REVIEW 3 major objections 3 minor 21 references
Measures of maximal entropy for suspension flows
T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read An arbitrarily small uniform perturbation of the roof of a suspension flow over a sub-shift of finite type can produce either uncountably many ergodic measures of maximal entropy or exactly one.
desk verdict Smart, short density result for MME of suspension flows, but Theorem 1.1 as stated is false for zero-entropy transitive SFTs — an easy positive-entropy hypothesis fixes it. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The carrying object is the dictionary between a suspension flow and its base: the Ambrose–Kakutani bijection $R(\mu)=(\mu\times \mathrm{Leb})|_Y/(\mu\times \mathrm{Leb})(Y)$ sends base invariant measures to flow invariant measures; Abramov's formula gives flow entropy $h(R(\mu))=h(\mu)/\int\tau\,d\mu$; and the flow's topological entropy is the unique root of $P(-t\tau)=0$, with flow measures of maximal entropy exactly $R(\mu)$ for $\mu$ an equilibrium state of $-h(\Phi)\tau$. The perturbation step takes a potential $\psi$ with $P(-\psi)=0$ and approximates it uniformly by potentials $\phi_n$ with $P(-\phi_n)=0$ whose equilibrium sets are, respectively, uncountable or a singleton; the roof is then changed to $\tau_n=\tau+(\phi_n-h\tau)/h$, preserving the original topological entropy while transferring the equilibrium cardinality to the flow.
What would settle it
Take the period-two shift on two symbols, with the transition matrix that has zeros on the diagonal and ones off the diagonal. It is topologically transitive and has alphabet size two, but it consists of exactly two points, has zero topological entropy, and has a unique invariant measure; the corresponding suspension flow therefore has a unique ergodic invariant measure for every continuous roof. This directly contradicts Theorem 1.1(a) as written, showing the theorem requires an explicit positive-entropy hypothesis.
Extended reading notes
Core claim
The paper's central claim is that, under its stated standing assumptions, any suspension flow over a one- or two-sided sub-shift of finite type with a positive continuous roof can have its roof perturbed by an arbitrarily small amount in the uniform norm so that the resulting flow has uncountably many ergodic measures of maximal entropy, and can also be perturbed so that it has a unique measure of maximal entropy. The perturbed flows are obtained by time reparametrization and have the same topological entropy as the original flow. The many-measures half is shown to extend to any base transformation whose thermodynamic formalism has finite entropy, entropy-dense ergodic measures, and an upper semi-continuous entropy map, while the uniqueness half extends to any expansive base map with the specification property.
Load-bearing premise
The proof needs the base sub-shift to have positive topological entropy, but the paper's standing assumption only says topologically transitive with at least two symbols; the period-two shift satisfies that assumption and has zero entropy, so Theorem 1.1(a) as stated is false for it.
Editorial extensions
If this is right
- Both sets—flows with uncountably many ergodic measures of maximal entropy and flows with a unique such measure—are dense in the space of suspension flows over a fixed sub-shift of finite type.
- Each perturbation can be made arbitrarily small in the uniform norm and preserves the topological entropy of the original flow, so the dichotomy is a property of nearby time changes rather than a special construction.
- The dichotomy transfers from one-sided semi-flows to two-sided sub-shifts and genuine flows.
- The uniqueness half extends to any expansive base map with specification, and the many-measures half to any compact base with finite entropy, entropy-dense ergodic measures, and an upper semi-continuous entropy map.
Reading between the lines
- Beyond the paper's claims: the stated standing assumptions are missing a positive-entropy hypothesis; the two-point period-two shift satisfies the stated assumptions but has zero entropy and a unique invariant measure, so Theorem 1.1(a) fails as written.
- Beyond the paper's claims: the same dictionary suggests a general principle—any dense class of base potentials with prescribed equilibrium-state cardinality should yield an analogous density statement for measures of maximal entropy of suspension flows over that base.
- Beyond the paper's claims: one testable extension is whether the same $C^0$-density dichotomy holds for suspension flows over countable Markov shifts or for roof regularities intermediate between continuous and Hölder.
- Beyond the paper's claims: the uncountably many measures in the dense examples are produced by a purely existential theorem; identifying them explicitly, or showing they can form a Cantor set, is left open.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies suspension flows over sub-shifts of finite type with continuous roof functions. The main theorem (Theorem 1.1) asserts that for any topologically transitive one- or two-sided SFT with at least two symbols and any positive continuous roof function, an arbitrarily small C0-perturbation of the roof produces a suspension flow with uncountably many ergodic measures of maximal entropy, and another small perturbation produces a flow with a unique measure of maximal entropy. The proofs rely on Israel's theorem that continuous functions with uncountably many equilibrium states are dense, Ruelle's theorem on uniqueness of equilibrium states for Hölder potentials, and the Abramov–Kac formulas for suspension flows. The paper also proves an analogous density statement for equilibrium measures of continuous potentials on the suspension flow.
