Pith. sign in

REVIEW 3 major objections 4 minor 38 references

Generic properties of planar symplectic billiards

T0 review · 3 major / 4 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 This paper proves that, for a residual set of C^∞ strongly convex planar domains, the symplectic billiard map has positive topological entropy.

desk verdict A serious, mostly correct transfer of generic billiard theory to symplectic billiards; the entropy theorem likely stands, but Proposition 7.3's proof is too quick at a coordinate singularity and should be fixed or replaced by the cited [3, Theorem 3]. read the letter →

arxiv 2607.21223 v1 pith:KZRKNY4B submitted 2026-07-23 math.DS

Luca Baracco , Olga Bernardi , José Pedro Gaivão This is my paper · ORCID
classification math.DS MSC 37D0537E4037C20
keywords symplecticbilliardsgenericdynamicspositivetopologicalentropyhyperbolicperiodicpointshomoclinicintersectionstwistmapsellipticislandsderivativerealizationlemma
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

The paper aims to show that planar symplectic billiards behave generically like classical billiards, despite being generated by symplectic area rather than length. Its central claim is that, for a residual set (a countable intersection of open dense subsets) of C^∞ strongly convex domains, the symplectic billiard map has positive topological entropy. Along the way it proves that, generically, periodic orbits are non-degenerate and in general position, elliptic periodic points form stable islands surrounded by invariant curves, and every hyperbolic periodic point has a transverse homoclinic point. The reason to care is that it is not obvious that the standard mechanisms for chaos in billiards survive the replacement of the length functional by the area functional; the paper says they do, generically.

What carries the argument

The central object is the symplectic billiard map, an area-preserving negative twist map on a phase cylinder, defined by the rule that the tangent at the middle of three consecutive boundary points is parallel to the chord joining the other two, and generated by the symplectic area between consecutive points. Four tools carry the argument: jet transversality forces periodic configurations into general position and non-degeneracy; a derivative-realization lemma lets small perturbations of the linearized map along any sufficiently long periodic orbit be produced by small boundary perturbations, with no no-focusing condition needed; a normal-form twist argument turns elliptic points into stable

What would settle it

Check the coordinate change used in the invariant-curve lemma: compute the derivative of the natural coordinate with respect to the angle coordinate at the boundary φ=π from the differential formula. If it is zero, the annulus coordinates in which the invariant-curve theorem is invoked are singular and that step of the proof does not apply; a concrete C^∞ strongly convex table for which the required invariant curves provably do not exist would directly falsify the key premise, while a proof that the twist-map hypotheses do hold at that boundary would close the gap.

Watch

Extended reading notes

Core claim

The paper proves that generic C^∞ strongly convex symplectic billiards are chaotic in a precise sense: the dynamics has positive topological entropy, meaning the number of distinguishable orbit segments grows exponentially with time. This is derived by showing that a generic table's symplectic billiard map contains a horseshoe, a compact invariant set on which the dynamics is a full shift on two symbols. The route is to make all periodic orbits non-degenerate and in general position via jet transversality, to obtain a derivative-realization lemma that controls the linearized map along periodic orbits, to show elliptic periodic points are stable, and finally to prove that hyperbolic periodic

Load-bearing premise

The argument that stable and unstable manifolds stay in a compact region, and hence that a homoclinic intersection must exist, depends on smooth invariant curves existing arbitrarily close to both boundary components of the phase space; the proof of this existence applies a twist-map invariant-curve theorem in angle coordinates that become singular at one boundary, so if those curves fail to exist the positive-entropy conclusion is not established by the stated argument.

Editorial extensions

If this is right

  • If the central theorem is right, a generic smooth strongly convex table supports exponentially many periodic orbits of each sufficiently large period, since positive topological entropy implies exponential orbit growth.
  • Generic symplectic billiards are neither integrable nor globally hyperbolic: stable elliptic islands and chaotic hyperbolic sets coexist.
  • The derivative-realization lemma, which needs no no-focusing condition, should make local perturbation constructions in symplectic billiards simpler than the analogous constructions for classical billiards.
  • Every hyperbolic periodic point of a generic table has a transverse homoclinic point, so the standard transversality picture of conservative dynamics holds in this family.

