American Institute of Mathematical Sciences

Advanced
  2019, 14: 179-226. doi: 10.3934/jmd.2019007

Tropical dynamics of area-preserving maps

Simion Filip

School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA

Dedicated to the memory of Bill Veech

Received  March 26, 2018 Revised  February 07, 2019 Published  March 2019

We consider a class of area-preserving, piecewise affine maps on the 2-sphere. These maps encode degenerating families of K3 surface automorphisms and are profitably studied using techniques from tropical and Berkovich geometries.

Keywords: Area-preserving maps, tropical geometry, Berkovich geometry.
Mathematics Subject Classification: Primary: 37P50; Secondary: 37E30.
Citation: Simion Filip. Tropical dynamics of area-preserving maps. Journal of Modern Dynamics, 2019, 14: 179-226. doi: 10.3934/jmd.2019007
References:
[1]

M. Baker, An introduction to Berkovich analytic spaces and non-Archimedean potential theory on curves, in p-adic Geometry, Univ. Lecture Ser., 45, Amer. Math. Soc., Providence, RI, 2008,123–174. doi: 10.1090/ulect/045/04.  Google Scholar

[2]

M. Baker and L. DeMarco, Preperiodic points and unlikely intersections, Duke Math. J., 159 (2011), 1-29.  doi: 10.1215/00127094-1384773.  Google Scholar

[3]

V. G. Berkovich, Spectral Theory and Analytic Geometry over non-Archimedean Fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, Providence, RI, 1990.  Google Scholar

[4]

S. Boucksom and M. Jonsson, Tropical and non-Archimedean limits of degenerating families of volume forms, J. Éc. Polytech. Math., 4 (2017), 87-139.  doi: 10.5802/jep.39.  Google Scholar

[5]

E. Bedford, M. Lyubich and J. Smillie, Polynomial diffeomorphisms of C2. Ⅳ. The measure of maximal entropy and laminar currents, Invent. Math., 112 (1993), 77-125.  doi: 10.1007/BF01232426.  Google Scholar

[6]

M. Baker, S. Payne and J. Rabinoff, Nonarchimedean geometry, tropicalization, and metrics on curves, Algebr. Geom., 3 (2016), 63-105.  doi: 10.14231/AG-2016-004.  Google Scholar

[7]

M. Baker and R. Rumely, Potential Theory and Dynamics on the Berkovich Projective Line, Mathematical Surveys and Monographs, 159, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/surv/159.  Google Scholar

[8]

S. Cantat, Dynamique des automorphismes des surfaces K3, Acta Math., 187 (2001), 1-57.  doi: 10.1007/BF02392831.  Google Scholar

[9]

S. Cantat and C. Dupont, Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy, https://perso.univ-rennes1.fr/serge.cantat/Articles/smooth-final.pdf. Google Scholar

[10]

A. Chambert-Loir and A. Ducros, Formes différentielles réelles et courants sur les espaces de Berkovich, arXiv: 1204.6277, (2012). Google Scholar

[11]

A. Chambert-Loir, Heights and measures on analytic spaces. A survey of recent results, and some remarks, in Motivic Integration and Its Interactions with Model Theory and NonArchimedean Geometry. Volume II, London Math. Soc. Lecture Note Ser., 384, Cambridge Univ. Press, Cambridge, 2011, 1–50.  Google Scholar

[12]

L. DeMarco, Dynamics of rational maps: A current on the bifurcation locus, Math. Res. Lett., 8 (2001), 57-66.  doi: 10.4310/MRL.2001.v8.n1.a7.  Google Scholar

[13]

M. Einsiedler, M. Kapranov and D. Lind, Non-Archimedean amoebas and tropical varieties, J. Reine Angew. Math., 601 (2006), 139-157.  doi: 10.1515/CRELLE.2006.097.  Google Scholar

[14]

C. Favre, Degeneration of endomorphisms of the complex projective space in the hybrid space, J. Inst. Math. Jussieu, to appear. doi: 10.1017/S147474801800035X.  Google Scholar

