Duke Mathematical Journal

Laminar currents and birational dynamics

Romain Dujardin

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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

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Duke Math. J. Volume 131, Number 2 (2006), 219-247.

First available in Project Euclid: 12 January 2006

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Zentralblatt MATH identifier

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04] 32H50: Iteration problems 32U40: Currents


Dujardin, Romain. Laminar currents and birational dynamics. Duke Mathematical Journal 131 (2006), no. 2, 219--247. doi:10.1215/S0012-7094-06-13122-8. http://projecteuclid.org/euclid.dmj/1137077884.

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  • 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.
  • —, Real Arnold complexity versus real topological entropy for birational transformations, J. Phys. A 33 (2000), 1465--1501.
  • 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.
  • E. Bedford and J. Diller, Energy and invariant measures for birational surface maps, Duke Math. J. 128 (2005), 331--368.
  • —, Real and complex dynamics of a family of birational maps of the plane: The golden mean subshift, Amer. J. Math. 127 (2005), 595--646.
  • E. Bedford, M. Ju. Lyubich, and J. Smillie, Distribution of periodic points of polynomial diffeomorphisms of $\,\cc^2$, Invent. Math. 114 (1993), 277--288.
  • —, Polynomial diffeomorphisms of $\,\cc^2$, IV: The measure of maximal entropy and laminar currents, Invent. Math. 112 (1993), 77--125.
  • B. Berndtsson and N. Sibony, The $\overline\fr$-equation on a positive current, Invent. Math. 147 (2002), 371--428.
  • 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.
  • S. Cantat, Dynamique des automorphismes des surfaces K$3$, Acta Math. 187 (2001) 1--57.
  • S. Cantat and C. Favre, Symétries birationnelles des surfaces feuilletées, J. Reine Angew. Math. 561 (2003), 199--235.
  • J. Diller, Dynamics of birational maps of $\,\mathbbP^2$, Indiana Univ. Math. J. 45 (1996), 721--772.
  • —, Invariant measure and Lyapunov exponents for birational maps of $\,\mathbbP^2$, Comment. Math. Helv. 76 (2001), 754--780.
  • J. Diller and C. Favre, Dynamics of bimeromorphic maps of surfaces, Amer. J. Math. 123 (2001), 1135--1169.
  • T.-C. Dinh and N. Sibony, Green currents for holomorphic automorphisms of compact Kähler manifolds, J. Amer. Math. Soc. 18 (2005), 291--312.
  • R. Dujardin, Laminar currents in $\mathbbP^2$, Math. Ann. 325 (2003), 745--765.
  • —, Sur l'intersection des courants laminaires, Publ. Mat. 48 (2004), 107--125.
  • —, Structure properties of laminar currents on $\mathbbP^2$, J. Geom. Anal. 15 (2005), 25--47.
  • —, 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.
  • C. Favre and V. Guedj, Dynamique des applications rationnelles des espaces multiprojectifs, Indiana Univ. Math. J. 50 (2001), 881--934.
  • M. Gromov, On the entropy of holomorphic maps, Enseign. Math. (2) 49 (2003), 217--235.
  • V. Guedj, Entropie topologique des applications méromorphes, to appear in Ergodic Theory Dynam. Systems.
  • A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl. 54, Cambridge Univ. Press, Cambridge, 1995.
  • 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.
  • M. Ju. Ljubich [Lyubich], Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems 3 (1983), 351--385.
  • D. S. Ornstein and B. Weiss, Statistical properties of chaotic systems, Bull. Amer. Math. Soc. (N.S.) 24 (1991), 11--116.
  • D. Ruelle and D. Sullivan, Currents, flows and diffeomorphisms, Topology 14 (1975), 319--327.
  • M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.