Pith. sign in

REVIEW 3 major objections 3 minor 22 references

From Markovian actions to Anosov flows

T0 review · 3 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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

Pith's one-line read An orientation-preserving group action on a bifoliated plane comes from a topological Anosov flow on a closed orientable 3-manifold exactly when it preserves a strong Markovian family of rectangles.

desk verdict The paper gives the cleanest converse yet for Barbot-Fenley: orientation-preserving strong Markovian actions are exactly what come from topological Anosov flows, with the corner-condition reduction the one step I want referees to push on. read the letter →

arxiv 2607.25604 v1 pith:E64SGDBZ submitted 2026-07-28 math.DS

François Béguin , Ioannis Iakovoglou This is my paper · ORCID
classification math.DS MSC 37D2037C8537B10
keywords strongMarkovianactionbifoliatedplanetopologicalAnosovflowMarkovpartitionorbitspace3-manifoldgroupcornercondition
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

This paper proves a complete characterization: an orientation-preserving group action on a bifoliated plane—a plane with two transverse foliations—arises from a topological Anosov flow on a closed orientable 3-manifold exactly when the action preserves a strong Markovian family of rectangles. Such a family is a group-invariant collection of rectangles satisfying finiteness, Markovian intersection, strong finite return, and expansivity axioms; it encodes the finite combinatorics of a Markov partition. The authors construct, from these planar data alone, a closed 3-manifold and a topological Anosov flow whose orbit space is the given plane, whose fundamental-group action is conjugate to the original action, and whose reduced Markov partition projects precisely to the given rectangle family. They also show that a technical 'corner condition' on the rectangle family is not a genuine restriction, since every strong Markovian family can be cut into one satisfying it. The result matters because it recasts Anosov flows, up to orbital equivalence, as finite combinatorial data on the plane.

What carries the argument

The load-bearing object is a strong Markovian family: a ρ-invariant collection of rectangles (trivially bifoliated disks) covering the plane, with finitely many group orbits, the Markovian intersection axiom (two rectangles meet in a horizontal subrectangle of one and a vertical subrectangle of the other), a strong finite return axiom (every small quadrant germ is contained in a triple intersection R ∩ R_h ∩ R_v), and an expansivity axiom (nested rectangle chains collapse to a single leaf). Within such a family, the 'big brother' B(R) of a rectangle R is defined as the unique rectangle intersecting R exactly along R's positive unstable boundary and minimal for this property; its existence (T

What would settle it

Exhibit a strong Markovian action (satisfying Definition 2.13) for which no invariant strong Markovian family satisfies the corner condition—for instance, a family whose every refinement forces a periodic point into the interior of a rectangle boundary—or, more directly, a strong Markovian family satisfying the corner condition for which the 'big brother' rectangle of Theorem-Definition 3.4 is not unique or does not exist (Remark 3.5 indicates this can happen without the corner condition). Such an example would contradict the claim that every strong Markovian action arises from an Anosov flow.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.5: given an orientation-preserving strong Markovian action ρ of a torsion-free countable group G on a bifoliated plane (P, F^s, F^u) that preserves a strong Markovian family R satisfying the corner condition, there exists a closed orientable 3-manifold M with a topological Anosov flow Φ such that the bifoliated plane of Φ is isomorphic to (P, F^s, F^u), the π1(M)-action on that plane is conjugate to ρ, and the projection of the lift of a reduced Markov partition of Φ is exactly R. Since every strong Markovian action admits a family satisfying the corner condition (Proposition 2.26), the advertised conclusion follows: orientation-preserving strong Markovian acti

Load-bearing premise

The proof's load-bearing premise is Proposition 2.26: that every strong Markovian action admits a strong Markovian family satisfying the corner condition (no periodic point lies in the interior of a rectangle boundary component); if this cutting construction fails to preserve the strong Markovian axioms for some action, the main theorem would be proved only under the extra corner-condition hypothesis.