[15]

C. Favre and M. Jonsson, The Valuative Tree, Lecture Notes in Mathematics, 1853, Springer-Verlag, Berlin, 2004. doi: 10.1007/b100262.  Google Scholar

[16]

C. Favre and J. Rivera-Letelier, Théorie ergodique des fractions rationnelles sur un corps ultramétrique, Proc. Lond. Math. Soc. (3), 100 (2010), 116–154. doi: 10.1112/plms/pdp022.  Google Scholar

[17]

S. Filip, Counting special Lagrangian fibrations in twistor families of K3 surfaces, Ann. ENS, to appear. Google Scholar

[18]

S. Filip and V. Tosatti, Smooth and rough positive currents, Annales de l'Institut Fourier, to appear. Google Scholar

[19]

S. Filip and V. Tosatti, Kummer rigidity for K3 surface automorphisms via Ricci-flat metrics, arXiv: 1808.08673, (2018). Google Scholar

[20]

R. L. Foote, Differential geometry of real Monge-Ampère foliations, Math. Z., 194 (1987), 331-350.  doi: 10.1007/BF01162241.  Google Scholar

[21] W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, 131, The William H. Roever Lectures in Geometry, Princeton University Press, Princeton, NJ, 1993.  doi: 10.1515/9781400882526.  Google Scholar
[22]

W. Gubler, Forms and current on the analytification of an algebraic variety (after Chambert-Loir and Ducros), in Nonarchimedean and Tropical Geometry, Simons Symp., Springer, [Cham], 2016, 1–30.  Google Scholar

[23]

P. Hartman and L. Nirenberg, On spherical image maps whose Jacobians do not change sign, Amer. J. Math., 81 (1959), 901-920.  doi: 10.2307/2372995.  Google Scholar

[24]

L. Hörmander, Notions of Convexity, Progress in Mathematics, 127, Birkhäuser Boston, Inc., Boston, MA, 1994.  Google Scholar

[25]

J. H. Hubbard and P. Papadopol, Superattractive fixed points in Cn, Indiana Univ. Math. J., 43 (1994), 321-365.  doi: 10.1512/iumj.1994.43.43014.  Google Scholar

[26]

I. Itenberg, G. Mikhalkin and E. Shustin, Tropical Algebraic Geometry, Second edition, Oberwolfach Seminars, 35, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-0346-0048-4.  Google Scholar

[27]

E. Katz, H. Markwig and T. Markwig, The tropical j-invariant, LMS J. Comput. Math., 12 (2009), 275-294.  doi: 10.1112/S1461157000001522.  Google Scholar

[28]

M. Kontsevich and Y. Soibelman, Homological mirror symmetry and torus fibrations, in Symplectic Geometry and Mirror Symmetry (Seoul, 2000), World Sci. Publ., River Edge, NJ, 2001,203–263. doi: 10.1142/9789812799821_0007.  Google Scholar

[29]

M. Kontsevich and Y. Soibelman, Affine structures and non-Archimedean analytic spaces, in The Unity of Mathematics, Progr. Math., 244, Birkhäuser Boston, Boston, MA, 2006,321–385. doi: 10.1007/0-8176-4467-9_9.  Google Scholar

[30]

A. Lagerberg, $L^2$-estimates for the $d$-operator acting on super forms, arXiv: 1109.3983, (2011). Google Scholar

[31]

A. Lagerberg, Super currents and tropical geometry, Math. Z., 270 (2012), 1011-1050.  doi: 10.1007/s00209-010-0837-8.  Google Scholar

[32]

R. Lozi, Un attracteur étrange du type attracteur de Hénon, Journal de Physique Colloques, 39 (1978), C5-9–C5-10. doi: 10.1051/jphyscol:1978505.  Google Scholar

[33]

D. Maclagan and B. Sturmfels, Introduction to tropical geometry, Graduate Studies in Mathematics, 161, American Mathematical Society, Providence, RI, 2015.  Google Scholar

