{"id":"68e0e78c-fc85-4d15-82b0-b3c310697d51","arxiv_id":"1908.01380","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Hyperbolic singularities can be surrounded by a finite-entropy countable partition whose atoms control scaled tubular neighborhoods, yielding entropy formulas and upper semi-continuity away from tangencies.","lead":"These authors construct a countable partition near hyperbolic singularities of a flow, with the structure of a Kakutani tower and finite entropy for every invariant probability measure. The construction yields upper semi-continuity results for metric entropy of singular flows away from homoclinic tangencies.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 7.6's key length estimate has a reversed inequality: replacing tilde-lambda^{-K0 n} by tilde-lambda^{-t_x^+} is invalid because t_x^+ >= K0 n, so Proposition 7.2's tail-entropy bound is unsupported.","rationale":"The strongest claims in the paper are Theorem F and its corollary Theorem G. Their proof route is: build the partition A, prove every invariant measure is A-expansive by showing h_tail(phi_1,x,A)=0 for mu-a.e. x, then apply Theorem E. The zero-tail-entropy check is Proposition 7.2. That proposition's only quantitative input about lengths near singularities is Lemma 7.6. The step in the proof of Lemma 7.6 that converts the initial length bound into a decay in t_x^+ is invalid: replacing K0 n by the larger t_x^+ in the exponent of a number less than 1 decreases the right-hand side. This is not a cosmetic placeholder; it is the step that allows the subsequent 'sum over each visit is bounded by a constant' assertion. With the inequality reversed, the only available bound is exponentially weaker at j=0. The reader's identified issue, the unproved Lemma 7.3 with placeholders, is also real and should be completed; both issues sit in the same section. But the Lemma 7.6 problem can be checked without external references and is therefore the more decisive test of the central claim. If the intended fix is to replace K0 with K1, the proof is likely repairable, so the manuscript still warrants a conditional verdict rather than rejection. Hence I recommend leaving the reader's verdict unchanged.","tokens_in":27406,"tokens_out":12392,"duration_ms":120126,"concrete_test":"Re-derive Lemma 7.6 at the endpoint j=0 with t_x^+ = K1 n, using the definitions K0=1/(2 log lambda'), K1=2/log lambda and tilde-lambda=(L K1 e)^(1/K0). Verify whether the bound length(I(x)) <= beta L0/e (L K1 e)^{-n} implies length(I(x)) <= C (L K1 e)^{-K1 n/K0}. Since K1/K0 > 1, it does not. Then rerun the proof of Proposition 7.2 with tilde-lambda=(L K1 e)^(1/K1); check that lambda' <= tilde-lambda and that each visit to O(sigma) contributes a constant, so the total length is O(n). If the corrected constant works, the gap is typographical; if not, Theorem F lacks a valid tail-entropy estimate.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's concern about Lemma 7.3 is legitimate, but the more immediately checkable gap is in Lemma 7.6, which is what makes Proposition 7.2's subexponential length estimate work. In the proof, tilde-lambda is defined as (L K1 e)^(1/K0), and Lemma 3.2 gives t_x^+ in [K0 n, K1 n]. The proof writes length(g^j(I(x))) <= beta L0/e * tilde-lambda^{-K0 n} tilde-lambda^j <= beta L0/e * tilde-lambda^{-t_x^+} tilde-lambda^j. Because tilde-lambda > 1 and t_x^+ >= K0 n, we have tilde-lambda^{-K0 n} >= tilde-lambda^{-t_x^+}; the second inequality has the wrong direction. For example, at j=0 and t_x^+ = K1 n, the claimed RHS is (L K1 e)^{-K1 n/K0}, exponentially smaller than the only bound available for the LHS, (L K1 e)^{-n}. Thus the claimed uniform decay C tilde-lambda^{-(t_x^+ - j)} is not established. Proposition 7.2 then sums the length of each visit to O(sigma) as a bounded constant; with the corrected direction, the sum is only controlled by t_x^+ of order n, and the total length bound O(n(r1 + C)) in the proof is no longer justified. Replacing K0 by K1 in the definition of tilde-lambda would repair the step, but as written Theorem F's tail-entropy collapse relies on an unproved estimate.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":27872,"tokens_out":12243,"duration_ms":119237,"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":[{"comment":"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":"Section 7.2, Lemma 7.6"},{"comment":"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.","section":"Section 7.2, Lemma 7.3"},{"comment":"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.","section":"Sections 4 and 5, Theorems B and C"}],"minor_comments":[{"comment":"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":"Section 7.2, proof of Lemma 7.6"},{"comment":"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.","section":"Section 8, Theorem 8.1"},{"comment":"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":"Appendix A, Proposition 7.5"},{"comment":"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).","section":"Section 3.3, proof of Lemma 3.6"},{"comment":"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.