bowen entropy. The upshot is that the passage to supported measures is monotone for packing entropy, whereas for upper capacity and Bowen entropy it acts as an amplifier that turns any positivity into infinitely many orbits.

Load-bearing premise

Everything rests on a known variational theorem that computes the entropy of a compact set from the entropies of the probability measures sitting on it, and the paper needs that theorem to work for the set of measures itself, even though that set is not required to be invariant; if the theorem fails there, the proofs collapse.

Editorial extensions

If this is right

For upper capacity entropy, the induced system on $M(K)$ obeys a strict zero-or-infinite law: $h^{\mathrm{UC}}_{\mathrm{top}}(T_*,M(K))$ is never a finite positive value, regardless of the compact set $K$.
Packing entropy gives a two-way test: deciding whether $M(K)$ has positive packing entropy is equivalent to deciding whether $K$ has positive packing entropy, so the two systems are indistinguishable at the level of packing-entropy positivity.
Positive Bowen entropy of $K$ always amplifies to infinite Bowen entropy of $M(K)$, so no compact set with positive Bowen entropy can have a measure-quiet supported space.
The zero-to-infinite jump for Bowen entropy is real: the paper's example shows that a compact non-invariant set can have zero Bowen entropy while its supported measures have infinite Bowen entropy.

Reading between the lines

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

Extension: because the proofs use only the product embedding and the variational principle, a similar amplification should hold for relative entropies of factor maps, where the same product-measure construction can be applied to the fibers.
Extension: for a $T$-invariant compact set the three subset entropies coincide on both sides, so the three theorems collapse into a clean zero-or-infinite law with no Bowen counterexample; the paper's example indicates that non-invariance is exactly what allows the Bowen jump.
Extension: the upper-capacity dichotomy suggests that separated-set entropy notions on convex measure spaces tend to be either zero or infinite, so packing entropy may be the right discriminator for finer local complexity; testing this on hyperspaces or spaces of invariant measures would be a natural next step.

Share X Bluesky LinkedIn Reddit HN

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

Two things you should know about arXiv:2608.09702. First, it proves the subset-level version of the Bauer-Sigmund/Glasner-Weiss dichotomy for upper capacity entropy, plus a packing-entropy equivalence and a striking counterexample where Bowen entropy jumps from 0 to infinity. That counterexample is the most valuable part; it is a real phenomenon and not what I would have guessed. Second, the proof of the packing equivalence has a genuine gap, but it is easily patched.

The new results are solid. Theorem 1.1 is proved with a clean spanning/separating argument and the Glasner-Weiss combinatorial lemma; the product-measure embedding gives the lower bound and the spanning-set argument gives the converse. Theorem 1.2 uses the Feng-Huang variational principle on both the original and induced systems, and the global strategy is sound. The counterexample in Section 5 is well constructed: the split into two zero-density coordinate sets gives K with zero Bowen entropy, while the family of product measures gives a large set of 'coordinates' in M(K) with exponentially small measure balls, forcing infinite Bowen entropy. This is genuinely new.

The soft spot is in the proof of Theorem 4.3. Lemma 4.1 produces a Borel set A with small outer measure and small upper capacity, and the proof then defines R_A(ν)=ν|_A/ν(A) and claims R_A(ν)∈M(A). That is false with the paper's own definition M(A)={µ: supp µ⊂A} when A is not closed; Lebesgue measure on (0,1) has support [0,1]. The subsequent application of Lemma 4.2, whose hypothesis is λ(M(A))=1, therefore does not go through as written. The stress-test note is right. The fix is routine: use inner regularity to find a compact A'⊂A with \bar µ(A\A') small and h^UC_top(T,A')≤h^UC_top(T,A), then work with R_{A'}(ν), which is genuinely supported in A'. With that change the argument works. This is a gap, not a fatal flaw.

Minor issues: the notation h_µ vs \bar h_µ for upper and lower local entropy is easy to misread, and the counterexample has typos (N_m=2^{2^m}, coordinate agreement from 0 through n+L). These are cosmetic.

The citation pattern is honest; the authors lean on Feng-Huang and Glasner-Weiss, which are the right tools, and they acknowledge discussions. No circularity.

