REVIEW 3 major objections 5 minor 28 references
A countable partition for singular flows, and its application on the entropy theory
T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read A countable partition computes metric entropy for singular flows away from homoclinic tangencies.
desk verdict Genuinely new local partition construction, but the proof of the main application contains a reversed inequality in Lemma 7.6 that as written breaks the tail-entropy estimate. 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 central object is the singularity-crossing section $D_\sigma = \{|v^s|=|v^u|\}$, layered into $D_n$ by $|v|\approx e^{-n}$. The coarse partition $C_\sigma = \{\varphi^{[0,1)}(D_n)\}$ has uniformly bounded entropy because the return-time estimate $t^\pm_x/n \in [K_0,K_1]$ makes $\sum_n n\,\mu(C_n)$ bounded. The refined partition $\mathscr A_\sigma$ is built by subdividing each $C_n$ using a maximal $r_n$-separated set with $r_n = \beta L^{-K_1 n} L_0 e^{-(n+1)}$, so the cardinality grows like $(L'')^n$ while the diameter shrinks like $(L')^{-n}$; the scaled tubular neighborhood theorem (sizes normalized by flow speed) then gives the shadowing property. For the entropy applications, the load-bearing dynamical input is a dominated splitting $E^1\oplus E^2\oplus E^3$ of the normal bundle for both the projected linear flow and its scaled version, with $\dim E^2 \le 1$, which yields one-dimensional fake foliations onto which $\mathscr A^\infty(x)$ projects; tail entropy vanishes because the total length of these curves over an orbit segment grows only linearly.
What would settle it
Take a $C^1$ flow away from homoclinic tangencies with hyperbolic singularities and an ergodic invariant measure whose support does not admit an $E^1\oplus E^2\oplus E^3$ dominated splitting of the normal bundle (for both the linear normal flow and its scaled version) with $\dim E^2 \le 1$; alternatively, find a positive-measure set of points $x$ for which the tail set $\mathscr A^\infty(x)$ has positive topological entropy under the partition of Theorem C. Either observation would disprove Theorems F and G.
Extended reading notes
Core claim
For a hyperbolic singularity $\sigma$, take the surface $D_\sigma = \exp_\sigma\{v \in T_\sigma M : |v^s| = |v^u|\}$ — the locus where the flow is turning and moving slowest. Cutting it into shells $D_n$ at distances $e^{-n}$, the flow boxes $C_n = \varphi^{[0,1)}(D_n)$ form a countable coarse partition $C_\sigma$ with $\sum_n n\,\mu(C_n)$ bounded by a universal constant, hence finite entropy for every invariant measure via a standard criterion for countable partitions. Refining each $C_n$ into about $O(L^n)$ small pieces yields $\mathscr A_\sigma$, whose atoms have diameter at most $c\,\beta\,(L')^{-n}$ and whose metric entropy is still uniformly finite; two points in the same atom stay in a $\beta$-scaled tubular neighborhood of each other until the orbit leaves $B_r(\sigma)$. Gluing these local partitions over all singularities with a finite regular partition gives a global countable partition $\mathscr A$ with $H_\mu(\mathscr A)<\infty$ for every invariant measure. The main applications are that for flows away from homoclinic tangencies $h_\mu(X) = h_\mu(\varphi_1, \mathscr A)$ for every invariant $\mu$, because the $\mathscr A^\infty$-classes are shadowed by one-dimensional curves whose total length grows only subexponentially, and that metric entropy is upper semi-continuous with defect at most $L_2\,\mu(\mathrm{Sing}(X))$.
Load-bearing premise
The proof that the partition computes entropy leans on an unproved lemma: for flows away from homoclinic tangencies, every ergodic invariant measure supports a decomposition of the normal directions into three invariant subbundles, the middle one of dimension at most one, with uniform exponential contraction on one side and expansion on the other for both the linear normal flow and its scaled version; if such a decomposition fails for some measure, the argument that the partition's tail has zero entropy collapses.
Editorial extensions
If this is right
- For star flows, the partition is almost generating: for every ergodic invariant measure, $\mathscr A^\infty(x)$ is contained in a finite orbit segment for almost every $x$, so $h_\mu(X)=h_\mu(\varphi_1,\mathscr A)$.
- For flows away from homoclinic tangencies with hyperbolic singularities, every invariant probability measure is $\mathscr A$-expansive; in particular the metric entropy of the flow is computed by the single countable partition $\mathscr A$.