Editorial extensions

If this is right

  • Every orientation-preserving strong Markovian action is realized by a topological Anosov flow on some closed orientable 3-manifold, so the finite rectangle combinatorics of a strong Markovian family completely determine the flow up to orbital equivalence.
  • The preserved strong Markovian family is precisely the projection of the lift of a reduced Markov partition of the flow, so Markov partitions can be reconstructed directly from planar data.
  • The construction works without assuming dense orbits or preservation of the orientations of the invariant foliations; the non-orientation-preserving case is handled by a fixed-point-free orbit-equivalence involution.
  • The same methods are stated to extend to actions preserving singular foliations, producing pseudo-Anosov flows; the authors defer that generalization to a subsequent work.
  • The theorem gives the converse to the known fact that every topological Anosov flow yields a strong Markovian action on its bifoliated plane, closing the circle between plane combinatorics and 3-dimensional dynamics.

Reading between the lines

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

  • An implicit consequence is that Anosov flows up to orbital equivalence can be encoded as finite equivalence classes of strong Markovian families, suggesting a purely combinatorial (symbolic) classification of such flows.
  • The corner-condition cutting procedure hints at a normal-form theorem: every strong Markovian family can be refined so periodic points lie exactly at rectangle corners; because Proposition 2.26 is what removes the corner hypothesis from Theorem 1.5, verifying that cutting preserves the strong Markovian axioms in all cases is the key checkpoint.
  • The big-brother minimality resembles a deterministic 'next rectangle' rule; a testable extension is that the directed graph of rectangles with first-generation predecessor/successor relations forms a finite-state coding of the flow's return map.
  • The non-fully-orientable construction suggests that flows with orientation-reversing symmetries arise as quotients of orientation-preserving ones by a free involution, which may yield new examples by taking such quotients of known Anosov flows.
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.

Referee Report

3 major / 3 minor

Summary. The paper claims a converse to the Barbot–Fenley construction: every orientation-preserving strong Markovian action on a bifoliated plane is realized by a topological Anosov flow on a closed orientable 3-manifold, and the preserved Markovian family lifts to a reduced Markov partition. The proof constructs the space W+ of positively oriented stable segments, proves that the group action on W+ is free, properly discontinuous, and cocompact in the fully orientable case, and obtains the flow as the quotient flow. A separate section treats non-fully orientable actions by passing to the index-two subgroup G+ that preserves the orientations of both foliations, constructing an involution on the resulting manifold, and taking a quotient. The corner condition appearing in Theorem 1.5 is supposedly removed by a cutting procedure in Proposition 2.26.

Significance. If the proof can be completed, this is a substantial contribution: it gives a purely planar/action-theoretic characterization of orbit spaces of topological Anosov flows, without assuming transitivity or preservation of foliation orientations, and it strengthens the strong-Markovian-family formalism to a two-way correspondence that includes Markov partitions. The construction is explicit, and many intermediate results—W+ ≅ R^3, the big-brother theorem, the cocompactness argument, and the cluster-separation procedure—are worked out in detail. The main reservations are the unverified corner-condition reduction and the heavy dependence on results imported from unpublished or very recent preprints by the second author.

major comments (3)
  1. [§2.6, Proposition 2.26] The proof verifies the Markovian intersection axiom for the cut family S in detail, but for the strong finite return and expansivity axioms it says only 'it is not difficult to check'. This is the sole step that lets Theorem 1.5 and the advertised equivalence apply to all strong Markovian actions rather than only those admitting a corner-condition family. The transfer is not automatic: after cutting a rectangle along an F^u-leaf through a periodic point, a nested sequence of vertical subrectangles of a cut rectangle could have intersection a proper subsegment of the F^u-leaf of that cut rectangle, violating Definition 2.13(4); and the strong finite return axiom requires an explicit argument for germs lying on the cut leaves. Please supply a complete verification of both axioms, or state Theorem 1.5 with the corner condition as an explicit hypothesis.
  2. [§2.5–2.6 (Theorems 2.22, Lemmas 2.15–2.18, 2.23)] Several load-bearing results are imported from [Ia2], listed as an unpublished arXiv preprint, and from [Ia]. Theorem 2.22(1) is used for freeness of eρ in Proposition 3.3 and for the absence of periodic rectangles in Section 5; Theorem 2.22(2)–(4) controls periodic points and gives the contraction/expansion used in Proposition 3.8 and Proposition 2.26; Lemmas 2.15–2.18 provide the predecessor/successor calculus throughout. Since these proofs are not reproduced, the referee cannot fully verify the foundation of the paper. Please either include the statements and proofs of these results, or make the dependence explicit and ensure the cited works are available in a verifiable form.
  3. [§6.1, Proposition 6.1] The reduction to the fully orientable case depends on the existence of a lift ef0 of ρ(h) to W+ satisfying π∘ef0 = ρ(h)∘π, invoked from [Ba1, Prop. 1.36]. This is a delicate point because for h ∈ G−G+ the orientation of F^s is reversed, whereas the fibers of π are the positive F^s rays. The reader needs more than the bare reference: either a precise statement adapted to the present conventions, or a short verification that such a lift exists. Without this, the non-fully orientable part of Theorem 1.5 is not established.
minor comments (3)
  1. [§6.2, (6.15)–(6.16)] The notation is overloaded: eR is used both for the original lifted rectangles and for the new invariant rectangles eRinv. Please use distinct notation for the two families to avoid confusion.
  2. [References] Reference [Ia3] contains a typo: 'Markov parititions' should be 'Markov partitions'.
  3. [Proposition 2.26, final paragraph] The paragraph says the construction can be repeated 'finitely many times', but the proof only performs one cut in the F^s direction and one in the F^u direction. If iteration is intended, please state it explicitly and justify termination.

