Rational Surface Automorphisms with Positive Entropy
Annales de l'Institut Fourier, Volume 66 (2016) no. 1, p. 377-432

The aim of this paper is to construct rational surface automorphisms with positive entropy by means of the concept of orbit data. The concept enables us to introduce some mild and verifiable condition, and to show that if an orbit data satisfies the condition, then there exists an automorphism realizing the orbit data. Applying this result, we describe the set of entropy values of the rational surface automorphisms in terms of Weyl groups.

Le but de ce travail est de construire des automorphismes de surfaces rationnelles d’entropie positive au moyen de la notion de donnée d’orbite. Celle-ci nous permet d’introduire une condition faible et vérifiable, et de démontrer que si une donnée d’orbite satisfait cette condition, alors il existe un automorphisme qui réalise la donnée d’orbite. En appliquant ce résultat, nous décrivons l’ensemble des valeurs d’entropie des automorphismes de surfaces rationnelles du point de vue des groupes de Weyl.

Received : 2013-07-25
Revised : 2015-05-14
Accepted : 2015-06-11
Published online : 2016-02-17
DOI : https://doi.org/10.5802/aif.3014
Classification:  14E07,  14J50,  37F99
Keywords: rational surface, automorphism, entropy, orbit data
@article{AIF_2016__66_1_377_0,
     author = {Uehara, Takato},
     title = {Rational Surface Automorphisms  with Positive Entropy},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {1},
     year = {2016},
     pages = {377-432},
     doi = {10.5802/aif.3014},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2016__66_1_377_0}
}
Uehara, Takato. Rational Surface Automorphisms  with Positive Entropy. Annales de l'Institut Fourier, Volume 66 (2016) no. 1, pp. 377-432. doi : 10.5802/aif.3014. http://www.numdam.org/item/AIF_2016__66_1_377_0/

[1] Alberich-Carramiñana, Maria Geometry of the plane Cremona maps, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1769 (2002), xvi+257 pages | Article

[2] Beauville, Arnaud Complex algebraic surfaces, Cambridge University Press, Cambridge, London Mathematical Society Student Texts, Tome 34 (1996), x+132 pages (Translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid) | Article

[3] Bedford, Eric; Kim, Kyounghee Periodicities in linear fractional recurrences: degree growth of birational surface maps, Michigan Math. J., Tome 54 (2006) no. 3, pp. 647-670 | Article

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

[5] Cantat, Serge Dynamique des automorphismes des surfaces projectives complexes, C. R. Acad. Sci. Paris Sér. I Math., Tome 328 (1999) no. 10, pp. 901-906 | Article

[6] Diller, Jeffrey Cremona transformations, surface automorphisms, and plane cubics, Michigan Math. J., Tome 60 (2011) no. 2, pp. 409-440 (With an appendix by Igor Dolgachev) | Article

[7] Dolgachev, Igor; Ortland, David Point sets in projective spaces and theta functions, Astérisque (1988) no. 165, 210 pp. (1989) pages

[8] Gromov, Mikhaïl On the entropy of holomorphic maps, Enseign. Math. (2), Tome 49 (2003) no. 3-4, pp. 217-235

[9] Harbourne, Brian Blowings-up of P 2 and their blowings-down, Duke Math. J., Tome 52 (1985) no. 1, pp. 129-148 | Article

[10] Harbourne, Brian Rational surfaces with infinite automorphism group and no antipluricanonical curve, Proc. Amer. Math. Soc., Tome 99 (1987) no. 3, pp. 409-414 | Article

[11] Mcmullen, Curtis T. Dynamics on blowups of the projective plane, Publ. Math. Inst. Hautes Études Sci. (2007) no. 105, pp. 49-89 | Article

[12] Nagata, Masayoshi On rational surfaces. I. Irreducible curves of arithmetic genus 0 or 1, Mem. Coll. Sci. Univ. Kyoto Ser. A Math., Tome 32 (1960), pp. 351-370

[13] Nagata, Masayoshi On rational surfaces. II, Mem. Coll. Sci. Univ. Kyoto Ser. A Math., Tome 33 (1960/1961), pp. 271-293

[14] Yomdin, Y. Volume growth and entropy, Israel J. Math., Tome 57 (1987) no. 3, pp. 285-300 | Article