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a promising construction, but the path from the local partition to the main entropy applications is not yet rigorous. The unresolved placeholders in Lemma 7.3 and the reversed inequality in Lemma 7.6 are concrete obstacles, and the overclaim in Theorems B(IV)/C(I) suggests the manuscript was not fully checked against its own proofs. The authors should be asked to provide complete proofs for these points. Given the heavy reliance on the authors' own preprint [19], the editor may also want to confirm the status of that preprint before further review."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core construction is genuinely new: a cross section Dσ that contains the singularity, a layer decomposition with exponentially controlled flight times, and a countable partition Aσ with uniformly finite entropy for every invariant measure. The authors remove the Lorenz-like and linearization hypotheses from earlier work and get a statement in arbitrary dimension. That is a real advance, and the entropy bounds in Theorems A and B via Mañé's lemma are clean and mostly check out. The applications to flows away from tangencies are ambitious and the overall strategy — control the tail entropy of A∞ by a family of one-dimensional curves — is sensible.\n\nThat said, there is a load-bearing gap in Section 7. In Lemma 7.6, the proof defines tilde-lambda = (L K1 e)^(1/K0) and then claims length(g^j(I(x))) is at most beta L0/e * tilde-lambda^{-t_x^+} tilde-lambda^j. But since t_x^+ ≥ K0 n and tilde-lambda > 1, the inequality tilde-lambda^{-K0 n} ≤ tilde-lambda^{-t_x^+} is false; it goes the other way. So the displayed bound in the lemma is not established. The issue is repairable by defining tilde-lambda with K1 in place of K0, but as written Proposition 7.2's subexponential length estimate — and therefore Theorem F's A-expansiveness — rests on an invalid step.\n\nI also share the reader's concern about Lemma 7.3. It is stated with unresolved placeholders (φY,?iL0(x), ≤ ?λ0) and its proof is delegated to another paper. That is a lot of weight to hang on an unverified splitting. The self-citation to [19] for cone estimates is not circular, but it does mean the paper's independence depends on material the authors have not included here. Minor point: the abstract overstates Theorem G, which only gives upper semi-continuity when μ(Sing(X)) = 0; otherwise there is an extra L2 μ(Sing(X)) term.\n\nOn balance, the local construction is likely correct and valuable, but the entropy applications need a serious check. The authors should fix the Lemma 7.6 inequality and supply a complete statement and proof or precise reference for Lemma 7.3. This deserves a serious referee, but not acceptance in its current form.","headline":"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.","tokens_in":28311,"tokens_out":2453,"would_cite":false,"duration_ms":25180,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37B40","37C10","37D30","37A35"],"pacs":[],"model":"deepseek-v4-flash","headline":"A countable partition computes metric entropy for singular flows away from homoclinic tangencies.","keywords":["singular flows","countable partitions","metric entropy","cross sections","homoclinic tangencies","upper semi-continuity","scaled linear Poincaré flow","dominated splittings"],"falsifier":"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.","tokens_in":27236,"feed_emoji":"🌀","tokens_out":11732,"duration_ms":106945,"temperature":0.7,"pith_summary":"The paper builds, for every $C^1$ vector field whose singularities are all hyperbolic, a countable partition $\\mathscr A$ of the phase space that is adapted to the singularities and has uniformly finite metric entropy for every invariant probability measure. The key is a new cross-section that actually contains each singularity and is stacked into layers of exponentially decreasing distance; the resulting partition forms a countable tower around the singularity, and its refinement places two points in the same atom inside a $\\beta$-scaled tubular neighborhood of each other's orbits. On flows away from homoclinic tangencies, the paper proves that every invariant measure is $\\mathscr A$-expansive, so the metric entropy of the time-one map equals the partition entropy, and that the metric-entropy function is upper semi-continuous in both the measure and the flow, with any possible