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DEGENERATION OF ENDOMORPHISMS OF THE COMPLEX PROJECTIVE SPACE IN THE HYBRID SPACE

Part of: Complex dynamical systems Classical measure theory Non-Archimedean analysis Holomorphic mappings and correspondences

Published online by Cambridge University Press:  31 August 2018

Charles Favre
Charles Favre*
Affiliation:
CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France (charles.favre@polytechnique.edu)
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Abstract

We consider a meromorphic family of endomorphisms of degree at least 2 of a complex projective space that is parameterized by the unit disk. We prove that the measure of maximal entropy of these endomorphisms converges to the equilibrium measure of the associated non-Archimedean dynamical system when the system degenerates. The convergence holds in the hybrid space constructed by Berkovich and further studied by Boucksom and Jonsson. We also infer from our analysis an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of endomorphisms.

Keywords

hybrid spacesequilibrium measuresendomorphisms of the complex projective planeholomorphic families of endomorphisms

MSC classification

Primary: 37F45: Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations
Secondary: 32P05: Non-Archimedean analysis (should also be assigned at least one other classification number from Section 32 describing the type of problem) 28A33: Spaces of measures, convergence of measures 32H50: Iteration problems
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 4 , July 2020 , pp. 1141 - 1183
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Copyright
© Cambridge University Press 2018

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Footnotes

The author is supported by the ERC-starting grant project ‘Nonarcomp’ no. 307856, and by the Brazilian project ‘Ciência sem fronteiras’ founded by the CNPq.

