American Institute of Mathematical Sciences

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  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.

[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.

[10]

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

[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.

[18]

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

[19]

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

[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).

[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. 

[38]

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

[39]

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

show all references

Dedicated to the memory of Bill Veech

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.

[10]

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

[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.

[18]

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

[19]

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

[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).

[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. 

[38]

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

[39]

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

Figure 1.  Orbits of a hyperbolic map on three randomly constructed tropical K3 surfaces
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Figure 2.  Pictures of currents in Rubik's cube example
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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
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Figure 4.  Stable and unstable currents in the Kummer examples have smooth potentials and are uniformly hyperbolic
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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 $
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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
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Figure 7.  The iterate of a segment under the twist map, an analogue of §6.2 in the present case
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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
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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:
$ABC: x+y-z+1 = 0 \quad \quad BCD: -(x+y+z) + 1 = 0$
$ABD: x-y+z+1 = 0 \quad \quad ACD: -x+y+z + 1 = 0$
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Figure 10.  Typical pictures at the corners of a tropical K3 surface
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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
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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
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Figure 13.  The lifted tent map, and its action on the section σ
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