[34]

C. T. McMullen, Dynamics on K3 surfaces: Salem numbers and Siegel disks, J. Reine Angew. Math., 545 (2002), 201-233.  doi: 10.1515/crll.2002.036.  Google Scholar

[35]

M. Mustaţă and J. Nicaise, Weight functions on non-Archimedean analytic spaces and the Kontsevich-Soibelman skeleton, Algebr. Geom., 2 (2015), 365-404.  doi: 10.14231/AG-2015-016.  Google Scholar

[36]

S. Payne, Analytification is the limit of all tropicalizations, Math. Res. Lett., 16 (2009), 543-556.  doi: 10.4310/MRL.2009.v16.n3.a13.  Google Scholar

[37]

P. Ramachandran and G. Varoquaux, Mayavi: 3D Visualization of Scientific Data, Computing in Science & Engineering, 13 (2011), 40-51.   Google Scholar

[38]

K. Spalding and A. P. Veselov, Tropical Markov dynamics and Cayley cubic, arXiv: 1707.01760, (2017). Google Scholar

[39]

The Sage Developers, SageMath, the Sage Mathematics Software System (Version 8.4), 2018. Available from: https://www.sagemath.org. Google Scholar

show all references

References:
[1]

M. Baker, An introduction to Berkovich analytic spaces and non-Archimedean potential theory on curves, in p-adic Geometry, Univ. Lecture Ser., 45, Amer. Math. Soc., Providence, RI, 2008,123–174. doi: 10.1090/ulect/045/04.  Google Scholar

[2]

M. Baker and L. DeMarco, Preperiodic points and unlikely intersections, Duke Math. J., 159 (2011), 1-29.  doi: 10.1215/00127094-1384773.  Google Scholar

[3]

V. G. Berkovich, Spectral Theory and Analytic Geometry over non-Archimedean Fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, Providence, RI, 1990.  Google Scholar

[4]

S. Boucksom and M. Jonsson, Tropical and non-Archimedean limits of degenerating families of volume forms, J. Éc. Polytech. Math., 4 (2017), 87-139.  doi: 10.5802/jep.39.  Google Scholar

[5]

E. Bedford, M. Lyubich and J. Smillie, Polynomial diffeomorphisms of C2. Ⅳ. The measure of maximal entropy and laminar currents, Invent. Math., 112 (1993), 77-125.  doi: 10.1007/BF01232426.  Google Scholar

[6]

M. Baker, S. Payne and J. Rabinoff, Nonarchimedean geometry, tropicalization, and metrics on curves, Algebr. Geom., 3 (2016), 63-105.  doi: 10.14231/AG-2016-004.  Google Scholar

[7]

M. Baker and R. Rumely, Potential Theory and Dynamics on the Berkovich Projective Line, Mathematical Surveys and Monographs, 159, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/surv/159.  Google Scholar

[8]

S. Cantat, Dynamique des automorphismes des surfaces K3, Acta Math., 187 (2001), 1-57.  doi: 10.1007/BF02392831.  Google Scholar

[9]

S. Cantat and C. Dupont, Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy, https://perso.univ-rennes1.fr/serge.cantat/Articles/smooth-final.pdf. Google Scholar

[10]

A. Chambert-Loir and A. Ducros, Formes différentielles réelles et courants sur les espaces de Berkovich, arXiv: 1204.6277, (2012). Google Scholar

[11]

A. Chambert-Loir, Heights and measures on analytic spaces. A survey of recent results, and some remarks, in Motivic Integration and Its Interactions with Model Theory and NonArchimedean Geometry. Volume II, London Math. Soc. Lecture Note Ser., 384, Cambridge Univ. Press, Cambridge, 2011, 1–50.  Google Scholar

[12]

L. DeMarco, Dynamics of rational maps: A current on the bifurcation locus, Math. Res. Lett., 8 (2001), 57-66.  doi: 10.4310/MRL.2001.v8.n1.a7.  Google Scholar

