Non-Archimedean Techniques and Dynamical Degenerations

Abstract

We develop non-Archimedean techniques to analyze the degeneration of a sequence of rational maps of the complex projective line. We provide an alternative to Luo’s method which was based on ultra-limits of the hyperbolic 3-space. We build hybrid spaces using Berkovich theory which enable us to prove the convergence of equilibrium measures, and to determine the asymptotics of Lyapunov exponents.

Keywords

Degeneration of rational maps / Non-standard analysis / Minimal resultant

References

[1] Baker, M., Rumely, R.: Potential Theory and Dynamics on the Berkovich Projective Line. Math. Surv. Monogr., vol. 159. American Mathematical Society, Providence, RI (2010)

[2] Beardon, A.F.: The Geometry of Discrete Groups. Grad. Texts Math., vol. 91. Springer-Verlag, New York (1983)

[3] Benedetto, R.L.: Dynamics in One Non-Archimedean Variable. Grad. Stud. Math., vol. 198. American Mathematical Society, Providence, RI (2019)

[4] Berkovich, V.G.: Spectral Theory and Analytic Geometry over Non-Archimedean Fields. Math. Surv. Monogr., vol. 33. American Mathematical Society, Providence, RI (1990)

[5] Berkovich, V.G.: A non-Archimedean interpretation of the weight zero subspaces of limit mixed Hodge structures. In: Algebra, Arithmetic, and Geometry, In honor of Yu. I. Manin, Prog. Math., vol. I, pp. 49–67. Birkhäuser, Boston, MA (2009)

[6] Berteloot, F.: Bifurcation currents in holomorphic families of rational maps. In: Pluripotential Theory. Lect. Notes Math., vol. 2075, pp. 1–93. Springer, Heiderberg (2013)

[7] Bosch, S., Güntzer, U., Remmert, R.: Non-archimedean Analysis, A Systematic Approach to Rigid Analytic Geometry. Grundlehren Math. Wiss., vol. 261. Springer-Verlag, Berlin (1984)

[8] BoucksomS, FavreC, JonssonM. Solution to a non-Archimedean Monge–Ampère equation. J. Am. Math. Soc., 2015, 283617-667.

[9] BoucksomS, FavreC, JonssonM. Singular semipositive metrics in non-Archimedean geometry. J. Algebr. Geom., 2016, 25177-139.

[10] BoucksomS, GublerW, MartinF. Differentiability of relative volumes over an arbitrary non-Archimedean field. Int. Math. Res. Not., 2022, 202286214-6242.

[11] BoucksomS, JonssonM. Tropical and non-Archimedean limits of degenerating families of volume forms. J. Éc. Polytech. Math., 2017, 4: 87-139.

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

[13] Comfort , W.W., Negrepontis, S.: The Theory of Ultrafilters. Grundlehren Math. Wiss., vol. 211. Springer-Verlag, New York (1974)

[14] Dang, N.-B., Mehmeti, V.: Variation of the Hausdorff dimension and degenerations of Schottky groups. arXiv:2401.06107 (2024)

[15] De Marco, L., Faber, X.: Degenerations of complex dynamical systems. Forum Math. Sigma 2, Paper No. e6, 36 pp.

[16] Demailly, J.-P.: Complex Analytic and Differential Geometry. https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf (2012)

[17] Demailly, J.-P.: On the Ohsawa–Takegoshi–Manivel L2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^2$$\end{document} extension theorem. In: Complex Analysis and Geometry (Paris, 1997), Progr. Math., vol. 188, pp. 47–82. Birkhäuser, Basel (2000)

[18] DeMarcoL. Iteration at the boundary of the space of rational maps. Duke Math. J., 2005, 1301169-197.

[19] DeMarcoL. The moduli space of quadratic rational maps. J. Am. Math. Soc., 2007, 202321-355.

[20] DeMarco, L., Faber, X.: Degenerations of complex dynamical systems II: analytic and algebraic stability. Math. Ann. 365(3–4), 1669–1699 (2016)

[21] DeMarcoL, KriegerH, YeH. Uniform Manin–Mumford for a family of genus 2 curves. Ann. Math. (2), 2020, 1913949-1001.

[22] DeMarcoLG, McMullenCT. Trees and the dynamics of polynomials. Ann. Sci. Éc. Norm. Supér. (4), 2008, 413337-383.