- The metric-entropy function is upper semi-continuous in the $C^1$ flow topology and weak* measure topology on this class; when the limit measure gives zero weight to the singular set, the entropy cannot jump up in the limit.
- The entropy loss when replacing $\mathscr A$ by a finite partition is controlled by the measure of a small neighborhood of the singularities: $h_\mu(\varphi_1,\mathscr A) - h_\mu(\varphi_1,\mathscr A_{0,N}) \le L_2\,\mu(O_N(\sigma)) + o(1)$, so the only possible drop is concentrated near singularities.
- If $\mu(\mathrm{Sing}(X))>0$, entropy can be lost in the limit by at most $L_2\,\mu(\mathrm{Sing}(X))$; this is a new mechanism for non-upper-semicontinuity that does not exist for diffeomorphisms.
Reading between the lines
- The same construction of a singularity-crossing section and layered partition should apply to other problems where the return time to a cross-section is unbounded, such as thermodynamic formalism for Lorenz-like and contracting-Lorenz systems, replacing ad hoc eigenvalue assumptions with the universal layer structure.
- If the quoted splitting lemma is extended to any singular flow whose supports admit such a dominated splitting of the normal bundle for the scaled linear flow, the tail-entropy argument would give $\mathscr A$-expansiveness and entropy upper semi-continuity beyond the away-from-tangencies class; checking this on known singular-hyperbolic examples is a direct test.
- The loss term $L_2\,\mu(\mathrm{Sing}(X))$ suggests a possible sharp formula for the defect in upper semi-continuity: the entropy drop may be realized by measures concentrating on the stable and unstable manifolds of the singularity, and identifying the optimal constant $L_2$ would connect it to the local eigenvalues.
- Question 1 in the paper — whether $\mathscr A$-expansiveness for every invariant measure implies pointwise tail entropy zero or $\varepsilon$-entropy expansiveness — might be resolved by the one-dimensional shadowing method: if $\mathscr A^\infty$-classes always admit subexponential-length curve families, then tail entropy vanishes at every point, not merely almost every point.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper constructs countable measurable partitions for C1 vector fields with hyperbolic singularities and applies them to the entropy theory of singular flows. Near each singularity σ, the authors introduce a cross section Dσ containing σ, cut it into layers Dn with exponentially decreasing flow speed, and form a coarse partition Cσ whose metric entropy is uniformly bounded for every invariant probability measure (Theorem A). Refining each layer into O(L^n) pieces yields a partition Aσ with uniformly bounded entropy and controlled geometry in scaled tubular neighborhoods (Theorem B). These local partitions are combined with a finite partition away from the singularities into a global partition A (Theorem C). The paper then proves a general criterion (Theorem E) relating A-expansiveness to equality of hμ(φ1) and hμ(φ1,A). For flows away from homoclinic tangencies, the authors claim that every invariant measure is A-expansive (Theorem F), so that hμ(X)=hμ(φ1,A), and that the metric entropy is upper semi-continuous with respect to both measures and flows, with a loss controlled by μ(Sing(X)) (Theorem G). A corollary for star flows is also stated (Corollary D).
Significance. If the main results are correct, this is a significant advance in the ergodic theory of singular flows: the construction of the partition avoids linearization and dimensional restrictions, works for every hyperbolic singularity, and yields uniformly bounded metric entropy for every invariant measure. The paper also provides a new mechanism for entropy loss near singularities, proportional to the measure of the singularity set, which does not occur for non-singular flows. Credit is due for the original local construction in Sections 3-4, the clean use of Mañé's lemma to bound Hμ(Cσ), and the modular formulation of Theorem E. However, the downstream arguments for the main applications contain serious gaps, including an unproved dominated-splitting lemma with unresolved placeholders and a concrete reversed inequality in the length estimate that underpins the tail-entropy control. These issues are load-bearing for Theorem F and Theorem G, so the paper is not yet ready for publication in its current form.
major comments (3)
- [Section 7.2, Lemma 7.6] The proof of Lemma 7.6 contains a reversed inequality. Starting from length(I(x)) ≤ βL0/e (LK1e)^{-n} and length(g^j(I(x))) ≤ βL0/e (LK1e)^{-n} λ'^j, the authors define tilde-lambda = (LK1e)^{1/K0} and write length(g^j(I(x))) ≤ βL0/e tilde-lambda^{-K0 n} tilde-lambda^j ≤ βL0/e tilde-lambda^{-t_x^+} tilde-lambda^j. Since Lemma 3.2 gives t_x^+ ≥ K0 n and tilde-lambda > 1, we have tilde-lambda^{-K0 n} ≥ tilde-lambda^{-t_x^+}, so the second inequality has the wrong direction. The claimed bound length(g^j(I(x))) ≤ C tilde-lambda^{-(t_x^+ - j)} is therefore not established; the available bound is typically exponentially larger, for instance at j=0 it is of order (LK1e)^{-n} rather than (LK1e)^{-(K1/K0)n}. Replacing K0 by K1 in the definition of tilde-lambda and in the exponent would repair the step, but as written Proposition 7.2's control of ∑ length(I_j) over a visit to O(σ) is unsupported, and the proof of Theorem F collapses at this point.
