July & October  2014, 8(3&4): 499-548. doi: 10.3934/jmd.2014.8.499

Rigidity of Julia sets for Hénon type maps

1. 

Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076, Singapore

2. 

Université Paris-Sud, Mathématique - Bâtiment 425, 91405 Orsay, France

Received  January 2013 Revised  August 2013 Published  April 2015

We prove that the Julia set of a Hénon type automorphism on $\mathbb{C}^2$ is very rigid: it supports a unique positive $dd^c$-closed current of mass 1. A similar property holds for the cohomology class of the Green current associated with an automorphism of positive entropy on a compact Kähler surface. Relations between this phenomenon, several quantitative equidistribution properties and the theory of value distribution will be discussed. We also survey some rigidity properties of Hénon type maps on $\mathbb{C}^k$ and of automorphisms of compact Kähler manifolds.
Citation: Tien-Cuong Dinh, Nessim Sibony. Rigidity of Julia sets for Hénon type maps. Journal of Modern Dynamics, 2014, 8 (3&4) : 499-548. doi: 10.3934/jmd.2014.8.499
References:
[1]

L. V. Ahlfors, Zur Theorie der Überlagerungsflächen,, Acta Math., 65 (1935), 157.  doi: 10.1007/BF02420945.  Google Scholar

[2]

L. V. Ahlfors and L. Sario, Riemann Surfaces,, Princeton Mathematical Series, (1960).   Google Scholar

[3]

E. Bedford and K. Kim, Dynamics of rational surface automorphisms: Rotation domains,, Amer. J. Math., 134 (2012), 379.  doi: 10.1353/ajm.2012.0015.  Google Scholar

[4]

E. Bedford, M. Lyubich and J. Smillie, Distribution of periodic points of polynomial diffeomorphisms of $C^2$,, Invent. Math., 114 (1993), 277.  doi: 10.1007/BF01232671.  Google Scholar

[5]

E. Bedford, M. Lyubich and J. Smillie, Polynomial diffeomorphisms of $C^2$. IV. The measure of maximal entropy and laminar currents,, Invent. Math., 112 (1993), 77.  doi: 10.1007/BF01232426.  Google Scholar

[6]

E. Bedford and J. Smillie, Polynomial diffeomorphisms of $C^2$: Currents, equilibrium measure and hyperbolicity,, Invent. Math., 103 (1991), 69.  doi: 10.1007/BF01239509.  Google Scholar

[7]

E. Bedford and J. Smillie, Polynomial diffeomorphisms of $C^2$. III. Ergodicity, exponents and entropy of the equilibrium measure,, Math. Ann., 294 (1992), 395.  doi: 10.1007/BF01934331.  Google Scholar

[8]

J.-B. Bost, H. Gillet and C. Soulé, Heights of projective varieties and positive Green forms,, J. Amer. Math. Soc., 7 (1994), 903.  doi: 10.1090/S0894-0347-1994-1260106-X.  Google Scholar

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D. Burns and N. Sibony, Limit currents and value distribution of holomorphic maps,, Ann. Inst. Fourier (Grenoble), 62 (2012), 145.  doi: 10.5802/aif.2703.  Google Scholar

[10]

S. Cantat, Dynamique des automorphismes des surfaces K3,, Acta Math., 187 (2001), 1.  doi: 10.1007/BF02392831.  Google Scholar

[11]

S. Cantat, Croissance des variétés instables,, Ergodic Theory Dynam. Systems, 23 (2003), 1025.  doi: 10.1017/S0143385702001591.  Google Scholar

[12]

D. Coman and V. Guedj, Invariant currents and dynamical Lelong numbers,, J. Geom. Anal., 14 (2004), 199.  doi: 10.1007/BF02922068.  Google Scholar

[13]

J.-P. Demailly, Regularization of closed positive currents and intersection theory,, J. Algebraic Geom., 1 (1992), 361.   Google Scholar

[14]

J.-P. Demailly, Complex Analytic and Differential Geometry., Available from: , ().   Google Scholar

[15]

J.-P. Demailly and M. Paun, Numerical characterization of the Kähler cone of a compact Kähler manifold,, Ann. of Math. (2), 159 (2004), 1247.  doi: 10.4007/annals.2004.159.1247.  Google Scholar

[16]

H. De Thélin, Sur la laminarité de certains courants,, Ann. Sci. École Norm. Sup. (4), 37 (2004), 304.  doi: 10.1016/j.ansens.2003.06.002.  Google Scholar

[17]

H. De Thélin, Sur les automorphismes réguliers de $\mathbbC^k$,, Publ. Mat., 54 (2010), 243.  doi: 10.5565/PUBLMAT_54110_14.  Google Scholar

[18]

H. De Thélin and T.-C. Dinh, Dynamics of automorphisms on compact Kähler manifolds,, Adv. Math., 229 (2012), 2640.  doi: 10.1016/j.aim.2012.01.014.  Google Scholar

[19]

T.-C. Dinh, Decay of correlations for Hénon maps,, Acta Math., 195 (2005), 253.  doi: 10.1007/BF02588081.  Google Scholar

[20]

T.-C. Dinh, Suites d'applications méromorphes multivaluées et courants laminaires,, J. Geom. Anal., 15 (2005), 207.  doi: 10.1007/BF02922193.  Google Scholar

[21]

T.-C. Dinh and V.-A. Nguyên, The mixed Hodge-Riemann bilinear relations for compact Kähler manifolds,, Geom. Funct. Anal., 16 (2006), 838.  doi: 10.1007/s00039-006-0572-9.  Google Scholar