[23] Donaldson, S.: Riemann Surfaces. Oxford Grad. Texts Math, vol. 22. Oxford University Press, Oxford (2011)

[24] DucrosA, HrushovskiE, LoeserF. Non-Archimedean integrals as limits of complex integrals. Duke Math. J., 2023, 1722313-386.

[25] DujardinR, FavreC. Degenerations of SL(2,C)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\rm SL}} (2, {\mathbb{C} })$$\end{document} representations and Lyapunov exponents. Ann. Henri Lebesgue, 2019, 2: 515-565.

[26] EpsteinAL. Bounded hyperbolic components of quadratic rational maps. Ergod. Theory Dyn. Syst., 2000, 203727-748.

[27] FavreC. Degeneration of endomorphisms of the complex projective space in the hybrid space. J. Inst. Math. Jussieu, 2020, 1941141-1183.

[28] Favre, C., Gauthier, T.: The Arithmetic of Polynomial Dynamical Pairs. Ann. Math. Stud., vol. 214. Princeton University Press, Princeton, NJ (2022)

[29] Favre, C., Jonsson, M.: The Valuative Tree. Lect. Notes Math., vol. 1853. Springer, Berlin (2004)

[30] FavreC, KiwiJ, TruccoE. A non-Archimedean Montel’s theorem. Compos. Math., 2012, 1483966-990.

[31] FavreC, Rivera-LetelierJ. Quantitative uniform distribution of points of small height on the projective line (in French). Math. Ann., 2006, 3352311-361.

[32] FavreC, Rivera-LetelierJ. Ergodic theory of rational maps over an ultrametric field (in French). Proc. Lond. Math. Soc. (3), 2010, 1001116-154.

[33] GauthierT, OkuyamaY, VignyG. Approximation of non-Archimedean Lyapunov exponents and applications over global fields. Trans. Am. Math. Soc., 2020, 373128963-9011.

[34] GrauertH, RemmertR. Plurisubharmonische Funktionen in komplexen Räumen. Math. Z., 1956, 65: 175-194.

[35] GublerW. Local heights of subvarieties over non-Archimedean fields. J. Reine Angew. Math., 1998, 498: 61-113.

[36] Hewitt, E., Stromberg, K.: Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable, 3rd printing. Grad. Texts Math., vol. 25. Springer-Verlag, New York (1975)

[37] Irokawa, R.: Hybrid dynamics of Hénon mappings. arXiv:2212.10851v2 (2023)

[38] Irokawa, R.: Hybrid dynamics of hyperbolic automorphisms of K3 surfaces. arXiv:2405.12517 (2024)

[39] JacobsK. Hyperbolic equivariants of rational maps. Int. Math. Res. Not., 2023, 202311-53.

[40] Ji, Z., Xie, J.: Homoclinic orbits, multiplier spectrum and rigidity theorems in complex dynamics. Forum Math. Pi 11, Paper No. e11, 37 pp. (2023)

[41] Jonsson, M.: Dynamics on Berkovich spaces in low dimensions. In: Berkovich Spaces and Applications, Lect. Notes Math., vol. 2119, pp. 205–366. Springer, Cham (2015)

[42] Kapovich, M.: Hyperbolic manifolds and discrete groups. Reprint of the 2001 Edition. Mod. Birkhäuser Class. Birkhäuser, Boston, MA (2009)

[43] KiwiJ. Puiseux series polynomial dynamics and iteration of complex cubic polynomials. Ann. Inst. Fourier, 2006, 5651337-1404.

[44] Kiwi, J.: Puiseux series dynamics of quadratic rational maps. Isr. J. Math. 201(2), 631–700 (2014)

[45] KiwiJ. Rescaling limits of complex rational maps. Duke Math. J., 2015, 16471437-1470.

[46] KiwiJ, NieH. Indeterminacy loci of iterate maps in moduli space. Indiana Univ. Math. J., 2023, 723969-1026.

[47] Lemanissier, T., Poineau, J.: Espaces de Berkovich globaux. Catégorie, topologie, cohomologie. Progr. Math., vol. 353. Birkhäuser, Cham (2024)

[48] LevyA. The space of morphisms on projective space. Acta Arith., 2011, 146113-31.

[49] Lightstone, A.H., Robinson, A.: Non-Archimedean Fields and Asymptotic Expansions, North-Holland Math. Libr., vol. 13. North-Holland, Amsterdam (1975)

[50] LuoY . Limits of rational maps, R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb{R} }$$\end{document}-trees and barycentric extension. Adv. Math., 2021, 393: 108075, 46.

