{"id":"bc528687-ec0a-4bf6-b569-4ce7e4ad06b5","arxiv_id":"2608.09702","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"For a compact subset K of a topological dynamical system, upper capacity entropy of K is zero exactly when the induced measure system on M(K) has zero entropy, otherwise the latter is infinite; packing entropies vanish together; but Bowen entropy of M(K) can be infinite while Bowen entropy of K is…","lead":"This paper proves precise relationships between the entropies of a compact set K and the entropies of the space of probability measures supported on K, under a dynamical map. For upper capacity and packing entropy the relationship is a clean dichotomy or equivalence, and for Bowen entropy the paper finds a surprising counterexample where the measure space has infinite entropy but the original set has zero.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof of Theorem 1.2 uses M(A) for Borel non-closed A although M(K) was defined via support; the restricted measure need not lie in M(A).","rationale":"The paper's central claims appear mathematically sound, and the Feng-Huang variational principles are likely applicable to compact K and to M(K) even when K is not invariant, so the reader's flagged weakest assumption is not the principal issue. The most concrete load-bearing gap is in the proof of Theorem 1.2: Lemma 4.1 produces an a priori non-closed Borel set A, while the paper's definition of M(A) uses support containment. The restriction map R_A(\\nu) can have support in the closure of A but not in A, so the constructed measure \\lambda may fail the hypothesis of Lemma 4.2. This is not a counterexample to the theorem; it is a fixable proof gap. The fix is to pass to a compact subset A' of A by inner regularity, which preserves the needed entropy bound and measure estimate. Given this repair, the reader's CONDITIONAL verdict remains appropriate: the manuscript should correct this step together with the noted typos and notational ambiguities before publication. The concern is therefore significant enough to report but does not overturn the central claims, so the verdict is unchanged.","tokens_in":13966,"tokens_out":38129,"duration_ms":326322,"concrete_test":"Under the §2 definition M(A)=\\{\\mu:\\mathrm{supp}\\,\\mu\\subset A\\}, test the claimed inclusion R_A(\\nu)\\in M(A) for A=(0,1)\\subset[0,1] and \\nu=Lebesgue measure; it fails because \\mathrm{supp}(\\nu|_A)=[0,1]\\not\\subset A. Then verify the proposed repair: use inner regularity to choose a compact A'\\subset A with \\bar\\mu(A\\setminus A')<\\theta\\tau(Q)/2; check that A'\\subset A gives h^UC_top(T,A')\\le h^UC_top(T,A)<c, that R_{A'}(\\nu)\\in M(A') for every \\nu with \\nu(A')>0, and that rerunning Theorem 4.3 with A' in place of A preserves \\lambda(M(A'))=1 and the final contradiction with h^UC_top(T,A')<c.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In §4, Theorem 4.3 invokes Lemma 4.1 to obtain a Borel set A (a G_delta set) with h^UC_top(T,A)<c and \\bar\\mu(A^c)<\\theta. It then defines R_A(\\nu)=\\nu|_A/\\nu(A) for \\nu with \\nu(A)>1/2 and claims R_A(\\nu)\\in M(A), so that \\lambda=(R_A)_*\\tau_H satisfies \\lambda(M(A))=1. However, §2 defines M(K)=\\{\\mu:\\mathrm{supp}\\,\\mu\\subset K\\}. For a non-closed Borel A this is false: for X=[0,1], A=(0,1), \\nu=Lebesgue, one has \\mathrm{supp}(\\nu|_A)=[0,1]\\not\\subset A. Thus \\lambda need not be a measure on M(M(A)), and Lemma 4.2, whose hypothesis is \\tau(M(A))=1, cannot be applied as written. This is a genuine gap in the proof of the packing-entropy equivalence (Theorem 1.2). The gap is repairable: by inner regularity choose a compact A'\\subset A with \\bar\\mu(A\\setminus A') small and h^UC_top(T,A')\\le h^UC_top(T,A)<c; then R_{A'}(\\nu) is genuinely supported in A' and the argument goes through unchanged. Because the repair is routine, the central claim is not invalidated, but the written proof is incomplete.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14260,"tokens_out":15139,"duration_ms":131354,"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":[{"comment":"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.","section":"§4, proof of Theorem 4.3 and Lemma 4.2"}],"minor_comments":[{"comment":"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.","section":"§5, Theorem 5.3"},{"comment":"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.","section":"§5, Proposition 5.1 and Theorem 1.3"},{"comment":"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.","section":"§4, Lemma 4.2"},{"comment":"The reference 'By Theorem 4.2' near the end of the proof of Theorem 4.3 should read 'By Lemma 4.2'.","section":"§4, Theorem 4.3"},{"comment":"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.","section":"§2.5 and §4, Lemma 4.1"}],"recommendation":"major_revision","confidential_remarks":"The gap in §4 is real but appears routine to fix by replacing the Borel set A with a compact subset. Once that repair is written, I expect the main results to stand. The paper is within scope and I have no concerns about citation or novelty."