Reading between the lines

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

  • The proof's reliance on invariant curves near the boundary is the point to test: if the coordinate singularity there can be repaired, the same strategy should work for C^k boundaries with k>5, not only for C^∞ boundaries.
  • Because the local perturbation lemma avoids the no-focusing condition, symplectic billiards may be a better testbed than classical billiards for higher-dimensional or outer-billiard versions of these genericity results.
  • Known integrable examples such as the circle have zero topological entropy; the theorem says they are isolated exceptions in the C^∞ topology, so the integrable–chaotic boundary in this family is topologically sharp.
  • A numerical scan of small Fourier perturbations of the circle could look for the predicted horseshoes and measure how entropy grows with perturbation size, giving a quantitative check of the generic behavior.
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 / 4 minor

Summary. The paper studies generic dynamical properties of planar symplectic billiards on C-infinity strongly convex domains. Its main claims are: (i) periodic symplectic billiard orbits are in general position and non-degenerate on a residual set of domains; (ii) a Franks' lemma holds for symplectic billiards; (iii) stably hyperbolic periodic points form a hyperbolic set; (iv) elliptic periodic points are generically stable; (v) hyperbolic periodic points generically have transverse homoclinic points; and (vi) generically the symplectic billiard map has positive topological entropy. The proofs use multijet transversality, normal perturbations of the boundary via curvature changes, and external theorems of Moser, Ruessmann, and Xia-Zhang.

Significance. If the stated results hold, this is a substantial contribution to the dynamical systems literature. It develops a generic theory for symplectic billiards parallel to classical billiards, and the positive topological entropy conclusion would be a notable extension of the classical generic entropy results. The paper is ambitious and contains many independent contributions: residual genericity of non-degenerate periodic orbits, a Franks' lemma, elliptic stability, and a Kupka-Smale-type statement. The main weakness is that the proof of the crucial Proposition 7.3 is incomplete in the presented form, and the Franks' lemma proof contains a rank verification gap; both are load-bearing for stated theorems.

major comments (3)
  1. [Section 7, Proposition 7.3] The proof of Proposition 7.3 is not valid as written. In the (theta, phi) coordinates the preserved area form is (partial s / partial phi) d theta wedge d phi, with s = -omega(gamma'(theta), gamma(theta+phi)); since partial s / partial phi = -omega(gamma'(theta), gamma'(theta+phi)) vanishes at phi = pi, the coordinate chart is singular at the boundary component where the invariant curves are being constructed. A theorem such as Ruessmann's [31, Theorem 2] requires a genuine annulus with a nondegenerate area form and verified twist hypotheses; the first-order expansion T(theta, pi - epsilon) = (theta + pi - epsilon, pi - epsilon + O(epsilon^2)) does not establish those hypotheses. Proposition 7.3 is the only step in Theorem 7.5 that confines W^s(x) and W^u(x) to a compact sub-annulus so that the Xia-Zhang theorem [38] can be applied. The result is stated to be already proved in [3, Theore
  2. [Section 5, Theorem 5.1] The local surjectivity claim in the proof of Franks' lemma is not established. The three derivative matrices partial M / partial k_1, partial M / partial k_2, partial M / partial k_3 are treated as rows of a 3 by 4 matrix, and it is asserted that rank 3 follows from A_2 B_3 != 0. This is not sufficient; for example, one of the 3 by 3 minors equals -A_2 (A_2 A_3 + B_2) B_3, so the derivative vectors are dependent if A_2 A_3 + B_2 = 0. No argument is given that A_2 A_3 + B_2 is nonzero for the periodic orbits in T_1, nor is it shown that a different triple of points can be chosen when it vanishes. Without this, the inverse-function-theorem step in Theorem 5.1 is unsupported, and the deduction of Theorem 5.5 from it is also unsupported.
  3. [Section 5, Theorem 5.5] Theorem 5.5 is stated for every gamma in int_{C^2}(H_{>= p}), but its proof invokes Theorem 5.1, which is only proved for gamma in T_1. No argument is given that int_{C^2}(H_{>= p}) is contained in T_1, nor that Franks' lemma extends to the non-generic case. The theorem should either be formulated for gamma in T_1 intersection int_{C^2}(H_{>= p}), or a separate argument avoiding the T_1 hypothesis should be supplied.
minor comments (4)
  1. [Section 4, Proposition 4.8] The proof refers to subsets [0,2 pi) times (0, epsilon) and [0,2 pi) times (pi - epsilon, pi), but the phase space P_gamma was defined with s in (psi_1(t), psi_2(t)), not with s in (0, pi). Please clarify the coordinate identification used in this argument.
  2. [Section 6, Lemma 6.3] In the definition of B(n), the trailing '= 0' appears to be a typo. As written, B(n) is set to zero, which would make the displayed Jacobian matrix singular and contradict the invertibility claim that follows.
  3. [Section 7, Corollary 7.6] The step 'all periodic points are non-hyperbolic implies there is an elliptic periodic point' uses the non-degeneracy property to exclude the eigenvalue -1, since that would produce a degenerate orbit of double period. This implication should be stated explicitly for completeness.
  4. [Section 5, proof of Theorem 5.1] The notation A_j, B_j is used before it is introduced in the display immediately following. Please reorder the definitions for readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: generic positive entropy is derived from external transversality, KAM, and Xia–Zhang results; self-citations are standard checkable lemmas.