Bottom line: this deserves a serious referee. The results are new and useful for people working on local entropy, and the one gap is repairable in a page. I'd send it to a good dynamics journal with a request for a minor revision. I would cite it if I work in this area.

Referee Report

1 major / 5 minor

Summary. This paper studies the relation between three subset entropies of a nonempty compact set K in a compact metric topological dynamical system and the corresponding entropies of the set M(K) of Borel probability measures supported on K, regarded as a subset of the induced system (M(X), T_*). Theorem 1.1 establishes a zero/infinite dichotomy for upper-capacity entropy: h^UC_top(T,K)=0 iff h^UC_top(T_*,M(K))=0, and h^UC_top(T,K)>0 iff h^UC_top(T_*,M(K))=+∞. Theorem 1.2 establishes h^P_top(T,K)>0 iff h^P_top(T_*,M(K))>0. Theorem 1.3 shows that h^B_top(T,K)>0 forces h^B_top(T_*,M(K))=+∞, and gives an explicit subshift example with h^B_top(T,K)=0 but h^B_top(T_*,M(K))=+∞. The proofs use the Feng-Huang variational principles, a Glasner-Weiss combinatorial lemma, and product-measure constructions.

Significance. If fully repaired, the paper gives a clean local-entropy counterpart to the Bauer-Sigmund and Glasner-Weiss induced-system dichotomy, and it identifies the different amplification behaviors of upper-capacity, packing, and Bowen entropies. The proof of Theorem 1.1 is careful and essentially self-contained, and the Bowen counterexample of Theorem 5.3 is explicit and checkable. The reliance on the published Feng-Huang variational principles is legitimate rather than circular. The main weakness is a genuine but repairable gap in the proof of Theorem 1.2, which must be fixed before the paper can be accepted.

major comments (1)

[§4, proof of Theorem 4.3 and Lemma 4.2] The proof of Theorem 4.3 defines R_A(ν)=ν|_A/ν(A) for the Borel set A produced by Lemma 4.1 and claims that R_A(ν)∈M(A), so that λ=(R_A)_*τ_H satisfies λ(M(A))=1. This is false when A is not closed, because M(A) was defined in §2 as {µ : supp µ⊂A}, and supp(ν|_A) need not be contained in A. For example, if X=[0,1], A=(0,1), and ν is Lebesgue measure, then supp(ν|_A)=[0,1]. Since Lemma 4.2 has the hypothesis τ(M(A))=1, the argument as written cannot be applied. This gap is load-bearing for the hard direction of Theorem 1.2. It is repairable: by inner regularity one may choose a compact A'⊂A with h^UC_top(T,A')≤h^UC_top(T,A)<c and with ar µ(A\A') small, and then run the same argument with R_{A'}; for compact A', supp(R_{A'}(ν))⊂A' is automatic. The proof should be amended accordingly.

minor comments (5)

[§5, Theorem 5.3] The inequality τ_r(B*_{D,n}(µ_p,1/(8r)))≤(r+1)^{-n} is verified only for µ_p∈C_r=Φ(F_r), whereas Lemma 5.2 is invoked with K=M(K). Since C_r⊂M(K), the clean fix is to apply Lemma 5.2 to C_r and then use monotonicity h^B_top(T_*,M(K))≥h^B_top(T_*,C_r), or to justify explicitly why the argument may be restricted to C_r.
[§5, Proposition 5.1 and Theorem 1.3] Theorem 1.3 states the first implication for any nonempty K, but the proof of Proposition 5.1 uses the compact-set variational principle and assumes K is compact. The statement should say 'non-empty compact K' to match the proof and the abstract, unless a separate argument for noncompact K is supplied.
[§4, Lemma 4.2] In the proof of Lemma 4.2, the closed Bowen balls of a spanning set are said to yield a partition A_1,...,A_{M_n} of K, but they should partition A. Also, in the displayed formula for Φ_n, the symbol x_i should be v_i.
[§4, Theorem 4.3] The reference 'By Theorem 4.2' near the end of the proof of Theorem 4.3 should read 'By Lemma 4.2'.
[§2.5 and §4, Lemma 4.1] The notation for upper and lower measure-theoretic entropies is hard to follow because the overline/underline distinctions are easily lost; in particular, Lemma 4.1 should state explicitly which entropy (upper or lower) is assumed to vanish and which one is used in the proof, since the choice matters for the inequality involving limsup and liminf.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main results are derived from external variational principles and explicit constructions.