- [Section 7.2, Lemma 7.3] Lemma 7.3 is stated with unresolved placeholders, including 'φY,?iL0(x)' and '≤ ?λ0', and its proof is not given; the text only says it follows from the proof of [15, Proposition 3.4]. This lemma provides the dominated splitting E1⊕E2⊕E3 for the scaled linear Poincaré flow over the support of every ergodic measure, with dim E2 ≤ 1 and uniform Lyapunov-type contraction/expansion, and it is the basis for reducing A∞(x) to a family of one-dimensional curves of controlled length in Proposition 7.2 and Theorem F. Without a precise statement and a complete proof, or a self-contained reference that exactly covers this setting, the central application to flows away from homoclinic tangencies is not rigorously established.
- [Sections 4 and 5, Theorems B and C] Theorem B(IV) and Theorem C(I) overclaim the scaled tubular neighborhood property. Proposition 4.2 establishes this property only for points in the same element of ~B_n, i.e., inside the refined pieces of ∪ C_n, not for the atoms B±(σ) and O(σ)^c of Aσ, which are coarse sets of positive size. The proof of Theorem C(I) invokes Proposition 4.2 for the entire visit to O(σ), but the text later explicitly concedes that the orbit segment in B±(σ) is not controlled ('we lose control ... for the orbit segment in B±(σ)', Section 5). Thus the stated theorems are not consequences of the proofs. The authors should either further refine B±(σ) into pieces respecting the scaled tubular neighborhoods or weaken the statements accordingly; the applications in Section 7 may only need the local estimates on the refined pieces, but as written the theorem statements are incorrect.
minor comments (5)
- [Section 7.2, proof of Lemma 7.6] The assertion that 'λ' ≤ tilde-lambda' follows from the choice of L≥N0 is not proved; it should be justified explicitly, since it uses the relation between K0 and λ' from Lemma 3.2.
- [Section 8, Theorem 8.1] The function u_{X,μ}(N) is said to converge to zero uniformly in μ and in a C1 neighborhood of X, but the proof does not spell out the uniformity in μ; a precise statement of the uniformity would help the reader verify the limit in Theorem G.
- [Appendix A, Proposition 7.5] The proof of Proposition 7.5 is very sketchy, especially the passage where 'A induces a natural order on each I_j' and the construction of the (n,ε)-spanning set from the covering of the curves; more details are needed to make the subexponential spanning argument fully rigorous.
- [Section 3.3, proof of Lemma 3.6] The displayed equality 'N·μ(∪ Cn) = 1/K0 μ(...)' is correct but confusing; it may be clearer to write K0N·μ(∪ Cn) = μ(...) before dividing, to show the role of the disjoint union of the images φ^k(C_n).
- [Throughout] The references to [19, Lemma 5.1] and [19, Lemma 5.2] are used for key cone estimates, and the paper would be more self-contained if these statements were recalled in detail or proved in an appendix.
Circularity Check
Entropy application leans on a co-authored splitting lemma; partition construction itself is independent.
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self citation load bearing
[Lemma 7.3 and Section 7.2, proof of Proposition 7.2]
"Furthermore, following the proof of [15, Proposition 3.4], which uses the result of [26] for diffeomorphisms away from homoclinic tangencies, we have the following lemma. Lemma 7.3. ... Nsuppμ =E1⊕E2⊕E3, with dim(E2)≤ 1, and ... lim_{n→∞} 1/n Σ log‖ψ∗ Y,J0|E1 φY,?iL0(x)‖≤ ?λ0"
Proposition 7.2's conclusion htail(φ1,x,A)=0, which is exactly the A-expansiveness needed for Theorem F, is derived by first importing Lemma 7.3. The manuscript does not prove Lemma 7.3; it says only 'following the proof of [15, Proposition 3.4]', and the displayed statement still contains unresolved '?' placeholders. Reference [15] is the authors' own co-authored work (J. Yang), so the load-bearing dominated splitting and the dim(E2)≤1 bound are carried into this paper through a self-citation rather than by a self-contained derivation. The one-dimensional fake-foliation control of A∞(x), and hence the zero tail entropy conclusion, rests entirely on that imported premise; without it the proof does not reduce A∞(x) to subexponentially many curves.