drop bounded by a constant times the measure of the singular set.","feed_headline":"Countable partition tames singular flows' entropy","feed_subtitle":"New cross sections at the singularities yield finite-entropy partitions, proving upper semi-continuity away from tangencies.","key_machinery":"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.","core_discovery":"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 $L_2\\,\\mu(\\mathrm{Sing}(X))$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the criterion that a countable partition has finite entropy when $\\sum n\\,\\mu(C_n)$ is bounded; used in Propositions 3.7 and 4.4.","marker":"[17]"},{"why":"Supplies the scaled tubular neighborhood theorem and the uniform estimates for the projected normal flow that give the exponential shrinking size of neighborhoods and the shadowing property.","marker":"[10]"},{"why":"Supplies the fake-foliation construction and the entropy argument away from tangencies that Lemma 7.3 transfers to flows, and the one-dimensional shadowing method of Proposition 7.5.","marker":"[15]"},{"why":"Used in Corollary D to ensure that every ergodic invariant measure of a star flow is hyperbolic with no nearly-zero exponents, so the middle subbundle can be taken trivial.","marker":"[9]"},{"why":"Supplies the entropy-expansiveness lemma and the spanning-set estimate used in the proof of Theorem E.","marker":"[4]"},{"why":"Gives the dominated-splitting result for diffeomorphisms away from homoclinic tangencies that Lemma 7.3 extends to the normal bundle of the flow.","marker":"[26]"},{"why":"Source of the cone estimates near hyperbolic singularities and of the entropy theory for sectional-hyperbolic flows; the template for the coarse partition near each singularity.","marker":"[19]"},{"why":"Provides the eigenvalue-control statement that makes the expansion/contraction constant $\\lambda_0$ arbitrarily close to zero in Remark 7.4 and Corollary D.","marker":"[25]"}],"fun_headline_variants":["New cross sections at singularities tame flow entropy","Countable partition gives finite entropy for singular flows","Entropy for singular flows tamed by countable partition","Singularity cross-sections yield finite entropy partitions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["New cross sections at singularities tame flow entropy","Countable partition gives finite entropy for singular flows","Entropy for singular flows tamed by countable partition","Singularity cross-sections yield finite entropy partitions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001234,"raw_usage":{"total_tokens":5084,"prompt_tokens":974,"completion_tokens":4110,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":590,"completion_tokens_details":{"reasoning_tokens":4049}},"tokens_in":590,"tokens_out":4110,"duration_ms":30875,"temperature":1.0,"reasoning_tokens":4049,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:14:35.105500+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Ma˜ n´ e","cited_arxiv_id":null,"evidence_quote":"Supplies the criterion that a countable partition has finite entropy when $\\sum n\\,\\mu(C_n)$ is bounded; used in Propositions 3.7 and 4.4."},{"cited_title":"Gan and D","cited_arxiv_id":null,"evidence_quote":"Supplies the scaled tubular neighborhood theorem and the uniform estimates for the projected normal flow that give the exponential shrinking size of neighborhoods and the shadowing property."},{"cited_title":"Liao and M","cited_arxiv_id":null,"evidence_quote":"Supplies the fake-foliation construction and the entropy argument away from tangencies that Lemma 7.3 transfers to flows, and the one-dimensional shadowing method of Proposition 7.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Used in Corollary D to ensure that every ergodic invariant measure of a star flow is hyperbolic with no nearly-zero exponents, so the middle subbundle can be taken trivial."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the entropy-expansiveness lemma and the spanning-set estimate used in the proof of Theorem E."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the dominated-splitting result for diffeomorphisms away from homoclinic tangencies that Lemma 7.3 extends to the normal bundle of the flow."},{"cited_title":"Entropy theory for sectional hyperbolic flows","cited_arxiv_id":"1901.07436","evidence_quote":"Source of the cone estimates near hyperbolic singularities and of the entropy theory for sectional-hyperbolic flows; the template for the coarse partition near each singularity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the eigenvalue-control statement that makes the expansion/contraction constant $\\lambda_0$ arbitrarily close to zero in Remark 7.4 and Corollary D."}],"review_version":1}