full rationale

The derivation chain is self-contained with respect to its own claims. Theorem 1.1 is proved from the explicit embedding Phi_m (Section 3, Lemma 3.1) and the external combinatorial lemma of Glasner-Weiss [9, Proposition 2.1]; no quantity being predicted is used in its own definition. Theorems 1.2 and 1.3 lean on the Feng-Huang variational principles [8, Theorems 1.2(i) and 1.3(i)] as stated in Theorem 2.2, which are published external results and are not derived from the present paper's assumptions; the authors' acknowledgement to Wen Huang does not make this citation load-bearing self-citation. The remaining arguments use explicit measure mappings (R_A and the product measure nu = mu tensor m) and explicit examples (the one-sided full shift example in Theorem 5.3). A separate repairable technical gap exists in Theorem 4.3, where R_A(nu) is asserted to lie in M(A) for a non-closed Borel set A although M(A) is defined by supp nu subset of A; this is a correctness issue, not a circularity, since replacing A by a compact subset of large mu-measure repairs the argument without changing the claimed result.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central theorems do not introduce free parameters or new entities; they are universal statements about compact metric topological dynamical systems. The only hand-chosen numerical construction is the block-length sequence in the counterexample, and the proofs lean on two external theorems (the Feng-Huang variational principle and the Glasner-Weiss lemma).

free parameters (1)

Block length sequence N_m (counterexample) = N_m = 2^{2^m} as intended; printed as N_m=2^{2m} in the text
Hand-chosen in the construction of Theorem 5.3 to make the ratio N_{m-1}/N_m tend to 0, giving J_0 and J_1 lower density 0. This is what forces Bowen entropy of K_i to be 0 while the measure space entropy is infinite.

assumptions (5)

standard math Feng-Huang variational principles: for every non-empty compact K, h^B_top(T,K)=sup_{mu in M(K)} h_mu(T) and h^P_top(T,K)=sup_{mu in M(K)} hbar_mu(T)
Imported from [8, Theorems 1.2(i), 1.3(i)]; used in Theorems 4.3, 4.4, and Proposition 5.1. Not proved in this paper.
standard math Glasner-Weiss combinatorial lemma (Lemma 3.3)
Imported from [9, Proposition 2.1]; used to convert separated sets in M(K) into lower bounds on spanning numbers of K in Theorem 3.4 and Lemma 4.2.
standard math Compactness and metrizability of M(X), continuity of T_*
Classical facts (Parthasarathy [15]); used throughout to define the induced system and its entropies.
standard math Uniform equivalence of compatible metrics on a compact space
Used in Theorem 4.4 to show local entropies are independent of the choice of metric; standard but not proved.
standard math Topological conjugacy invariance of subset entropies
Used in Proposition 3.2 and Proposition 5.1 to transfer entropies from X^m to Phi_m(X^m); not stated in Proposition 2.1 nor proved.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Entropies of compact subsets and supported measures." pith.science (2026). https://pith.science/paper/QYE6IYWK

@misc{pith2026260809702,
  author       = {Pith},
  title        = {Pith review of: Entropies of compact subsets and supported measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QYE6IYWK}},
  note         = {Machine review of arXiv:2608.09702}
}

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{equation*} \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} \end{equation*} 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.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

R. L. Adler, A. G. Konheim and M. H. McAndrew.Topological entropy, Trans. Amer. Math. Soc.114(1965), 309–319

work page 1965
[2]

Bauer and K

W. Bauer and K. Sigmund,Topological dynamics of transformations induced on the space of probability measures, Monatsh. Math.79(1975), 81–92

work page 1975
[3]

Ben Ovadia and F

S. Ben Ovadia and F. Rodriguez-Hertz,Neutralized local entropy and dimension bounds for invariant measures, Int. Math. Res. Not. IMRN 2024, no. 11, 9469–9481

work page 2024
[4]

Blanchard,A disjointness theorem involving topological entropy, Bull

F. Blanchard,A disjointness theorem involving topological entropy, Bull. Soc. Math. France121(1993), no. 4, 465–478

work page 1993
[5]

