no. G2
Article de recherche - Géométrie algébrique, Systèmes dynamiques
The ordinal of dynamical degrees of birational maps of the projective plane
[L’ordinal des degrés dynamiques des transformations birationelles du plan projectif]
Bot, Anna1
1Department of Mathematics and Computer Science, University of Basel, 4051 Basel, Switzerland
Comptes Rendus. Mathématique, Tome 362 (2024) no. G2, pp. 117-134

We show that the ordinal of the dynamical degrees of all birational maps of the complex projective plane is ω ω .

Nous démontrons que l’ordinal des degrés dynamiques de toutes les transformations birationnelles du plan projectif complexe est ω ω .

Reçu le :
Révisé le :
Accepté le :
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DOI : 10.5802/crmath.540
Classification : 37F10, 32H50, 14E07, 14E05
Keywords: Dynamical degree, rational projective surfaces, ordinals

Bot, Anna  1

1 Department of Mathematics and Computer Science, University of Basel, 4051 Basel, Switzerland
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {The ordinal of dynamical degrees  of birational maps of the projective plane},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {117--134},
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Bot, Anna. The ordinal of dynamical degrees  of birational maps of the projective plane. Comptes Rendus. Mathématique, Tome 362 (2024) no. G2, pp. 117-134. doi: 10.5802/crmath.540

[1] Bedford, Eric Dynamics of rational surface automorphisms, Holomorphic dynamical systems (Lecture Notes in Mathematics), Volume 1998, Springer, 2010, pp. 57-104 | DOI | MR

[2] Bedford, Eric; Kim, Kyounghee Dynamics of rational surface automorphisms: linear fractional recurrences, J. Geom. Anal., Volume 19 (2009) no. 3, pp. 553-583 | DOI | MR | Zbl

[3] Bellon, Marc P.; Viallet, Claude-Michel Algebraic entropy, Commun. Math. Phys., Volume 204 (1999) no. 2, pp. 425-437 | DOI | MR | Zbl

[4] Bisi, Cinzia; Calabri, Alberto; Mella, Massimiliano On plane Cremona transformations of fixed degree, J. Geom. Anal., Volume 25 (2015) no. 2, pp. 1108-1131 | DOI | MR | Zbl

[5] Blanc, Jérémy; Cantat, Serge Dynamical degrees of birational transformations of projective surfaces, J. Am. Math. Soc., Volume 29 (2016) no. 2, pp. 415-471 | DOI | MR | Zbl

[6] Blanc, Jérémy; Furter, Jean-Philippe Topologies and structures of the Cremona groups, Ann. Math., Volume 178 (2013) no. 3, pp. 1173-1198 | DOI | MR | Zbl

[7] Bourbaki, Nicolas Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer, 2002, xii+300 pages (Translated from the 1968 French original by Andrew Pressley) | DOI | MR

[8] Bourbaki, Nicolas Theory of sets, Elements of Mathematics (Berlin), Springer, 2004, viii+414 pages (Reprint of the 1968 English translation) | DOI | MR

[9] Carruth, Philip W. Arithmetic of ordinals with applications to the theory of ordered Abelian groups, Bull. Am. Math. Soc., Volume 48 (1942), pp. 262-271 | DOI | MR | Zbl

[10] Déserti, Julie; Grivaux, Julien Automorphisms of rational surfaces with positive entropy, Indiana Univ. Math. J., Volume 60 (2011) no. 5, pp. 1589-1622 | DOI | MR | Zbl

[11] Diller, Jeffrey Cremona transformations, surface automorphisms, and plain cubics, Mich. Math. J., Volume 60 (2011) no. 2, pp. 409-440 (with an appendix by Igor Dolgachev) | DOI | MR | Zbl

[12] Diller, Jeffrey; Favre, Charles Dynamics of bimeromorphic maps of surfaces, Am. J. Math., Volume 123 (2001) no. 6, pp. 1135-1169 | MR | DOI | Zbl

[13] Dolgachev, Igor; Ortland, David Point sets in projective spaces and theta functions, Astérisque, 165, Société Mathématique de France, 1988, 210 pages | MR | Numdam

[14] Gromov, Mikhaïl On the entropy of holomorphic maps, Enseign. Math., Volume 49 (2003) no. 3-4, pp. 217-235 | MR | Zbl

[15] Humphreys, James E. Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, 29, Cambridge University Press, 1990, xii+204 pages | DOI | MR

[16] McMullen, Curtis T. Dynamics on blowups of the projective plane, Publ. Math., Inst. Hautes Étud. Sci. (2007) no. 105, pp. 49-89 | DOI | MR | Zbl | Numdam

[17] Nagata, Masayoshi On rational surfaces. II, Mem. Coll. Sci., Univ. Kyoto, Ser. A, Volume 33 (1960/61), pp. 271-293 | DOI | MR | Zbl

[18] Uehara, Takato Rational surface automorphisms preserving cuspidal anticanonical curves, Math. Ann., Volume 365 (2016) no. 1-2, pp. 635-659 | DOI | MR | Zbl

[19] Uehara, Takato Rational surface automorphisms with positive entropy, Ann. Inst. Fourier, Volume 66 (2016) no. 1, pp. 377-432 | MR | DOI | Numdam | Zbl

[20] Urech, Christian Remarks on the degree growth of birational transformations, Math. Res. Lett., Volume 25 (2018) no. 1, pp. 291-308 | DOI | MR | Zbl

[21] Xie, Junyi Periodic points of birational transformations on projective surfaces, Duke Math. J., Volume 164 (2015) no. 5, pp. 903-932 | DOI | MR | Zbl

[22] Yomdin, Yosef Volume growth and entropy, Isr. J. Math., Volume 57 (1987) no. 3, pp. 285-300 | DOI | MR

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