On the complex dynamics of birational surface maps defined over number fields
-
Mattias Jonsson
and
Paul Reschke
Abstract
We show that any birational selfmap of a complex projective surface that has dynamical degree greater than one and is defined over a number field automatically satisfies the Bedford–Diller energy condition after a suitable birational conjugacy. As a consequence, the complex dynamics of the map is well behaved. We also show that there is a well-defined canonical height function.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1266207
Award Identifier / Grant number: DMS-0943832
Award Identifier / Grant number: DMS-1045119
Funding statement: The first author was supported by NSF grant DMS-1266207. The second author was supported by NSF grants DMS-0943832 and DMS-1045119.
Acknowledgements
The authors would like to thank Jeff Diller, Romain Dujardin, Joe Silverman, and the referees for useful comments.
References
[1] E. Amerik, Existence of non-preperiodic algebraic points for a rational self-map of infinite order, Math. Res. Lett. 18 (2011), 251–256. 10.4310/MRL.2011.v18.n2.a5Search in Google Scholar
[2] E. Bedford, On the dynamics of birational mappings of the plane, J. Korean Math. Soc. 40 (2003), 373–390. 10.4134/JKMS.2003.40.3.373Search in Google Scholar
[3] E. Bedford and J. Diller, Energy and invariant measures for birational surface maps, Duke Math. J. 128 (2005), 331–368. 10.1215/S0012-7094-04-12824-6Search in Google Scholar
[4] E. Bedford and J. Diller, Dynamics of a two parameter family of plane birational maps: Maximal entropy, J. Geom. Anal. 16 (2006), 409–430. 10.1007/BF02922060Search in Google Scholar
[5]
E. Bedford, M. Lyubich and J. Smillie,
Distribution of periodic points of polynomial diffeomorphisms of
[6]
E. Bedford, M. Lyubich and J. Smillie,
Polynomial diffeomorphisms of
[7]
E. Bedford and J. Smillie,
Polynomial diffeomorphisms of
[8]
E. Bedford and J. Smillie,
Polynomial diffeomorphisms of
[9]
E. Bedford and J. Smillie,
Polynomial diffeomorphisms of
[10] E. Bombieri and W. Gubler, Heights in Diophantine geometry, Cambridge University Press, Cambridge 2006. Search in Google Scholar
[11] S. Boucksom, C. Favre and M. Jonsson, Degree growth of meromorphic surface maps, Duke Math. J. 141 (2008), 519–538. 10.1215/00127094-2007-004Search in Google Scholar
[12] X. Buff, Courants dynamiques pluripolaires, Ann. Fac. Sci. Toulouse Math. (6) 20 (2011), 203–214. 10.5802/afst.1290Search in Google Scholar
[13] S. Cantat, Dynamique des automorphismes des surfaces K3, Acta Math. 187 (2001), 1–57. 10.1007/BF02392831Search in Google Scholar
[14] G. Call and J. Silverman, Canonical heights on varieties with morphisms, Compos. Math. 89 (1993), 163–205. Search in Google Scholar
[15] H. De Thélin and G. Vigny, Entropy of meromorphic maps and dynamics of birational maps, Mém. Soc. Math. Fr. (N.S.) 122 (2010). 10.24033/msmf.434Search in Google Scholar
[16]
J. Diller,
Dynamics of birational maps of
[17] J. Diller, R. Dujardin and V. Guedj, Dynamics of meromorphic maps II: Energy and invariant measure, Comment. Math. Helv. 86 (2011), 277–316. 10.4171/CMH/224Search in Google Scholar
[18] J. Diller and C. Favre, Dynamics of bimeromorphic maps of surfaces, Amer. J. Math. 123 (2001), 1135–1169. 10.1353/ajm.2001.0038Search in Google Scholar
[19] J. Diller, D. Jackson and A. Sommese, Invariant curves for birational surface maps, Trans. Amer. Math. Soc. 359 (2007), 2973–2991. 10.1090/S0002-9947-07-04162-1Search in Google Scholar
[20] T.-C. Dinh and N. Sibony, Une borne supérieure pour l’entropie topologique d’une application rationelle, Ann. of Math. (2) 161 (2005), 1637–1644. 10.4007/annals.2005.161.1637Search in Google Scholar
[21] R. Dujardin, Laminar currents and birational dynamics, Duke Math. J. 131 (2001), 219–247. 10.1215/S0012-7094-06-13122-8Search in Google Scholar
[22]
R. Dujardin,
Laminar currents in
[23]
C. Favre,
Points périodiques d’applications birationelles de