full rationale
Most of the paper is not circular by construction. Theorems A and B define Dσ, Cσ, and Aσ using only the linearized flow near a hyperbolic singularity, the scaled tubular neighborhood theorem, and Mañé's finite-entropy lemma; the finite entropy bounds are proved directly. Theorem E is a standard tail-entropy criterion, and Theorem 8.1 estimates the entropy loss when gluing tail elements in a self-contained way. The main load-bearing external input is Lemma 7.3, whose proof is delegated to [15, Proposition 3.4] from co-authored prior work and whose displayed statement still contains placeholders. That is a self-citation supporting the central Theorem F; however, [15] is an independent published theorem and the partition construction does not define its conclusions into its hypotheses. I also note the reviewer-identified inequality reversal in Lemma 7.6 (tilde-lambda^{-K0 n} is not ≤ tilde-lambda^{-t_x^+} because t_x^+ ≥ K0 n); this is a correctness gap in the tail-length estimate, not a circularity, so I do not score it under circularity. Overall score 4: some load-bearing self-citation in the application section, with independent content in the core partition construction.
Assumptions & free parameters
free parameters (2)
- n0 = n0(r)
- L and β
assumptions (5)
- domain assumption The vector field is C1 and every singularity is hyperbolic; near each singularity the flow is a C1 small perturbation of the linear flow e^{At}.
- standard math Scaled linear Poincaré flow estimates (Propositions 2.2, 2.3, 2.5) from [10] hold uniformly in a C1 neighborhood.
- ad hoc to paper Cone lemmas [19, Lemma 5.1] and [19, Lemma 5.2] give a uniform time T_{α0} to enter the unstable cone and relate manifold and tangent cones.
- domain assumption For flows away from tangencies, Lemma 7.3 gives a dominated splitting with dim E2 ≤ 1 and Lyapunov bounds for ψt and ψt*.
- standard math Fake foliations with local product structure (Lemma 7.1) exist for dominated splittings.
Cite this review
Pith. "Pith review of A countable partition for singular flows, and its application on the entropy theory." pith.science (2026). https://pith.science/paper/A7GO64FY
@misc{pith2026190801380,
author = {Pith},
title = {Pith review of: A countable partition for singular flows, and its application on the entropy theory},
year = {2026},
howpublished = {\url{https://pith.science/paper/A7GO64FY}},
note = {Machine review of arXiv:1908.01380}
}
abstract
In this paper, we construct a countable partition $\mathscr{A}$ for flows with hyperbolic singularities by introducing a new cross section at each singularity. Such partition forms a Kakutani tower in a neighborhood of the singularity, and turns out to have finite metric entropy for every invariant probability measure. Moreover, each element of $\mathscr{A}^\infty$ will stay in a scaled tubular neighborhood for arbitrarily long time. This new construction enables us to study entropy theory for singular flows away from homoclinic tangencies, and show that the entropy function is upper semi-continuous with respect to both invariant measures and the flows.