Bowen,Topological entropy for noncompact sets, Trans

R. Bowen,Topological entropy for noncompact sets, Trans. Amer. Math. Soc.184 (1973), 125–136

work page 1973
[6]

Brin and A

M. Brin and A. Katok,On local entropy, Geometric dynamics (Rio de Janeiro, 1981), 30–38, Lecture Notes in Math.1007, Springer, Berlin, 1983

work page 1981
[7]

D. Dou, D. Zheng and X. Zhou,Packing topological entropy for amenable group actions, Ergodic Theory Dynam. Systems43(2023), no. 2, 480–514

work page 2023
[8]

Feng and W

D.-J. Feng and W. Huang,Variational principles for topological entropies of subsets, J. Funct. Anal.263(2012), no. 8, 2228–2254

work page 2012

Show all 18 references

[9]

Glasner and B

E. Glasner and B. Weiss,Quasi-factors of zero-entropy systems, J. Amer. Math. Soc. 8(1995), no. 3, 665–686

1995
[10]

Huang and X

W. Huang and X. Ye,A local variational relation and applications, Israel J. of Math. 151(2006), 237–280

2006
[11]

Kerr and H

D. Kerr and H. Li,Dynamical entropy in Banach spaces, Invent. math.162(2005), 649–686

2005
[12]

Liu and Y

K. Liu and Y. Qiao,Relative topological entropy and relative mean dimension of induced factors, arXiv:2511.18040, 2025. 18 QIANG HUO AND XIANGTONG W ANG

2025 arXiv
[13]

K. Liu, Y. Qiao and L. Xu,Topological entropy of nonautonomous dynamical systems, J. Differential Equations268(2020), 5353–5365

2020
[14]

Liu and Y

Z. Liu and Y. Zheng,Local and global relative entropy via preimage structure, J. Differential Equations475(2026), Paper No. 114479, 32 pp

2026
[15]

K. R. Parthasarathy,Probability measures on metric spaces, Academic Press, New York-London, 1967

1967
[16]

Qiao and X

Y. Qiao and X. Zhou,Zero sequence entropy and entropy dimension, Discrete Contin. Dyn. Syst.37(2017), 435–448

2017
[17]

Walters,An Introduction to Ergodic Theory, Graduate Texts in Mathematics, vol

P. Walters,An Introduction to Ergodic Theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982

1982
[18]

X. Wang, W. Wu and Y. Zhu,Local entropy via preimage structure, J. Differential Equations317(2022), 639–684. 1 School of Mathematical Sciences & State Key Laboratory of Cognitive Intelligence, University of Science and Technology of China, Hefei, Anhui, 230026, P. R. China Ema...

2022

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

R. L. Adler, A. G. Konheim and M. H. McAndrew.Topological entropy, Trans. Amer. Math. Soc.114(1965), 309–319

work page 1965

[2] [2]

Bauer and K

W. Bauer and K. Sigmund,Topological dynamics of transformations induced on the space of probability measures, Monatsh. Math.79(1975), 81–92

work page 1975

[3] [3]

Ben Ovadia and F

S. Ben Ovadia and F. Rodriguez-Hertz,Neutralized local entropy and dimension bounds for invariant measures, Int. Math. Res. Not. IMRN 2024, no. 11, 9469–9481

work page 2024

[4] [4]

Blanchard,A disjointness theorem involving topological entropy, Bull

F. Blanchard,A disjointness theorem involving topological entropy, Bull. Soc. Math. France121(1993), no. 4, 465–478

work page 1993

[5] [5]

Bowen,Topological entropy for noncompact sets, Trans

R. Bowen,Topological entropy for noncompact sets, Trans. Amer. Math. Soc.184 (1973), 125–136

work page 1973

[6] [6]

Brin and A

M. Brin and A. Katok,On local entropy, Geometric dynamics (Rio de Janeiro, 1981), 30–38, Lecture Notes in Math.1007, Springer, Berlin, 1983

work page 1981

[7] [7]

D. Dou, D. Zheng and X. Zhou,Packing topological entropy for amenable group actions, Ergodic Theory Dynam. Systems43(2023), no. 2, 480–514

work page 2023

[8] [8]

Feng and W

D.-J. Feng and W. Huang,Variational principles for topological entropies of subsets, J. Funct. Anal.263(2012), no. 8, 2228–2254

work page 2012