full rationale

I walked the derivation chain from the residual-set constructions through the positive-entropy conclusion. The central theorems (Theorems 3.1, 3.2, 4.4, 4.7, 4.10, 5.1, 5.5, 6.4, 7.1, 7.5, and Corollary 7.6) are obtained by multijet transversality, dimension-counting submanifolds, Franks' lemma style perturbations, Moser's twist theorem, Rüssman's theorem, the Xia–Zhang homoclinic argument, and Angenent's entropy criterion. These are external benchmarks whose hypotheses do not include the paper's target conclusion. There are no fitted parameters and no 'prediction' that is equal to an input by construction: residual sets are countable intersections of open dense sets, and the arguments are perturbative rather than statistically forced. The paper cites its own prior work in a few places: [8] is used for the derivative formula Lemma 2.2 and for phase-space conventions, and [12] is cited as background/context for classical billiards and in Remark 5.6. Those uses are not load-bearing reductions: Lemma 2.2 is a direct computation from the defining generating function and is independently checkable, and [12] concerns classical billiards, not the symplectic case under study. The closest concern is Proposition 7.3, whose proof applies Rüssman's theorem in tangent-angle coordinates where the area form is degenerate at one boundary; that is a possible gap in verifying hypotheses, not a circularity, since the existence of invariant curves is not assumed in the input but is cited from external works [3] and [31]. Therefore I find no self-definitional step, no fitted input renamed as prediction, and no self-citation chain that forces the result.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The paper's central claim rests on a cascade of standard theorems from dynamical systems and singularity theory that are imported as black boxes: multijet transversality, Moser/Rüssman invariant-curve theorems, Contreras' hyperbolicity theorem, Xia-Zhang homoclinic criteria, Angenent's rotation-number criterion, and Birkhoff-Smale. None of these are proved in the paper. They are not free parameters and no invented entities are introduced. The load-bearing assumptions are standard for the genre, so they are listed as axioms.