Circularity Check

1 steps flagged · score 4.0 of 10

Construction is independent, but the advertised iff characterization and several load-bearing dynamical lemmas are imported from the second author's prior work.

  1. self citation load bearing [Section 2.6, Theorem 2.22 and Lemma 2.23; used throughout Sections 3–6]
    "Theorem 2.22 (Theorem 1.5, Lemmas 3.3 and 3.6, [Ia2]). Let ρ:G→Homeo(P) be a strong Markovian action on the bifoliated plane (P,F^s,F^u) preserving a strong Markovian family R. We have that: (1) For every R∈R and every g∈G, if ρ(g)(R)=R, then g=e."

    The proof of Theorem 1.5 relies on Theorem 2.22 and Lemma 2.23 for freeness and proper discontinuity of the W^+-action, for Proposition 2.26 (corner-condition reduction), and for the big-brother/cocompactness arguments. These results are not proved in this paper but are cited to [Ia2], an unpublished work by the second author. The advertised 'if and only if' characterization also uses Proposition 2.14, citing [Ia], for the Anosov-flow-to-Markovian direction. Thus the central claim is partly carried by a self-citation chain, though the forward construction itself is not fitted or definitionally circular.

full rationale

The construction of the Anosov flow from a strong Markovian family is a genuine derivation: W^+ is built from positively oriented F^s-segments, the flow is the fiber foliation of the projection π, and the stable/unstable foliations together with expansivity are verified using the strong Markovian family axioms. No fitted parameter is later renamed as a prediction, and no step assumes the existence of the target flow. So the paper is not circular in the sense of 'prediction reduces to input by construction'. The main circularity concern is the heavy reliance on the second author's earlier works. Proposition 2.14 (Anosov flows produce strong Markovian families) is cited to [Ia], and Theorem 2.22 plus Lemma 2.23 (periodic-point structure, topological contraction/expansion) are cited to [Ia2]. These are load-bearing: they are used to prove Prop. 2.26, freeness and proper discontinuity, the existence of big brothers, cocompactness, and the Anosov verification. They are not proved or machine-checked in the present manuscript, so the self-citation burden is real. However, these cited results are parameter-free theorems about strong Markovian actions, not restatements of the target flow; the central construction still has independent mathematical content. Separately, the proof of Proposition 2.26 leaves the strong finite return and expansivity axioms for the cut family to the sentence 'it is not difficult to check'. This is an omitted verification rather than a circular step, but it means the advertised removal of the corner-condition hypothesis rests on the least secure internal link. Overall, the derivation is not circular by definition or by fit, but it is not fully self-contained and leans on self-citations in a load-bearing way.