References

Barlet, D., Développement asymptotique des fonctions obtenues par intégration sur les fibres, Invent. Math. 68 (1982), 129–174.Google Scholar
Bassanelli, G. and Berteloot, F., Bifurcation currents in holomorphic dynamics on ℙk, J. Reine Angew. Math. 608 (2007), 201–235.Google Scholar
Bedford, E. and Jonsson, M., Dynamics of regular polynomial endomorphisms of Ck, Amer. J. Math. 122 (2000), 153–212.Google Scholar
Bedford, E. and Taylor, B. A., The Dirichlet problem for the complex Monge–Ampère equation, Invent. Math. 37 (1976), 1–44.Google Scholar
Berkovich, V. G., Spectral Theory and Analytic Geometry over Non-Archimedean Fields, Mathematical Surveys and Monographs, Volume 33, (American Mathematical Society, Providence, RI, 1990).Google Scholar
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, Vol. I, Progress in Mathematics, Volume 269, pp. 49–67 (Birkhäuser Boston, Inc., Boston, MA, 2009).Google Scholar
Boucksom, S., Favre, C. and Jonsson, M., Solution to a non-Archimedean Monge–Ampère equation, J. Amer. Math. Soc. 28 (2015), 617–667.Google Scholar
Boucksom, S., Favre, C. and Jonsson, M., Singular semipositive metrics in non-Archimedean geometry, J. Algebraic Geom. 25 (2016), 77–139.Google Scholar
Boucksom, S., Favre, C. and Jonsson, M., The non-Archimedean Monge–Ampère equation, in Nonarchimedean and Tropical Geometry. Simons Symposia (ed. Baker, M. and Payne, S.), (Springer, Cham, 2016).Google Scholar
Boucksom, S. and Jonsson, M., Tropical and non-Archimedean limits of degenerating families of volume forms, Journal de l’École polytechnique – Mathématiques 4 (2017), 87–139.Google Scholar
Briend, J.-Y. and Duval, J., Exposants de Liapounoff et distribution des points périodiques d’un endomorphisme de CPk, Acta Math. 182 (1999), 143–157.Google Scholar
Briend, J.-Y. and Duval, J., Deux caractérisations de la mesure d’équilibre d’un endomorphisme de ℙk(ℂ), Publ. Math. Inst. Hautes Études Sci. 93 (2001), 145–159. Erratum. Publ. Math. Inst. Hautes Études Sci. 109 (2009), 295–296.Google Scholar
Burgos Gil, J. I., Gubler, W., Jell, P., Künnemann, K. and Martin, F., Differentiability of non-Archimedean volumes and non-Archimedean Monge–Ampère equations. arXiv:1608.01919.Google Scholar
Chambert-Loir, A., Mesures et équidistribution sur des espaces de Berkovich, J. Reine Angew. Math. 595 (2006), 215–235.Google Scholar
Chambert-Loir, A., Heights and measures on analytic spaces. A survey of recent results, and some remarks, in Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry: Volume II (ed. Cluckers, R., Nicaise, J. and Sebag, J.), (Cambridge University Press, Cambridge, 2011).Google Scholar
Chambert-Loir, A. and Thuillier, A., Mesures de Mahler et équidistribution logarithmique, Ann. Inst. Fourier 59 (2009), 977–1014.Google Scholar
Conrad, B., Relative ampleness in rigid geometry, Ann. Inst. Fourier 56(4) (2006), 1049–1126.Google Scholar
Demailly, J.-P., Monge–Ampère operators, Lelong numbers and intersection theory, in Complex Analysis and Geometry (ed. Ancona, V. and Silva, A.), Univ. Series in Math., (Plenum Press, New-York, 1993).Google Scholar
Demailly, J.-P., $L^{2}$vanishing theorems for positive line bundles and adjunction theory. Lecture Notes of the CIME Session Transcendental Methods in Algebraic Geometry, Cetraro, Italy, July 1994 (International Press, Somerville, MA; Higher Education Press, Beijing, 2012). viii+231 pp.Google Scholar
Demailly, J.-P., Analytic Methods in Algebraic Geometry, Surveys of Modern Mathematics Series, Volume 1 (International Press, Cambridge, 2012).Google Scholar
DeMarco, L., Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity, Math. Ann. 326(1) (2003), 43–73.Google Scholar
DeMarco, L., Bifurcations, intersections, and heights, Algebra Number Theory 10 (2016), 1031–1056.Google Scholar
DeMarco, L. and Faber, X., Degenerations of complex dynamical systems, Forum Math. Sigma 2 (2014), e6, 36 pp.Google Scholar
DeMarco, L. and Ghioca, D., Rationality of dynamical canonical height. Ergodic Theory Dynam. Systems (to appear). arXiv:1602.05614.Google Scholar
DeMarco, L. and Okuyama, Y., Discontinuity of a degenerating escape rate. Conform. Geom. Dyn. (to appear). arXiv:1710.01660.Google Scholar
DeMarco, L. G. and McMullen, C. T., Trees and the dynamics of polynomials, Ann. Sci. Éc. Norm. Supér. (4) 41(3) (2008), 337–382.Google Scholar
Di Nezza, E. and Favre, C., Regularity of push-forwards of Monge–Ampère measures. Prepublication. hal-01672332.Google Scholar
Dinh, T. C. and Sibony, N., Dynamique des applications d’allure polynomiale, J. Math. Pures et Appl. 82 (2003), 367–423.Google Scholar
Dujardin, R. and Favre, C., Degenerations of $\text{SL}(2,\mathbb{C})$ representations and Lyapunov exponents. Prepublication. hal-01736453.Google Scholar
Favre, C. and Gauthier, T., Distribution of postcritically finite polynomials, Israel J. Math. 209(1) (2015), 235–292.Google Scholar
Favre, C. and Gauthier, T., Continuity of the Green function in meromorphic families of polynomials. Algebra Number Theory (to appear). arXiv:1706.04676.Google Scholar
Favre, C. and Rivera-Letelier, J., Théorie ergodique des fractions rationnelles sur un corps ultramétrique, Proc. Lond. Math. Soc. (3) 100(1) (2010), 116–154.Google Scholar
Favre, C. and Rivera-Letelier, J., Expansion et entropie en dynamique non-archimédienne. In preparation.Google Scholar
Fornæss, J. E. and Sibony, N., Oka’s inequality for currents and applications, Math. Ann. 301(3) (1995), 399–419.Google Scholar
Fulton, W., Intersection Theory, 2nd ed. (Springer, New York, NY, 1998).Google Scholar
Gauthier, T., Okuyama, Y. and Vigny, G., Hyperbolic components of rational maps: quantitative equidistribution and counting. arXiv:1705.05276.Google Scholar
Ghioca, D. and Ye, H., The dynamical André–Oort conjecture for cubic polynomials IMRN (to appear). arXiv:1603.05303.Google Scholar
Grauert, H. and Remmert, R., Bilder und Urbilder analytischer Garben (German), Ann. of Math. (2) 68 (1958), 393–443.Google Scholar
Griffiths, P. and Harris, J., Principles of Algebraic Geometry (Wiley, New York, 1978).Google Scholar
Gubler, W., Local heights of subvarieties over non-Archimedean fields, J. Reine Angew. Math. 498 (1998), 61–113.Google Scholar
Gubler, W. and Martin, F., On Zhang’s semipositive metrics. arXiv:1608.08030.Google Scholar
Jacobs, K., A lower bound for non-Archimedean Lyapunov exponents. Trans. AMS (to appear). arXiv:1510.02440.Google Scholar
Kiwi, J., Puiseux series polynomial dynamics and iteration of complex cubic polynomials, Ann. Inst. Fourier (Grenoble) 56(5) (2006), 1337–1404.Google Scholar
Morgan, J. W. and Shalen, P. B., An introduction to compactifying spaces of hyperbolic structures by actions on trees, in Geometry and Topology (College Park, MD, 1983/84), Lecture Notes in Mathematics, Volume 1167, pp. 228–240 (Springer, Berlin, 1985).Google Scholar
Nakayama, N., The lower-semi continuity of the plurigenera of complex varieties, in Algebraic Geometry, Sendai 1985 (ed. Oda, T.), Advanced Studies in Pure Mathematics, Volume 10, (North-Holland, Amsterdam, 1987).Google Scholar
Nakayama, N., Zariski decomposition and abundance. MSJ memoir 14 (2004).Google Scholar
Nicaise, J., Berkovich skeleta and birational geometry, in Nonarchimedean and Tropical Geometry (ed. Baker, M. and Payne, S.), Simons Symposia, pp. 179–200 (2016).Google Scholar
Norguet, F., Images de faisceaux analytiques cohérents (d’après H. Grauert et R. Remmert). (French) 1959 Séminaire P. Lelong, 1957/58 exp. 11, 17 pp. Faculté des Sciences de Paris.Google Scholar
Okuyama, Y., Repelling periodic points and logarithmic equidistribution in non-Archimedean dynamics, Acta Arith. 152(3) (2012), 267–277.Google Scholar
Okuyama, Y., Quantitative approximations of the Lyapunov exponent of a rational function over valued fields, Math. Z. 280(3) (2015), 691–706.Google Scholar
Poineau, J., La droite de Berkovich sur ℤ, Astérisque 334 (2010), 45–50.Google Scholar
Poineau, J., Les espaces de Berkovich sont angéliques, Bull. de la SMF 141(2) (2013), 267–297.Google Scholar
Sibony, N., Dynamique des applications rationnelles de ℙk, Panorama et Synthèses, Volume 8 (Soc. Math. France, Paris, 1999).Google Scholar
Stoll, W., The continuity of the fiber integral, Math. Z. 95(2) (1966), 87–138.Google Scholar
Zhang, S.-W., Small points and adelic metrics, J. Algebraic Geom. 4(2) (1995), 281–300.Google Scholar