Significance. If the result holds as stated, it gives a strong dichotomy between the smooth (Hölder) and merely continuous thermodynamic formalism for suspension flows: both unique and uncountably many measures of maximal entropy are dense in the space of roof functions. The proof is transparent and uses external theorems without introducing fitted parameters, and the flow-level extension in Theorem 3.13 is a useful complement. However, the main theorem as stated is false without a positive-entropy hypothesis on the base, so the central claim needs correction before the result can be accepted.
major comments (3)
- [§2.1 and Proposition 3.4] The standing assumption is inconsistent and does not imply the positive-entropy claim used later. The definition of 'topologically transitive' as existence of m with all entries of A^m positive is actually a mixing condition, and the stated equivalence with existence of a dense orbit is false: the two-symbol shift with transition matrix [[0,1],[1,0]] has a dense orbit in the usual transitive sense but no power of A is positive. Under the dense-orbit reading, this base satisfies the standing assumption, has zero topological entropy, and has exactly one invariant measure; by the Ambrose–Kakutani correspondence every suspension flow over it has a unique invariant measure, so Theorem 1.1(a) is false for that base. Proposition 3.4 explicitly asserts 'by our standing assumption (Σ,σ) has positive topological entropy', but that assertion is not a consequence of the stated assumptions under the standard meaning of topological transitivity. The theorem requires an explicit hypothesis that h(σ)>0 (equivalently h(Φ_τ)>0), and the definition of transitivity should be corrected or the erroneous equivalence removed.
- [Lemma 3.6] The construction in Lemma 3.6 divides by h(Φ_τ) when defining τ_n(x) = (τ(x) + φ_n(x) - h(Φ_τ)τ(x)) / h(Φ_τ). If the base shift has zero entropy, Abramov's formula gives h(Φ_τ)=0 for every invariant measure, so this expression is ill-defined. This is the precise technical step that requires a positive-entropy hypothesis on the base; adding h(σ)>0 to the hypotheses of Lemma 3.6 and Theorem 1.1 restores the argument, since then h(Φ_τ)>0 as well.
- [Remark 3.9] The claimed sufficient conditions for Theorem 1.1a are not sufficient. The period-two shift with transition matrix [[0,1],[1,0]] has finite topological entropy (zero), its ergodic measures are entropy dense in the space of invariant measures (trivially, since the space is a singleton), and the entropy map is upper semi-continuous. Nevertheless, the conclusion of Theorem 1.1a fails for this base because the set A(F,ε) defined in the remark contains only the unique ergodic measure and cannot support the non-atomic measure ν_ε used in Israel's construction. The remark therefore needs an additional hypothesis, such as positive topological entropy or a condition ensuring that A(F,ε) is non-atomic.
minor comments (3)
- [Proof of Proposition 3.4] The sentence 'we can conclude that the the suspension flow' contains a duplicated article; it should read 'the suspension flow'.
- [Proof of Theorem 1.1, after Lemma 3.6] The notation 'pYτ 1m, Φ τmq' for the first sequence should be 'pYτm, Φ τmq' to be consistent with the definition of τ_n in Lemma 3.6.
- [References] Several reference entries contain typographical errors, for example 'Addis on-W esley' and 'F acultad'; these should be corrected.
Circularity Check
No circularity: the proof chain rests on external theorems of Israel and Ruelle plus standard pressure and Abramov–Kac formulas; the zero-entropy hypothesis gap is a correctness issue, not circularity.
full rationale
The derivation is not circular. Theorem 1.1 is obtained by applying Israel's theorem (Theorem 3.3) and Ruelle's theorem (Theorem 3.1) to perturb potentials, then translating base equilibrium states into flow measures of maximal entropy through the standard Abramov–Kac dictionary and the pressure identity PΦ(g)=0 iff P(Δg−tτ)=0 (Remark 2.2). The normalization P(−φ)=0 in Lemma 3.5 is a pressure-scaling device, not a fit of the target conclusion: the uncountably-many or unique equilibrium states come from the externally cited theorems, not from the normalization. The construction τ=P(τ0)−τ0 in Proposition 3.4 makes P(−τ)=0 by definition, but the uncountably many equilibrium measures are supplied by Israel's theorem, an independent external result. No fitted parameter is renamed as a prediction, and no definition presupposes the flow's measure-of-maximal-entropy structure. The only self-citation, [CI], is cited merely for a related discussion of time-change size and is not load-bearing. There is a genuine mathematical flaw that should be flagged as a correctness risk rather than circularity: the standing assumption of topological transitivity with alphabet size at least two does not imply positive topological entropy, yet Proposition 3.4 says 'by our standing assumption (Σ,σ) has positive topological entropy' and Lemma 3.6 divides by h(Φτ); the period-two shift is a counterexample to Theorem 1.1(a) as stated. That is a missing hypothesis, not a circular reduction. Consequently, the circularity score is 0.
Assumptions & free parameters
assumptions (4)
- domain assumption The base shift has positive topological entropy.
- standard math Israel's theorem: continuous potentials with uncountably many ergodic equilibrium states are dense in C(Σ) for the relevant SFT.
- standard math Ruelle's theorem: Hölder continuous potentials on an SFT have a unique equilibrium state, and Hölder potentials are uniformly dense in C(Σ).
- standard math Suspension-flow pressure dictionary: the flow entropy h(Φ) is the unique root of P(-tτ)=0, and flow MME are R(μ) for μ an equilibrium state of -h(Φ)τ.
Cite this review
Pith. "Pith review of Measures of maximal entropy for suspension flows." pith.science (2026). https://pith.science/paper/VWVR2LEL
@misc{pith2026190807020,
author = {Pith},
title = {Pith review of: Measures of maximal entropy for suspension flows},
year = {2026},
howpublished = {\url{https://pith.science/paper/VWVR2LEL}},
note = {Machine review of arXiv:1908.07020}
}
read the original abstract
We study suspension flows defined over sub-shifts of finite type with continuous roof functions. We prove the existence of suspension flows with uncountably many ergodic measures of maximal entropy. More generally, we prove that any suspension flow defined over a sub-shift of finite type can be perturbed (by an arbitrarily small perturbation) so that the resulting flow has uncountably many ergodic measures of maximal entropy, and that the same can be arranged so that the new flow has a unique measure of maximal entropy.
Reference graph
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Reviewed August 14, 2026 · model on record in the stance chip above.
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