[24] C. Favre, Multiplicity of holomorphic functions, Math. Ann. 316 (2000), 355–378. 10.1007/s002080050016Search in Google Scholar
[25] S. Friedland, Entropy of polynomial and rational maps, Ann. of Math. (2) 133 (1991), 359–368. 10.2307/2944341Search in Google Scholar
[26] M. Jonsson and E. Wulcan, Canonical heights for plane polynomial maps of small topological degree, Math. Res. Lett. 19 (2012), 1207–1217. 10.4310/MRL.2012.v19.n6.a3Search in Google Scholar
[27] S. Kawaguchi, Canonical height functions for affine plane automorphisms, Math. Ann. 335 (2006), 285–310. 10.1007/s00208-006-0750-ySearch in Google Scholar
[28] S. Kawaguchi, Projective surface automorphisms of positive entropy from an arithmetic viewpoint, Amer. J. Math. 130 (2008), 159–186. 10.1353/ajm.2008.0008Search in Google Scholar
[29] S. Kawaguchi, Local and global canonical height functions for affine space regular automorphisms, Algebra Number Theory 7 (2013), 1225–1252. 10.2140/ant.2013.7.1225Search in Google Scholar
[30] S. Kawaguchi and J. H. Silverman, On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties, J. reine angew. Math. 713 (2016), 21–48. 10.1515/crelle-2014-0020Search in Google Scholar
[31] R. Lazarsfeld, Positivity in algebraic geometry I, Ergeb. Math. Grenzgeb. (3) 48, Springer, Berlin 2004. 10.1007/978-3-642-18808-4Search in Google Scholar
[32] J. Milnor, Dynamics in one complex variable, 3rd ed., Ann. of Math. Stud. 160, Princeton University Press, Princeton 2006. Search in Google Scholar
[33] A. Russakovskii and B. Shiffman, Value distribution for sequences of rational mappings and complex dynamics, Indiana Univ. Math. J. 46 (1997), 897–932. 10.1512/iumj.1997.46.1441Search in Google Scholar
[34]
N. Sibony,
Dynamique des applications rationnelles de
[35] J. H. Silverman, Rational points on K3 surfaces: A new canonical height, Invent. Math. 105 (1991), 343–373. 10.1007/BF01232270Search in Google Scholar
[36] J. H. Silverman, Integer points, Diophantine approximation and iteration of rational maps, Duke Math. J. 71 (1993), 793–829. 10.1215/S0012-7094-93-07129-3Search in Google Scholar
[37] J. H. Silverman, Geometric and arithmetic properties of the Hénon map, Math. Z. 215 (1994), 237–250. 10.1007/BF02571713Search in Google Scholar
[38] J. H. Silverman, Dynamical degrees, arithmetic entropy, and canonical heights for dominant rational self-maps of projective space, Ergodic Theory Dynam. Systems 34 (2014), 647–678. 10.1017/etds.2012.144Search in Google Scholar
[39]
G. Vigny,
Exponential decay of correlations for generic regular birational maps of
[40] J. Xie, Periodic points of birationals maps on projective surfaces, Duke Math. J. 164 (2015), 903–932. 10.1215/00127094-2877402Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- The spectral decomposition of shifted convolution sums over number fields
- Hodge theory on Cheeger spaces
- Geometric structures of collapsing Riemannian manifolds II
- Combinatorics and topology of proper toric maps
- On the long time behaviour of the conical Kähler–Ricci flows
- On invariance of plurigenera for foliations on surfaces
- Behavior of canonical divisors under purely inseparable base changes
- Algebraic nature of singular Riemannian foliations in spheres
- On the complex dynamics of birational surface maps defined over number fields
Articles in the same Issue
- Frontmatter
- The spectral decomposition of shifted convolution sums over number fields
- Hodge theory on Cheeger spaces
- Geometric structures of collapsing Riemannian manifolds II
- Combinatorics and topology of proper toric maps
- On the long time behaviour of the conical Kähler–Ricci flows
- On invariance of plurigenera for foliations on surfaces
- Behavior of canonical divisors under purely inseparable base changes
- Algebraic nature of singular Riemannian foliations in spheres
- On the complex dynamics of birational surface maps defined over number fields