[13]

M. Einsiedler, M. Kapranov and D. Lind, Non-Archimedean amoebas and tropical varieties, J. Reine Angew. Math., 601 (2006), 139-157.  doi: 10.1515/CRELLE.2006.097.  Google Scholar

[14]

C. Favre, Degeneration of endomorphisms of the complex projective space in the hybrid space, J. Inst. Math. Jussieu, to appear. doi: 10.1017/S147474801800035X.  Google Scholar

[15]

C. Favre and M. Jonsson, The Valuative Tree, Lecture Notes in Mathematics, 1853, Springer-Verlag, Berlin, 2004. doi: 10.1007/b100262.  Google Scholar

[16]

C. Favre and J. Rivera-Letelier, Théorie ergodique des fractions rationnelles sur un corps ultramétrique, Proc. Lond. Math. Soc. (3), 100 (2010), 116–154. doi: 10.1112/plms/pdp022.  Google Scholar

[17]

S. Filip, Counting special Lagrangian fibrations in twistor families of K3 surfaces, Ann. ENS, to appear. Google Scholar

[18]

S. Filip and V. Tosatti, Smooth and rough positive currents, Annales de l'Institut Fourier, to appear. Google Scholar

[19]

S. Filip and V. Tosatti, Kummer rigidity for K3 surface automorphisms via Ricci-flat metrics, arXiv: 1808.08673, (2018). Google Scholar

[20]

R. L. Foote, Differential geometry of real Monge-Ampère foliations, Math. Z., 194 (1987), 331-350.  doi: 10.1007/BF01162241.  Google Scholar

[21] W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, 131, The William H. Roever Lectures in Geometry, Princeton University Press, Princeton, NJ, 1993.  doi: 10.1515/9781400882526.  Google Scholar
[22]

W. Gubler, Forms and current on the analytification of an algebraic variety (after Chambert-Loir and Ducros), in Nonarchimedean and Tropical Geometry, Simons Symp., Springer, [Cham], 2016, 1–30.  Google Scholar

[23]

P. Hartman and L. Nirenberg, On spherical image maps whose Jacobians do not change sign, Amer. J. Math., 81 (1959), 901-920.  doi: 10.2307/2372995.  Google Scholar

[24]

L. Hörmander, Notions of Convexity, Progress in Mathematics, 127, Birkhäuser Boston, Inc., Boston, MA, 1994.  Google Scholar

[25]

J. H. Hubbard and P. Papadopol, Superattractive fixed points in Cn, Indiana Univ. Math. J., 43 (1994), 321-365.  doi: 10.1512/iumj.1994.43.43014.  Google Scholar

[26]

I. Itenberg, G. Mikhalkin and E. Shustin, Tropical Algebraic Geometry, Second edition, Oberwolfach Seminars, 35, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-0346-0048-4.  Google Scholar

[27]

E. Katz, H. Markwig and T. Markwig, The tropical j-invariant, LMS J. Comput. Math., 12 (2009), 275-294.  doi: 10.1112/S1461157000001522.  Google Scholar

[28]

M. Kontsevich and Y. Soibelman, Homological mirror symmetry and torus fibrations, in Symplectic Geometry and Mirror Symmetry (Seoul, 2000), World Sci. Publ., River Edge, NJ, 2001,203–263. doi: 10.1142/9789812799821_0007.  Google Scholar

[29]

M. Kontsevich and Y. Soibelman, Affine structures and non-Archimedean analytic spaces, in The Unity of Mathematics, Progr. Math., 244, Birkhäuser Boston, Boston, MA, 2006,321–385. doi: 10.1007/0-8176-4467-9_9.  Google Scholar

[30]

A. Lagerberg, $L^2$-estimates for the $d$-operator acting on super forms, arXiv: 1109.3983, (2011). Google Scholar

[31]