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Works this paper leans on
-
[19]
Entropy theory for sectional hyperbolic flows
M.J. Pacifico, F. Yang and J. Yang. Entropy theory for sectional hyperbolic flows. Preprint. Available at arXiv:1901.07436
work page Pith review arXiv 1901
-
[1]
V. Araujo, M.J. Pacifico, E. Pujals and M. Viana. Singular-hyperbolic attractors are chaotic. Trans. Amer. Math. Soc. 361 (2009), no. 5, 2431-C2485
work page 2009
-
[2]
V. Ara´ ujo and M. J. Pacifico;Three-dimensional flows, volume 53 of Ergebnisse der Mathe- matik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] . Springer, Heidelberg, 2010. With a foreword by Marcelo Viana
work page 2010
-
[3]
Star fows and multisingular hyperbolicity
C. Bonatti and A. da Luz. Star flows and multisingular hyperbolicity. Preprint. Available at arXiv:1705.05799
-
[4]
R. Bowen. Entropy expansive maps. Trans. Amer. Math. Soc., 164:323–331, 1972
work page 1972
-
[5]
R. Bowen. Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Second revised edition, Lecture Notes in Mathematics, vol. 470, Springer-Verlag, Berlin, 2008. With a preface by David Ruelle; Edited by Jean-Ren´ e Chazottes. ENTROPY THEORY FOR SINGULAR FLOWS 35
work page 2008
-
[6]
K. Burns and A. Wilkinson. On the ergodicity of partially hyperbolic systems. Annals of Math., 171: 451–489, 2010
work page 2010
-
[7]
Y. Cao, G. Liao and Z. You. Upper bounds on measure theoretic tail entropy for dominated splittings. Preprint. Available at arXiv:1901.01694
work page Pith review arXiv 1901
Show all 28 references
-
[8]
Galatolo and M.J
S. Galatolo and M.J. Pacifico. Lorenz-like flows: exponential decay of correlations for the Poincar´ e map, logarithm law, quantitative recurrence.Ergodic Theory Dynam. Systems 30 (2010), no. 6, 1703–1737
2010
-
[9]
S. Gan, Y. Shi and L. Wen. On the singular hyperbolicity of star flows. J. Mod. Dyn. 8:191– 219, 2014
2014
-
[10]
Gan and D
S. Gan and D. Yang. Morse-Smale systems and horseshoes for three dimensional singular flows. Ann. Sci. ´Ec. Norm. Sup´ er., 51:39–112, 2018
2018
-
[11]
S. Gan, J. Yang and R. Zheng. Lyapunov stable chain recurrent class for singular flows. Preprint
-
[12]
M. Li, S. Gan and L. Wen. Robustly transitive singular sets via approach of extended linear Poincar´ e flow.Discrete Contin. Dyn. Syst. , 13:239–269, 2005
2005
-
[13]
S. T. Liao. On ( η,d )-contractible orbits of vector fields. Systems Science and Mathematica Sciences., 2: 193–227, 1989
1989
-
[14]
S. T. Liao. The qualitative theory of differential dynamical systems. Science Press, 1996
1996
-
[15]
Liao and M
G. Liao and M. Viana and J. Yang. The entropy conjecture for diffeomorphisms away from tangencies. J. Eur. Math. Soc. (JEMS) 15: 6, 2043–2060, 2013
2013
-
[16]
Lima and O
Y. Lima and O. Sarig. Symbolic dynamics for three-dimensional flows with positive topological entropy. J. Eur. Math. Soc. (JEMS) 21 (2019), no. 1, 199–256
2019
-
[17]
Ma˜ n´ e
R. Ma˜ n´ e. A proof of Pesin’s formula,Ergod. Th. & Dynam. Sys. , (1981), 1, 95–102
1981
-
[18]
Pacifico and M
M.J. Pacifico and M. Todd. Thermodynamic formalism for contracting Lorenz flows. J. Stat. Phys. 139 (2010), no. 1, 159–176
2010
-
[20]
M. Ratner. Markov partitions for Anosov flows on n-dimensional manifolds. Israel J. Math. 15 (1973), 92–114
1973
-
[21]
A. Rovella. The dynamics of perturbations of the contracting Lorenz attractor. Bol. Soc. Brasil. Mat. (N.S.) 24 (1993), no. 2, 233–259
1993
-
[22]
Y. Sinai. Construction of Markov partitionings. Funkcional. Anal. i Prilˇ ozen. 2 (1968), no. 3, 70-C80 (Loose errata) (Russian)
1968
-
[23]
W. Sun, T. Young and Y. Zhou. Topological entropies of equivalent smooth flows. Trans. Amer. Math. Soc. 361 (2009), no. 6, 3071–3082
2009
-
[24]
Walters: An Introduction to Ergodic Theory; Springer 1982
P. Walters: An Introduction to Ergodic Theory; Springer 1982
1982
-
[25]
L. Wen. Homoclinic tangencies and dominated splittings. Nonlinearity, 15 (2002), 1445–1469
2002
-
[26]
L. Wen. Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Brazil. Math. Soc. 35, 419–452, 2004
2004
-
[27]
Wen and L
X. Wen and L. Wen. A rescaled expansiveness for flows. Trans. Amer. Math. Soc. 371 (2019), no. 5, 3179–3207
2019
-
[28]
J. Yang. Entropy along expanding foliations. Preprint. Available at arXiv:1601.05504. School of Mathematical Sciences, Peking Univesity, Beijing, China. E-mail address: shiyi@math.pku.edu.cn Department of Mathematics, University of Oklahoma, Norman, Oklahoma, USA. E-mail addre...
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