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

No free parameters or invented entities. The central claim rests on importing several results: Whitney's foliation-flow theorem, Barbot-Fenley correspondence, Markov-partition existence, and, most importantly, self-cited theorems from [Ia2] and [Ia] supplying hyperbolic dynamics and predecessor combinatorics.

assumptions (5)
  • domain assumption Theorem 2.22 ([Ia2]): strong Markovian actions have saddle-type periodic points, unique periodic point on each periodic leaf, and topological contraction/expansion along separatrices.
    Invoked repeatedly (Prop 2.26, Prop 3.3, Section 6); it is a self-cited prior result not proved in this paper, and it supplies the hyperbolic dynamics used to prove freeness and proper discontinuity.
  • domain assumption Lemma 2.15 and Lemmas 2.17-2.18 ([Ia]): predecessors/successors of first generation are finite, well-defined, and generate a partial order.
    The combinatorial backbone of §3.2–3.4; the paper says proofs carry over verbatim from [Ia].
  • domain assumption Barbot's Proposition 1.36 ([Ba1]): a homeomorphism of the bifoliated plane that conjugates the action lifts to a self-orbit equivalence of the associated flow.
    Used in §6.1 to extend the action from G+ to G in the non-fully-orientable case; cited from a habilitation thesis.
  • standard math Whitney's Theorem 27A ([Whi]): every C^0 foliation of the plane is the orbit foliation of a continuous flow.
    Used in Prop 3.2 to identify W+ with R^3 and in §4 to parametrize the foliation on M by a non-singular flow.
  • domain assumption Theorem-Def 2.7 (Barbot-Fenley): the orbit space of a topological Anosov flow is a bifoliated plane and π1(M) acts on it preserving foliations.
    Gives the necessary direction and the target structure; cited as extending verbatim to topological flows.

how reviews work

0 comments
Cite this review

Pith. "Pith review of From Markovian actions to Anosov flows." pith.science (2026). https://pith.science/paper/E64SGDBZ

@misc{pith2026260725604,
  author       = {Pith},
  title        = {Pith review of: From Markovian actions to Anosov flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E64SGDBZ}},
  note         = {Machine review of arXiv:2607.25604}
}
abstract

In this paper, we establish a necessary and sufficient condition for a group action on a bifoliated plane to arise from a topological Anosov flow supported by an orientable $3$-manifold. More precisely, we show that a group action on a non-singular bifoliated plane arises from such a flow if and only if it is an orientation preserving strong Markovian action, that is, an orientation preserving action of the plane preserving a family of rectangles satisfying suitable Markov-type properties. Furthermore, we prove that this family of rectangles can be lifted to a Markov partition of the associated flow.

Figures

Figures reproduced from arXiv: 2607.25604 by the authors.

