Laminar currents and birational dynamics
Romain Dujardin
Source: Duke Math. J. Volume 131, Number 2
(2006), 219-247.
Abstract
We study the dynamics of a bimeromorphic map $X\rightarrow X$, where $X$ is a compact complex Kähler surface. Under a natural geometric hypothesis, we construct an invariant probability measure, which is mixing, hyperbolic, and of maximal entropy. The proof relies heavily on the theory of laminar currents and is new even in the case of polynomial automorphisms of $\mathbb{C}^2$. This extends recent results by E. Bedford and J. Diller
First Page:
Show
Hide
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.dmj/1137077884
Digital Object Identifier: doi:10.1215/S0012-7094-06-13122-8
Mathematical Reviews number (MathSciNet): MR2219241
Zentralblatt MATH identifier: 1099.37037
References
N. Abarenkova, J.-Ch. AnglèS D'Auriac, S. Boukraa, S. Hassani, and J.-M. Maillard, Topological entropy and Arnold complexity for two-dimensional mappings, Phys. Lett. A 262 (1999), 44--49.
Mathematical Reviews (MathSciNet): MR1732080
Digital Object Identifier: doi:10.1016/S0375-9601(99)00662-3
Zentralblatt MATH: 0936.37005
—, Real Arnold complexity versus real topological entropy for birational transformations, J. Phys. A 33 (2000), 1465--1501.
Mathematical Reviews (MathSciNet): MR1749025
Digital Object Identifier: doi:10.1088/0305-4470/33/8/301
Zentralblatt MATH: 0983.37021
N. Abarenkova, J.-Ch. AnglèS D'Auriac, S. Boukraa, and J.-M. Maillard, Real topological entropy versus metric entropy for birational measure-preserving transformations, Phys. D 144 (2000), 387--433.
Mathematical Reviews (MathSciNet): MR1779810
Digital Object Identifier: doi:10.1016/S0167-2789(00)00079-8
E. Bedford and J. Diller, Energy and invariant measures for birational surface maps, Duke Math. J. 128 (2005), 331--368.
Mathematical Reviews (MathSciNet): MR2140266
Digital Object Identifier: doi:10.1215/S0012-7094-04-12824-6
Project Euclid: euclid.dmj/1117728418
Zentralblatt MATH: 1076.37031
—, Real and complex dynamics of a family of birational maps of the plane: The golden mean subshift, Amer. J. Math. 127 (2005), 595--646.
Mathematical Reviews (MathSciNet): MR2141646
Digital Object Identifier: doi:10.1353/ajm.2005.0015
Zentralblatt MATH: 1083.37038
E. Bedford, M. Ju. Lyubich, and J. Smillie, Distribution of periodic points of polynomial diffeomorphisms of $\,\cc^2$, Invent. Math. 114 (1993), 277--288.
Mathematical Reviews (MathSciNet): MR1240639
Digital Object Identifier: doi:10.1007/BF01232671
Zentralblatt MATH: 0799.58039
—, Polynomial diffeomorphisms of $\,\cc^2$, IV: The measure of maximal entropy and laminar currents, Invent. Math. 112 (1993), 77--125.
Mathematical Reviews (MathSciNet): MR1207478
Digital Object Identifier: doi:10.1007/BF01232426
B. Berndtsson and N. Sibony, The $\overline\fr$-equation on a positive current, Invent. Math. 147 (2002), 371--428.
Mathematical Reviews (MathSciNet): MR1881924
Digital Object Identifier: doi:10.1007/s002220100178
Zentralblatt MATH: 1031.32005
J.-Y. Briend and J. Duval, Deux caractérisations de la mesure d'équilibre d'un endomorphisme de $\mathbbP^\,k(\cc)$, Publ. Math. Inst. Hautes Études Sci. 93 (2001), 145--159.
Mathematical Reviews (MathSciNet): MR1863737
Digital Object Identifier: doi:10.1007/s10240-001-8190-4
S. Cantat, Dynamique des automorphismes des surfaces K$3$, Acta Math. 187 (2001) 1--57.
Mathematical Reviews (MathSciNet): MR1864630
Digital Object Identifier: doi:10.1007/BF02392831
Zentralblatt MATH: 1045.37007
S. Cantat and C. Favre, Symétries birationnelles des surfaces feuilletées, J. Reine Angew. Math. 561 (2003), 199--235.
Mathematical Reviews (MathSciNet): MR1998612
Digital Object Identifier: doi:10.1515/crll.2003.066
J. Diller, Dynamics of birational maps of $\,\mathbbP^2$, Indiana Univ. Math. J. 45 (1996), 721--772.
Mathematical Reviews (MathSciNet): MR1422105
Digital Object Identifier: doi:10.1512/iumj.1996.45.1331
Zentralblatt MATH: 0874.58022
—, Invariant measure and Lyapunov exponents for birational maps of $\,\mathbbP^2$, Comment. Math. Helv. 76 (2001), 754--780.