assumptions (9)
  • standard math Multijet Transversality Theorem (Golubitsky–Guillemin [22])
    Used to obtain residual sets of tables for which jet extensions avoid the incidence submanifolds Σσ and Σn (Theorems 3.1, 3.2, 4.4, 4.10).
  • domain assumption Contreras' Theorem: every bounded stably hyperbolic family of periodic sequences in Sp(1) is uniformly hyperbolic ([16, Thm 8.1])
    Used in the proof of Theorem 5.5 to promote stable hyperbolicity of the family of derivative sequences to uniform hyperbolicity and hence to a hyperbolic set.
  • domain assumption Bunimovich–Grigo Theorem 3 (sufficient condition for non-vanishing first Birkhoff coefficient) [13, Thm 3]
    Used in Theorem 6.4 to conclude stability of elliptic periodic points from a C≠0 term in the ε-derivative of the generating function.
  • standard math Moser's twist theorem [27, Thm 2.13]
    Used after showing the first Birkhoff coefficient is non-zero to produce invariant curves surrounding elliptic points (Theorem 6.4, Corollary 7.6).
  • standard math Rüssman's invariant-curve theorem [31, Thm 2]
    Used in Proposition 7.3 to show invariant curves accumulate on the boundaries of the phase space.
  • domain assumption Lazutkin invariant-curve theorem for symplectic billiards [3, Thm 3]
    Used in Proposition 7.3 for the boundary component φ=0.
  • domain assumption Xia–Zhang propositions on recurrent branches and homoclinic points for twist maps [38, Prop 3.3, 4.2, Thm 4.3]
    Used in Theorem 7.5 to obtain the existence of a transverse homoclinic point from the invariant-curve confinement.
  • domain assumption Angenent's criterion on essential invariant curves for zero-entropy area-preserving twist maps [5, Thm A]
    Used in Corollary 7.6 to force the existence of a hyperbolic periodic point in the zero-entropy contradiction argument.
  • standard math Birkhoff–Smale homoclinic theorem
    Used in Corollary 7.6 to convert a transverse homoclinic point into a horseshoe and positive topological entropy.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Generic properties of planar symplectic billiards." pith.science (2026). https://pith.science/paper/KZRKNY4B

@misc{pith2026260721223,
  author       = {Pith},
  title        = {Pith review of: Generic properties of planar symplectic billiards},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KZRKNY4B}},
  note         = {Machine review of arXiv:2607.21223}
}
abstract

We study generic properties of planar symplectic billiards, a symplectic analogue of classical Birkhoff billiards introduced by P. Albers and S. Tabachnikov. We prove that, for a residual set of $C^\infty$ strongly convex domains, periodic symplectic billiard trajectories are in general position and non-degenerate. We also prove a Franks' lemma for symplectic billiards and deduce that stably hyperbolic periodic points form a hyperbolic set given by the closure of the set of hyperbolic periodic points. We further show that, for a residual set of $C^\infty$ strongly convex domains, elliptic periodic points are stable and hyperbolic periodic points have transverse homoclinic points. Based on these results, we conclude that --generically-- $C^\infty$ strongly convex symplectic billiard has positive topological entropy.

Figures

Figures reproduced from arXiv: 2607.21223 by the authors.

Figure 1
Figure 1. det(x1 − O, x2 − O) is the area of the parallelogram in figure. Since Ω is strictly convex, for every point γ(t) ∈ Γ, there exists a unique point γ(t ∗ ) (t ∗ ̸= t) such that ω(γ ′ (t), γ ′ (t ∗ )) = 0 . (In the above formula, γ ′ (t) and γ ′ (t ∗ ) denote the tangent vectors at Γ at the points γ(t) and γ(t ∗ ) respectively). We refer to Pˆγ = {(γ(t1), γ(t2)) ∈ Γ × Γ : γ(t1) < γ(t2) < γ(t ∗ 1)} (2.1) as the (open, p… view at source ↗
Figure 2
Figure 2. The symplectic billiard map reflection. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. The phase space Pγ. Proposition 2.1. The following properties hold. 1. Tˆγ is C k−1 and it extends continuously to the closure of Pˆγ so that Tˆγ(γ(t), γ(t)) = ((γ(t), γ(t)) and Tˆγ(γ(t), γ(t ∗ )) = ((γ(t ∗ ), γ(t)). 2. For every (γ(t1), γ(t2)) ∈ Pˆγ one has Tˆγ(γ(t1), γ(t2)) = (γ(t2), γ(t3)) ⇐⇒ ω(γ(t1), γ ′ (t2)) + ω(γ ′ (t2), γ(t3)) = 0 . (2.2) In other words, ω is a generating function for Tˆγ. 3. The map Tˆγ doe… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

38 extracted references · 3 linked inside Pith

  1. [3]

    Albers and S

    P. Albers and S. Tabachnikov. Introducing symplectic billiards.Adv. Math., 333:822–867, 2018

    2018

  2. [38]