Figure 2.1
Figure 2.1. A strong Markovian family in P (4) (Expansivity axiom) If (Ri)i∈N is a sequence of rectangles in R such that Ri+1∩Ri is a non-trivial vertical (resp. horizontal) subrectangle of Ri for every i ∈ N, then ∩ i≥0 Ri is an F u -leaf (resp. F s -leaf) of R0. ♠ Strong Markovian families were originally introduced in [Ia] as analogues of Markov partitions for group actions on the plane. Similarly to Markov partitions, these… view at source ↗
Figure 2.2
Figure 2.2. In the above figure, R′′ is a predecessor of R′ of some gener￾ation and R′ is a predecessor of R of some generation. We finish this section by remarking that the relation “being a predecessor of some generation” defines a partial order on R: Remark 2.19. Take R, R′ , R′′, R′′′ ∈ R such that R′ is a predecessor of n-th generation of R, R′′ is a predecessor of k-th generation of R′ and R′′′ is a successor of l-th gene… view at source ↗
Figure 2.3
Figure 2.3. A periodic point in the boundary of R1 can not belong in the interior of R2. Finally, assume that p ∈ ∂ sR2 (see [PITH_FULL_IMAGE:figures/full_fig_p012_2_3.png] view at source ↗
Figures from the paper (9 more)
Figure 2.4
Figure 2.4. Figure 2.4: Cutting R1 along two F u -leaves. Case 3: The interior of ∂ sR1 contains exactly two periodic points of ρ Denote by p and p ′ the previous two periodic points. Recall that by our previous arguments, p, p′ ∈/ R2 − ∂ sR2. Therefore, by exchanging the roles of p and p ′…
Figure 3.1
Figure 3.1. Figure 3.1: Under the assumptions of Subcase 1, ρ(gn)([xn, yn] s ) can not converge to [X, Y ] s . Since ρ(gm)(r) is a successor of R of generation greater than M + 1 and ρ(gm)(V ) is a predecessor of ρ(gm)(r) of generation M that intersects Int(R), Remark 2.19 implies that ρ(gm…
Figure 3.2
Figure 3.2. Figure 3.2: Under the assumptions of Subcase 2, ρ(gn)([xn, yn] s ) can not converge to [X, Y ] s . Passing once again to a subsequence, we may assume that the above properties hold for every n ∈ N. Since the rectangles ρ(gn)(r) are successors of progressively larger generations …
Figure 3.3
Figure 3.3. Figure 3.3: The non-existence of a minimal rectangle with property (⋆), contradicts the expansivity axiom. Next, for the sake of simplicity, and for the remainder of this proof, we say that a rectangle r ∈ R satisfies property (⋆) if r ∩ R = ∂ u +R. Let us now prove that there e…
Figure 3.4
Figure 3.4. Figure 3.4: The set I(x, y) is non-empty. □ Proof of Item (2). Consider I, J ∈ I(x, y) with I ̸= J. Recall that by definition, [x, y] s ⊂ I ∩ J. Fix R, S ∈ R such that (R ∪ B(R)) ∩ L = I, (S ∪ B(S)) ∩ L = J, R ∩ Int(D) ̸= ∅, S ∩ Int(D) ̸= ∅ and x ∈ (R − ∂ u +R) ∩ (S − ∂ u +S). T…
Figure 3.5
Figure 3.5. Figure 3.5: The configuration of R, S, and their big brothers in the case where F u (∂ u +R) ̸= F u (∂ u +S). B(S) or B(R) ∩ B(S) is a vertical subrectangle of B(R). In both of the previous cases we have that S ∩ L ⊂ R ∩ L and B(S) ∩ L ⊆ B(R) ∩ L which implies that J ⊂ I [PITH_…
Figure 3.6
Figure 3.6. Figure 3.6: The configuration of R, S, and their big brothers in the case where F u (∂ u +R) = F u (∂ u +S). □ Proof of Item (3). Assume by contradiction that I(x, y) does not contain a minimal ele￾ment. Since I(x, y) ̸= ∅, this implies that there exists (In)n∈N a sequence of se…
Figure 5.1
Figure 5.1. Figure 5.1: If the positive Φ-orbit of ( e x, y) intersects the interior of Se, then (x, y) can not lie in ∂ sRe. Consider R = π(Re), S = π(Se), x = π(xe), and let y ∈ P denote the unique point of intersection of F s +(x) with ∂ u +B(R) (see [PITH_FULL_IMAGE:figures/full_fig_p0…
Figure 5.2
Figure 5.2. Figure 5.2: In both of the above cases, we have that Re ∩ Se = ∅. Indeed, if R does not intersect ∂ u +S, then Int(B(R)) ∩ Int(S) ̸= ∅ (see the left panel of [PITH_FULL_IMAGE:figures/full_fig_p032_5_2.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 3 linked inside Pith

  1. [1]

    G\'eom\'etrie transverse des flots d’Anosov, Thesis, Ecole Normale Sup\'erieure de Lyon (1992)

    Thierry Barbot. G\'eom\'etrie transverse des flots d’Anosov, Thesis, Ecole Normale Sup\'erieure de Lyon (1992)

    1992

  2. [2]

    De l’hyperbolique au globalement hyperbolique, Habilitation à diriger des recherches, Université Claude Bernard - Lyon I

    Thierry Barbot. De l’hyperbolique au globalement hyperbolique, Habilitation à diriger des recherches, Université Claude Bernard - Lyon I. (2005)

    2005

  3. [3]

    Orbit equivalences for Pseudo-Anosov flows, Inventiones mathematicae, Vol

    Thomas Barthelmé, Steven Frankel, Kathryn Mann. Orbit equivalences for Pseudo-Anosov flows, Inventiones mathematicae, Vol. 240, pp. 1119–1192 (2025)

    2025

  4. [4]

    Thomas Barthelmé, Christian Bonatti, Kathryn Mann Non-transitive pseudo-Anosov flows, arXiv: 2411.03586

    arXiv

  5. [5]