A. Lagerberg, Super currents and tropical geometry, Math. Z., 270 (2012), 1011-1050.  doi: 10.1007/s00209-010-0837-8.  Google Scholar

[32]

R. Lozi, Un attracteur étrange du type attracteur de Hénon, Journal de Physique Colloques, 39 (1978), C5-9–C5-10. doi: 10.1051/jphyscol:1978505.  Google Scholar

[33]

D. Maclagan and B. Sturmfels, Introduction to tropical geometry, Graduate Studies in Mathematics, 161, American Mathematical Society, Providence, RI, 2015.  Google Scholar

[34]

C. T. McMullen, Dynamics on K3 surfaces: Salem numbers and Siegel disks, J. Reine Angew. Math., 545 (2002), 201-233.  doi: 10.1515/crll.2002.036.  Google Scholar

[35]

M. Mustaţă and J. Nicaise, Weight functions on non-Archimedean analytic spaces and the Kontsevich-Soibelman skeleton, Algebr. Geom., 2 (2015), 365-404.  doi: 10.14231/AG-2015-016.  Google Scholar

[36]

S. Payne, Analytification is the limit of all tropicalizations, Math. Res. Lett., 16 (2009), 543-556.  doi: 10.4310/MRL.2009.v16.n3.a13.  Google Scholar

[37]

P. Ramachandran and G. Varoquaux, Mayavi: 3D Visualization of Scientific Data, Computing in Science & Engineering, 13 (2011), 40-51.   Google Scholar

[38]

K. Spalding and A. P. Veselov, Tropical Markov dynamics and Cayley cubic, arXiv: 1707.01760, (2017). Google Scholar

[39]

The Sage Developers, SageMath, the Sage Mathematics Software System (Version 8.4), 2018. Available from: https://www.sagemath.org. Google Scholar

Figure 1.  Orbits of a hyperbolic map on three randomly constructed tropical K3 surfaces
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Pictures of currents in Rubik's cube example
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Stable and unstable currents of a perturbed Kummer example, viewed from different angles. The perturbed Kummers exhibit tangency of the stable and unstable manifolds
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Stable and unstable currents in the Kummer examples have smooth potentials and are uniformly hyperbolic
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Left: The monomials that are minimized in each region of the plane, together with the tropical elliptic curve in the $ (e_1, e_2) $-plane. Right: The dual subdivision of the Newton polytope. The Legendre transform of the function on the left determines the subdivision on the right. In the picture, all affine linear functions in the definition of $ h $ are minimized for some value of $ e_1, e_2 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  A tropical elliptic curve with the skeleton in bold and dashed reflection lines. The dotted vertical and horizontal lines denote the points where the reflection lines change slope
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  The iterate of a segment under the twist map, an analogue of §6.2 in the present case
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Left: The invariant curves of the rotation and the lines of reflection for the involutions. Right: The break lines of the function h° which determines the core pencil
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  A fundamental domain in the $(a, b)$ plane $\mathbb{R}^2$ for the $\mathbb{Z}^2$ and $\pm 1$ action, and its image under the map to $\mathbb{R}^3$. The domain is divided into $4$ triangles where the embedding is affine, with corresponding affine maps to $\mathbb{R}^3$ indicated on each triangle. The face and equations of the image tetrahedron are:
$ABC: x+y-z+1 = 0 \quad \quad BCD: -(x+y+z) + 1 = 0$
$ABD: x-y+z+1 = 0 \quad \quad ACD: -x+y+z + 1 = 0$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10.  Typical pictures at the corners of a tropical K3 surface
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 11.  Tropical K3 surfaces in the Rubik's cube family. Left: level set in $[\frac{1}{2}, 1]$. Right: level set $>1$. The surfaces are not drawn to scale, i.e. in the $\mathbb{R}^3$ that contains both, the one on the left is much smaller
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 12.  Forward (red) and backward (blue) iterates of the triangle face on the tropical K3. Left: for a small value of t. Right: for a large value of t. Figure 2 contains further examples of iterates of the triangle face for a Rubik's cube example for large t
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 13.  The lifted tent map, and its action on the section σ
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Denis Gaidashev, Tomas Johnson. Spectral properties of renormalization for area-preserving maps. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3651-3675. doi: 10.3934/dcds.2016.36.3651
[2]