    Xia and P

    Z. Xia and P. Zhang. Homoclinic points for convex billiards.Nonlinearity, 27(6):1181–1192, 2014. 26

    2014

  3. [1]

    Albers, G

    P. Albers, G. Banhatti, F. Sadlo, R. E. Schwartz, and S. Tabachnikov. Polygonal symplectic billiards.Journal of Experimental Mathematics, 1(1):1–22, 2025

    2025

  4. [2]

    Albers, A

    P. Albers, A. C. Caliz, and S. Tabachnikov. Symplectic billiards as minkowski billiards. Preprint, arXiv:2607.05986 [math.DS] (2026), 2026

    arXiv 2026

  5. [4]

    H. N. Alishah and J. L. Dias. Realization of tangent perturbations in discrete and continuous time conservative systems.Discrete and Continuous Dynamical Systems, 34(12):5359–5374, 2014

    2014

  6. [5]

    S. B. Angenent. A remark on the topological entropy and invariant circles of an area pre- serving twist map. InTwist Mappings and Their Applications, volume 44 ofIMA Volumes in Mathematics and its Applications, pages 1–5, New York, 1992. Springer

    1992

  7. [6]

    Baldomá, A

    I. Baldomá, A. Florio, M. Leguil, and T. M. Seara. Chaoticity of generic analytic convex billiards.Preprint, 2026. https://arxiv.org/abs/2605.19897

    arXiv 2026

  8. [7]

    Baracco and O

    L. Baracco and O. Bernardi. Totally integrable symplectic billiards are ellipses.Adv. Math., 454:Paper No. 109873, 17, 2024

    2024

Show all 38 references
  1. [8]

    Birkhoffattractorsfordissipativesymplectic billiards.Journal of Dynamics and Differential Equations, 2026

    L.Baracco, O.Bernardi, A.Florio, andA.Nardi. Birkhoffattractorsfordissipativesymplectic billiards.Journal of Dynamics and Differential Equations, 2026. To appear

    2026

  2. [9]

    Baracco, O

    L. Baracco, O. Bernardi, and A. Nardi. Bialy-Mironov type rigidity for centrally symmetric symplectic billiards.Nonlinearity, 37(12):Paper No. 125025, 12, 2024

    2024

  3. [10]

    Baracco, O

    L. Baracco, O. Bernardi, and A. Nardi. Higher order terms of Mather’sβ-function for sym- plectic and outer billiards.J. Math. Anal. Appl., 537(2):Paper No. 128353, 20, 2024

    2024

  4. [11]

    Baracco, O

    L. Baracco, O. Bernardi, and A. Nardi. Area spectral rigidity for axially symmetric domains. Journal of Modern Dynamics, 2026. To appear

    2026

  5. [12]

    Bessa, G

    M. Bessa, G. Del Magno, J. L. Dias, J. P. Gaivão, and M. J. Torres. Billiards in generic convex bodies have positive topological entropy.Advances in Mathematics, 442:109592, 2024

    2024

  6. [13]

    L. A. Bunimovich and A. Grigo. Focusing components in typical chaotic billiards should be absolutely focusing.Comm. Math. Phys., 293(1):127–143, 2010

    2010

  7. [14]

    Buzzi, S

    J. Buzzi, S. Crovisier, and T. Fisher. Local perturbations of conservative c1 diffeomorphisms. Nonlinearity, 30(9):3613, aug 2017

    2017

  8. [15]

    J. Cheng. Variational approach to homoclinic orbits in twist maps and an application to billiard systems.Z. Angew. Math. Phys., 55(3):400–419, 2004

    2004

  9. [16]

    Contreras

    G. Contreras. Geodesic flows with positive topological entropy, twist maps and hyperbolicity. Annals of Mathematics, 172(2):761–808, 2010. 25

    2010

  10. [17]

    M. J. Dias Carneiro, S. Oliffson Kamphorst, and S. Pinto-de Carvalho. Elliptic islands in strictly convex billiards.Ergodic Theory and Dynamical Systems, 23(3):799–812, 2003

    2003

  11. [18]