    Reconstructing flows from the orbit space, arXiv:2509.01594

    Thomas Barthelmé, Sergio Fenley, Kathryn Mann. Reconstructing flows from the orbit space, arXiv:2509.01594

    arXiv

  6. [6]

    Pseudo-Anosov flows: a plane approach, arXiv:2509.15375

    Thomas Barthelmé, Kathryn Mann. Pseudo-Anosov flows: a plane approach, arXiv:2509.15375

    arXiv

  7. [7]

    Theory of Lie groups, Princeton Univ

    Claude Chevalley. Theory of Lie groups, Princeton Univ. Press (1946)

    1946

  8. [8]

    Anosov flows in 3-manifolds, Ann

    Sergio Fenley. Anosov flows in 3-manifolds, Ann. of Math., Vol. 139, No. 1, pp. 79-115 (1994)

    1994

Show all 22 references
  1. [9]

    Quasigeodesic flows in hyperbolic 3-manifolds, Topology, Vol

    Sergio Fenley, Lee Mosher. Quasigeodesic flows in hyperbolic 3-manifolds, Topology, Vol. 40, No. 3, pp. 503-537 (2001)

    2001

  2. [10]

    Markovian families for pseudo-Anosov flows, arXiv:2509.19530

    Ioannis Iakovoglou. Markovian families for pseudo-Anosov flows, arXiv:2509.19530

  3. [11]

    London Math

    Ioannis Iakovoglou, A new combinatorial invariant characterizing Anosov flows on 3-manifolds, J. London Math. Soc., Vol. 113 (2026)

    2026

  4. [12]

    Ioannis Iakovoglou, Markovian actions are Anosov-like, arXiv:2606.19246

    arXiv

  5. [13]

    Markov parititions for non-transitive pseudo-Anosov flows, Comptes Rendus Mathématique (accepted 2024)

    Ioannis Iakovoglou. Markov parititions for non-transitive pseudo-Anosov flows, Comptes Rendus Mathématique (accepted 2024)

    2024

  6. [14]

    Nonsingular expansive flows on 3-manifolds and foliations with circle prong singularities, Japan

    Takashi Inaba and Shigenori Matsumoto. Nonsingular expansive flows on 3-manifolds and foliations with circle prong singularities, Japan. J. Math.Vol. 16, No. 2 (1990)

    1990

  7. [15]

    Taylor, A characterization of pseudo-Anosov orbit spaces via bifoliated planes, arXiv:2606.16588

    Kathryn Mann and Samuel J. Taylor, A characterization of pseudo-Anosov orbit spaces via bifoliated planes, arXiv:2606.16588

  8. [16]

    Lee Mosher, Laminations and flows transverse to finite depth foliations, manuscript available online here https://homepages.warwick.ac.uk/ masgar/Maths/1996lamina-tions and flows transverse to finite depth foliations OCRed.pdf

  9. [17]

    Survey on aspherical manifolds, arXiv:0902.2480

    Wolfgang Lueck. Survey on aspherical manifolds, arXiv:0902.2480

    arXiv

  10. [18]

    M. Ratner. Markov partitions for C-flows on 3-dimensional manifolds, Mat. Zametki, Vol. 6, No. 6, pp. 693-704 (1969)

    1969

  11. [19]

    P. Scott. The geometries of 3-manifolds, Bull. London Math. Soc., Vol.15, No.5, pp. 401–487 (1983)

    1983

  12. [20]

    Hyperbolic models of transitive topologically Anosov flows in dimension three, Annales Scientifiques de l'École Normale Supérieure, Series 4, Vol

    Mario Shannon. Hyperbolic models of transitive topologically Anosov flows in dimension three, Annales Scientifiques de l'École Normale Supérieure, Series 4, Vol. 58, No. 5, pp. 1243--1318 (2025)

    2025

  13. [21]

    V. V. Solodov. Topological problems in the theory of dynamical systems, Uspekhi Mat. Nauk, Volume 46, Issue 4, pp.107 134 (1991)

    1991

  14. [22]

    Hassler Whitney, Regular families of curves, Annals of Mathematics, No.34 (1933)

    1933

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.