Hans Koch. On hyperbolicity in the renormalization of near-critical area-preserving maps. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7029-7056. doi: 10.3934/dcds.2016106
[3]

Mário Bessa, César M. Silva. Dense area-preserving homeomorphisms have zero Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1231-1244. doi: 10.3934/dcds.2012.32.1231
[4]

Giovanni Forni. The cohomological equation for area-preserving flows on compact surfaces. Electronic Research Announcements, 1995, 1: 114-123.

[5]

Denis Gaidashev, Tomas Johnson. Dynamics of the universal area-preserving map associated with period-doubling: Stable sets. Journal of Modern Dynamics, 2009, 3 (4) : 555-587. doi: 10.3934/jmd.2009.3.555
[6]

Weiyuan Qiu, Fei Yang, Yongcheng Yin. Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3375-3416. doi: 10.3934/dcds.2016.36.3375
[7]

Daniel N. Dore, Andrew D. Hanlon. Area preserving maps on $\boldsymbol{S^2}$: A lower bound on the $\boldsymbol{C^0}$-norm using symplectic spectral invariants. Electronic Research Announcements, 2013, 20: 97-102. doi: 10.3934/era.2013.20.97
[8]

Alex L Castro, Wyatt Howard, Corey Shanbrom. Bridges between subriemannian geometry and algebraic geometry: Now and then. Conference Publications, 2015, 2015 (special) : 239-247. doi: 10.3934/proc.2015.0239
[9]

Joachim Escher, Boris Kolev, Marcus Wunsch. The geometry of a vorticity model equation. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1407-1419. doi: 10.3934/cpaa.2012.11.1407
[10]

Janina Kotus, Mariusz Urbański. The dynamics and geometry of the Fatou functions. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 291-338. doi: 10.3934/dcds.2005.13.291
[11]

Jean-Marc Couveignes, Reynald Lercier. The geometry of some parameterizations and encodings. Advances in Mathematics of Communications, 2014, 8 (4) : 437-458. doi: 10.3934/amc.2014.8.437
[12]

Yong Lin, Gábor Lippner, Dan Mangoubi, Shing-Tung Yau. Nodal geometry of graphs on surfaces. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1291-1298. doi: 10.3934/dcds.2010.28.1291
[13]

Katarzyna Grabowska, Paweƚ Urbański. Geometry of Routh reduction. Journal of Geometric Mechanics, 2019, 11 (1) : 23-44. doi: 10.3934/jgm.2019002
[14]

Len G. Margolin, Roy S. Baty. Conservation laws in discrete geometry. Journal of Geometric Mechanics, 2019, 11 (2) : 187-203. doi: 10.3934/jgm.2019010
[15]

Klas Modin. Geometry of matrix decompositions seen through optimal transport and information geometry. Journal of Geometric Mechanics, 2017, 9 (3) : 335-390. doi: 10.3934/jgm.2017014
[16]

Matteo Novaga, Enrico Valdinoci. The geometry of mesoscopic phase transition interfaces. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 777-798. doi: 10.3934/dcds.2007.19.777
[17]

Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040
[18]

Abbas Bahri. Recent results in contact form geometry. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 21-30. doi: 10.3934/dcds.2004.10.21
[19]

François Lalonde, Yasha Savelyev. On the injectivity radius in Hofer's geometry. Electronic Research Announcements, 2014, 21: 177-185. doi: 10.3934/era.2014.21.177
[20]

Manuel Gutiérrez. Lorentz geometry technique in nonimaging optics. Conference Publications, 2003, 2003 (Special) : 386-392. doi: 10.3934/proc.2003.2003.386

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (59)
  • HTML views (296)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences