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Tropical dynamics of area-preserving maps
School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA |
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.
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. |
[2] |
M. Baker and L. DeMarco,
Preperiodic points and unlikely intersections, Duke Math. J., 159 (2011), 1-29.
doi: 10.1215/00127094-1384773. |
[3] |
V. G. Berkovich, Spectral Theory and Analytic Geometry over non-Archimedean Fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, Providence, RI, 1990. |
[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. |
[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. |
[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. |
[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. |
[8] |
S. Cantat,
Dynamique des automorphismes des surfaces K3, Acta Math., 187 (2001), 1-57.
doi: 10.1007/BF02392831. |
[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. |
[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. |
[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. |
[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. |
[15] |
C. Favre and M. Jonsson, The Valuative Tree, Lecture Notes in Mathematics, 1853, Springer-Verlag, Berlin, 2004.
doi: 10.1007/b100262. |
[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. |
[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. |
[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.![]() ![]() |
[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. |
[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. |
[24] |
L. Hörmander, Notions of Convexity, Progress in Mathematics, 127, Birkhäuser Boston, Inc., Boston, MA, 1994. |
[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. |
[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. |
[27] |
E. Katz, H. Markwig and T. Markwig,
The tropical j-invariant, LMS J. Comput. Math., 12 (2009), 275-294.
doi: 10.1112/S1461157000001522. |
[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. |
[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. |
[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. |
[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. |
[33] |
D. Maclagan and B. Sturmfels, Introduction to tropical geometry, Graduate Studies in Mathematics, 161, American Mathematical Society, Providence, RI, 2015. |
[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. |
[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. |
[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. |
[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. |
[2] |
M. Baker and L. DeMarco,
Preperiodic points and unlikely intersections, Duke Math. J., 159 (2011), 1-29.
doi: 10.1215/00127094-1384773. |
[3] |
V. G. Berkovich, Spectral Theory and Analytic Geometry over non-Archimedean Fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, Providence, RI, 1990. |
[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. |
[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. |
[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. |
[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. |
[8] |
S. Cantat,
Dynamique des automorphismes des surfaces K3, Acta Math., 187 (2001), 1-57.
doi: 10.1007/BF02392831. |
[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. |
[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. |
[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. |
[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. |
[15] |
C. Favre and M. Jonsson, The Valuative Tree, Lecture Notes in Mathematics, 1853, Springer-Verlag, Berlin, 2004.
doi: 10.1007/b100262. |
[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. |
[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. |
[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.![]() ![]() |
[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. |
[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. |
[24] |
L. Hörmander, Notions of Convexity, Progress in Mathematics, 127, Birkhäuser Boston, Inc., Boston, MA, 1994. |
[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. |
[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. |
[27] |
E. Katz, H. Markwig and T. Markwig,
The tropical j-invariant, LMS J. Comput. Math., 12 (2009), 275-294.
doi: 10.1112/S1461157000001522. |
[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. |
[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. |
[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. |
[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. |
[33] |
D. Maclagan and B. Sturmfels, Introduction to tropical geometry, Graduate Studies in Mathematics, 161, American Mathematical Society, Providence, RI, 2015. |
[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. |
[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. |
[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. |
[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 |













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