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nMinor 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.\n\nThe citation pattern is honest; the authors lean on Feng-Huang and Glasner-Weiss, which are the right tools, and they acknowledge discussions. No circularity.\n\nBottom 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.","headline":"Solid, genuinely new subset-level entropy dichotomy with a striking Bowen counterexample; one repairable gap in the packing-entropy proof.","tokens_in":14790,"tokens_out":3649,"would_cite":true,"duration_ms":30496,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37B40","37A35","28A78","60B05","54H20"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["upper capacity entropy","Bowen entropy","packing entropy","local entropy","induced system on measures","supported measures","variational principle","compact subsets"],"falsifier":"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.","tokens_in":13758,"feed_emoji":"📈","tokens_out":13769,"duration_ms":115238,"temperature":0.7,"pith_summary":"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.","feed_headline":"Supported measures amplify a set's positive entropy","feed_subtitle":"Upper capacity and Bowen entropies explode to infinity; packing entropy stays in step with the set.","key_machinery":"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 $r$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the variational principles (Theorem 2.2) identifying Bowen and packing entropies of a compact set with measure-theoretic local entropies; the core tool for Theorems 1.2 and 1.3.","marker":"[8]"},{"why":"Supplies the combinatorial separation lemma (Lemma 3.3 here) used to transfer positive upper capacity entropy from M(K) back to K in Theorem 3.4.","marker":"[9]"},{"why":"Establishes the classical zero-or-infinite dichotomy for induced systems that this paper localizes to arbitrary compact subsets.","marker":"[2]"},{"why":"Gives the original definition of Bowen topological entropy for noncompact sets, the notion generalized and used throughout.","marker":"[5]"}],"fun_headline_variants":["Packing entropy preserved; upper and Bowen entropies go infinite","Measure support sends positive Bowen entropy to infinity","Supported measures: upper and Bowen blow up, packing stays","Set's positive entropy makes measure space's upper capacity infinite"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Packing entropy preserved; upper and Bowen entropies go infinite","Measure support sends positive Bowen entropy to infinity","Supported measures: upper and Bowen blow up, packing stays","Set's positive entropy makes measure space's upper capacity infinite"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000363,"raw_usage":{"total_tokens":2079,"prompt_tokens":1187,"completion_tokens":892,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":803,"completion_tokens_details":{"reasoning_tokens":826}},"tokens_in":803,"tokens_out":892,"duration_ms":8880,"temperature":1.0,"reasoning_tokens":826,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:26:59.777582+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Feng and W","cited_arxiv_id":null,"evidence_quote":"Supplies the variational principles (Theorem 2.2) identifying Bowen and packing entropies of a compact set with measure-theoretic local entropies; the core tool for Theorems 1.2 and 1.3."},{"cited_title":"Glasner and B","cited_arxiv_id":null,"evidence_quote":"Supplies the combinatorial separation lemma (Lemma 3.3 here) used to transfer positive upper capacity entropy from M(K) back to K in Theorem 3.4."},{"cited_title":"Bauer and K","cited_arxiv_id":null,"evidence_quote":"Establishes the classical zero-or-infinite dichotomy for induced systems that this paper localizes to arbitrary compact subsets."},{"cited_title":"Bowen,Topological entropy for noncompact sets, Trans","cited_arxiv_id":null,"evidence_quote":"Gives the original definition of Bowen topological entropy for noncompact sets, the notion generalized and used throughout."}],"review_version":1}