[51] LuoY. Trees, length spectra for rational maps via barycentric extensions, and Berkovich spaces. Duke Math. J., 2022, 171142943-3001.

[52] Maculan, M.: Diophantine Applications of Geometric Invariant Theory. Mém. Soc. Math. Fr. (N.S.) 152, 149 pp. (2017)

[53] MiasnikovN, StoutB, WilliamsP. Automorphism loci for the moduli space of rational maps. Acta Arith., 2017, 1803267-296.

[54] MilnorJ . Geometry and dynamics of quadratic rational maps (with an Appendix by J. Milnor and Tan Lei). Exp. Math., 1993, 2137-83.

[55] Milnor, J.: Dynamics in One Complex Variable, 3rd edn. Ann. Math. Stud., vol. 160. Princeton University Press, Princeton, NJ (2006)

[56] Morgan, J.W., Shalen, P.B.: An introduction to compactifying spaces of hyperbolic structures by actions on trees. In: Geometry and Topology (College Park, Md., 1983–1984), Lect. Note Math., vol. 1167, pp. 228–240

[57] Mumford, D., Fogarty, J.: Geometric Invariant Theory, 2nd edn. Ergeb. Math. Grenzgeb., vol. 34. Springer, Berlin (1982)

[58] NieH, PilgrimKM. Bounded hyperbolic components of bicritical rational maps. J. Mod. Dyn., 2022, 18: 533-553.

[59] Novacoski, J: Valuations centered at a two-dimensional regular local ring: infima and topologies. arXiv:1205.5625 (2012)

[60] OkuyamaY. Quantitative approximations of the Lyapunov exponent of a rational function over valued fields. Math. Z., 2015, 2803–4691-706.

[61] Petersen, C.L.: Conformally natural extensions revisited. arXiv:1102.1470 (2011)

[62] Poineau, J.: La droite de Berkovich sur Z\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\bf Z}$$\end{document}, Astérisque 334, viii+xii+284 pp. (2010)

[63] PoineauJ. Berkovich spaces are angelic (in French). Bull. Soc. Math. Fr., 2013, 1412267-297.

[64] Poineau, J.: Berkovich spaces over Z\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb{Z}}$$\end{document}: local study (in French). Invent. Math. 194(3), 535–590 (2013)

[65] Poineau, J.: Dynamique analytique sur Z\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{Z}$$\end{document}, I: Mesures d’équilibre sur une droite projective relative. arXiv:2201.08480v4 (2024)

[66] Poineau, J.: Dynamique analytique sur Z\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{Z}$$\end{document}, II: Écart uniforme entre Lattès et conjecture de Bogomolov–Fu–Tschinkel. arXiv:2207.01574v3 (2024)

[67] Rudin, W.: Real and Complex Analysis, 2nd edn. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York (1974)

[68] RumelyR. The minimal resultant locus. Acta Arith., 2015, 1693251-290.

[69] SilvermanJH. The space of rational maps on P1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\textbf{P}}^1$$\end{document}. Duke Math. J., 1998, 94141-77.

[70] Silverman, J.H.: The Arithmetic of Dynamical Systems. Grad. Texts Math., Vol. 241. Springer, New York (2007)

[71] StevensonM. A non-Archimedean Ohsawa–Takegoshi extension theorem. Math. Z., 2019, 2911–2279-302.

[72] Szpiro, L., Tepper, M., Williams, P.: Resultant and conductor of geometrically semi-stable self maps of the projective line over a number field or function field. Publ. Mat. 58(2), 295–329 (2014)

[73] Thuillier, A.: Théorie du potentiel sur les courbes en géométrie analytique non archim édienne, Applications à la théorie d’Arakelov. Ph.D. thesis, Université Rennes 1 (2005)

Funding

China Sponsorship Council(SC-202108070159)

RIGHTS & PERMISSIONS

Peking University

Just Accepted

This article has successfully passed peer review and final editorial review, and will soon enter typesetting, proofreading and other publishing processes. The currently displayed version is the accepted final manuscript. The officially published version will be updated with format, DOI and citation information upon launch. We recommend that you pay attention to subsequent journal notifications and preferentially cite the officially published version. Thank you for your support and cooperation.