    M. J. Dias Carneiro, S. Oliffson Kamphorst, and S. Pinto-de Carvalho. Periodic orbits of generic oval billiards.Nonlinearity, 20(10):2453–2462, 2007

    2007

  12. [19]

    V. J. Donnay. Creating transverse homoclinic connections in planar billiards.Journal of Mathematical Sciences, 128(2):2747–2753, 2005

    2005

  13. [20]

    Fierobe, A

    C. Fierobe, A. Sorrentino, and A. Vig. Deformational spectral rigidity of axially-symmetric symplectic billiards. Preprint, arXiv:2410.13777 [math.DS] (2024), 2024

    2024 arXiv

  14. [21]

    J. Franks. Necessary conditions for stability of diffeomorphisms.Transactions of the American Mathematical Society, 158(2):301–308, 1971

    1971

  15. [22]

    Golubitsky and V

    M. Golubitsky and V. Guillemin.Stable mappings and their singularities, volume Vol. 14 of Graduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1973

    1973

  16. [23]

    P. M. Gruber. Convex billiards.Geometriae Dedicata, 33(2):205–226, 1990

    1990

  17. [24]

    Katok and B

    A. Katok and B. Hasselblatt.Introduction to the Modern Theory of Dynamical Systems, volume 54 ofEncyclopedia of Mathematics and its Applications. Cambridge University Press, 1997

    1997

  18. [25]

    V. F. Lazutkin. Convex billiard and eigenfunctions of the laplace operator.Leningrad Univ., Leningrad, 1981

    1981

  19. [26]

    R. Mañé. An ergodic closing lemma.Annals of Mathematics, 116(3):503–540, 1982

    1982

  20. [27]

    Moser.Stable and random motions in dynamical systems

    J. Moser.Stable and random motions in dynamical systems. Princeton Landmarks in Math- ematics. Princeton University Press, Princeton, NJ, 2001. With special emphasis on celestial mechanics, Reprint of the 1973 original, With a foreword by Philip J. Holmes

    2001

  21. [28]

    Petkov and L

    V. Petkov and L. Stojanov. Periods of multiple reflecting geodesics and inverse spectral results.American Journal of Mathematics, 109(4):619–668, 1987

    1987

  22. [29]

    Petkov and L

    V. Petkov and L. Stojanov. Spectrum of the poincaré map for periodic reflecting rays in generic domains.Mathematische Zeitschrift, 194(4):505–518, 1987

    1987

  23. [30]

    Clark Robinson

    R. Clark Robinson. Generic properties of conservative systems.American Journal of Mathe- matics, 92(3):562–603, 1970

    1970

  24. [31]

    H. Rüssman. On the existence of invariant curves of twist mappings of an annulus. In Geometric dynamics (Rio de Janeiro, 1981), volume 1007 ofLecture Notes in Math., pages 677–718. Springer, Berlin, 1983

    1981

  25. [32]

    K. F. Siburg.The principle of least action in geometry and dynamics.Lecture Notes in Mathematics. Springer Berlin, Heidelberg, 2004

    2004

  26. [33]

    Stojanov

    L. Stojanov. Generic properties of periodic reflecting rays.Ergodic Theory Dynam. Systems, 7(4):597–609, 1987

    1987

  27. [34]

    Tabachnikov.Geometry and billiards, volume 30

    S. Tabachnikov.Geometry and billiards, volume 30. American Mathematical Society (AMS), 2005

    2005

  28. [35]

    Treves.Analytic partial differential equations, volume 359 ofGrundlehren der mathema- tischen Wissenschaften [Fundamental Principles of Mathematical Sciences]

    F. Treves.Analytic partial differential equations, volume 359 ofGrundlehren der mathema- tischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Cham, [2022]©2022

    2022

  29. [36]

    Tsodikovich

    D. Tsodikovich. Local rigidity for symplectic billiards.J. Geom. Anal., 35(10):26, 2025. Id/No 306

    2025

  30. [37]

    Afranks’lemmaforconvexplanarbilliards.Dynamical Systems

    D.Visscher. Afranks’lemmaforconvexplanarbilliards.Dynamical Systems. An International Journal, 30(3):333–340, 2015

    2015

Pith tools

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