Mathematical Reviews (MathSciNet): MR1881705
Digital Object Identifier: doi:10.1007/s00014-001-8327-6
Zentralblatt MATH: 1061.32012
J. Diller and C. Favre, Dynamics of bimeromorphic maps of surfaces, Amer. J. Math. 123 (2001), 1135--1169.
Mathematical Reviews (MathSciNet): MR1867314
Digital Object Identifier: doi:10.1353/ajm.2001.0038
Zentralblatt MATH: 1112.37308
T.-C. Dinh and N. Sibony, Green currents for holomorphic automorphisms of compact Kähler manifolds, J. Amer. Math. Soc. 18 (2005), 291--312.
Mathematical Reviews (MathSciNet): MR2137979
Digital Object Identifier: doi:10.1090/S0894-0347-04-00474-6
R. Dujardin, Laminar currents in $\mathbbP^2$, Math. Ann. 325 (2003), 745--765.
Mathematical Reviews (MathSciNet): MR1974567
Digital Object Identifier: doi:10.1007/s00208-002-0402-9
Zentralblatt MATH: 1021.37018
—, Sur l'intersection des courants laminaires, Publ. Mat. 48 (2004), 107--125.
Mathematical Reviews (MathSciNet): MR2044640
—, Structure properties of laminar currents on $\mathbbP^2$, J. Geom. Anal. 15 (2005), 25--47.
Mathematical Reviews (MathSciNet): MR2132264
—, Dynamique d'applications nonpolynomiales et courants laminaires, Ph.D. dissertation, Université Paris-Sud, Orsay, France, 2002.
C. Favre, Multiplicity of holomorphic functions, Math. Ann. 316 (2000), 355--378.
Mathematical Reviews (MathSciNet): MR1741274
Digital Object Identifier: doi:10.1007/s002080050016
Zentralblatt MATH: 0948.32020
C. Favre and V. Guedj, Dynamique des applications rationnelles des espaces multiprojectifs, Indiana Univ. Math. J. 50 (2001), 881--934.
Mathematical Reviews (MathSciNet): MR1871393
Digital Object Identifier: doi:10.1512/iumj.2001.50.1880
Zentralblatt MATH: 1046.37026
M. Gromov, On the entropy of holomorphic maps, Enseign. Math. (2) 49 (2003), 217--235.
Mathematical Reviews (MathSciNet): MR2026895
V. Guedj, Entropie topologique des applications méromorphes, to appear in Ergodic Theory Dynam. Systems.
Mathematical Reviews (MathSciNet): MR2183297
Digital Object Identifier: doi:10.1017/S0143385705000192
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl. 54, Cambridge Univ. Press, Cambridge, 1995.
Mathematical Reviews (MathSciNet): MR1326374
F. Ledrappier and J.-M. Strelcyn, A proof of the estimation from below in Pesin's entropy formula, Ergodic Theory Dynam. Systems 2 (1982), 203--219.
Mathematical Reviews (MathSciNet): MR0693976
Digital Object Identifier: doi:10.1017/S0143385700001528
Zentralblatt MATH: 0533.58022
M. Ju. Ljubich [Lyubich], Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems 3 (1983), 351--385.
Mathematical Reviews (MathSciNet): MR0741393
Digital Object Identifier: doi:10.1017/S0143385700002030
D. S. Ornstein and B. Weiss, Statistical properties of chaotic systems, Bull. Amer. Math. Soc. (N.S.) 24 (1991), 11--116.
Mathematical Reviews (MathSciNet): MR1023980
Digital Object Identifier: doi:10.1090/S0273-0979-1991-15953-7
Project Euclid: euclid.bams/1183556246
Zentralblatt MATH: 0718.58038
D. Ruelle and D. Sullivan, Currents, flows and diffeomorphisms, Topology 14 (1975), 319--327.
Mathematical Reviews (MathSciNet): MR0415679
Digital Object Identifier: doi:10.1016/0040-9383(75)90016-6
Zentralblatt MATH: 0321.58019
M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.
Mathematical Reviews (MathSciNet): MR0114894
Duke Mathematical Journal