[22]

T.-C. Dinh, V.-A. Nguyên and N. Sibony, Heat equation and ergodic theorems for Riemann surface laminations,, Math. Ann., 354 (2012), 331.  doi: 10.1007/s00208-011-0730-8.  Google Scholar

[23]

T.-C. Dinh, V.-A. Nguyên and N. Sibony, Dynamics of horizontal-like maps in higher dimension,, Adv. Math., 219 (2008), 1689.  doi: 10.1016/j.aim.2008.07.006.  Google Scholar

[24]

T.-C. Dinh and N. Sibony, Groupes commutatifs d'automorphismes d'une variété kählérienne compacte,, Duke Math. J., 123 (2004), 311.  doi: 10.1215/S0012-7094-04-12323-1.  Google Scholar

[25]

T.-C. Dinh and N. Sibony, Regularization of currents and entropy,, Ann. Sci. École Norm. Sup. (4), 37 (2004), 959.  doi: 10.1016/j.ansens.2004.09.002.  Google Scholar

[26]

T.-C. Dinh and N. Sibony, Green currents for holomorphic automorphisms of compact Kähler manifolds,, J. Am. Math. Soc., 18 (2005), 291.  doi: 10.1090/S0894-0347-04-00474-6.  Google Scholar

[27]

T.-C. Dinh and N. Sibony, Dynamics of regular birational maps in $P^k$,, J. Funct. Anal., 222 (2005), 202.  doi: 10.1016/j.jfa.2004.07.018.  Google Scholar

[28]

T.-C. Dinh and N. Sibony, Geometry of currents, intersection theory and dynamics of horizontal-like maps,, Ann. Inst. Fourier (Grenoble), 56 (2006), 423.  doi: 10.5802/aif.2188.  Google Scholar

[29]

T.-C. Dinh and N. Sibony, Super-potentials of positive closed currents, intersection theory and dynamics,, Acta Math., 203 (2009), 1.  doi: 10.1007/s11511-009-0038-7.  Google Scholar

[30]

T.-C. Dinh and N. Sibony, Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms,, J. Algebraic Geom., 19 (2010), 473.  doi: 10.1090/S1056-3911-10-00549-7.  Google Scholar

[31]

T.-C. Dinh and N. Sibony, Dynamics in several complex variables: Endomorphisms of projective spaces and polynomial-like mappings,, in Holomorphic Dynamical Systems, (1998), 165.  doi: 10.1007/978-3-642-13171-4_4.  Google Scholar

[32]

T.-C. Dinh and N. Sibony, Exponential mixing for automorphisms on compact Kähler manifolds,, in Dynamical Numbers-Interplay Between Dynamical Systems and Number Theory, (2010), 107.  doi: 10.1090/conm/532/10486.  Google Scholar

[33]

T.-C. Dinh and N. Sibony, Equidistribution of saddle periodic points for Hénon-type automorphisms of $\mathbbC^k$,, , ().   Google Scholar

[34]

R. Dujardin, Hénon-like mappings in $\mathbbC^2$,, Amer. J. Math., 126 (2004), 439.   Google Scholar

[35]

J. E. Fornæss and N. Sibony, Complex Hénon mappings in $C^2$ and Fatou-Bieberbach domains,, Duke Math. J., 65 (1992), 345.  doi: 10.1215/S0012-7094-92-06515-X.  Google Scholar

[36]

J. E. Fornæss and N. Sibony, Complex dynamics in higher dimensions. Notes partially written by Estela A. Gavosto,, in Complex Potential Theory (Montreal, (1993), 131.   Google Scholar

[37]

J. E. Fornæss and N. Sibony, Harmonic currents of finite energy and laminations,, Geom. Funct. Anal., 15 (2005), 962.  doi: 10.1007/s00039-005-0531-x.  Google Scholar

[38]

J. E. Fornæss and N. Sibony, Unique ergodicity of harmonic currents on singular foliations of $P^2$,, Geom. Funct. Anal., 19 (2010), 1334.  doi: 10.1007/s00039-009-0043-1.  Google Scholar

[39]

S. Friedland and J. Milnor, Dynamical properties of plane polynomial automorphisms,, Ergodic Theory Dynam. Systems, 9 (1989), 67.  doi: 10.1017/S014338570000482X.  Google Scholar

[40]

M. Gromov, Convex sets and Kähler manifolds,, in Advances in Differential Geometry and Topology, (1990), 1.   Google Scholar

[41]

M. Gromov, On the entropy of holomorphic maps,, Enseign. Math. (2), 49 (2003), 217.   Google Scholar

[42]

R. C. Gunning, Introduction to Holomorphic Functions of Several Variables. Vol. I, II. Function Theory,, The Wadsworth & Brooks/Cole Mathematics Series, (1990).   Google Scholar

[43]

L. Hörmander, The Analysis of Linear Partial Differential Operators. II. Differential Operators with Constant Coefficients,, Reprint of the 1983 original, (1983).   Google Scholar

[44]

H. W. E. Jung, Über ganze birationale Transformationen der Ebene,, J. Reine Angew. Math., 184 (1942), 161.   Google Scholar

[45]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, With a supplementary chapter by Katok and L. Mendoza, (1995).  doi: 10.1017/CBO9780511809187.  Google Scholar

[46]

A. G. Khovanskii, The geometry of convex polyhedra and algebraic geometry,, Uspehi Mat. Nauk., 34 (1979), 160.   Google Scholar

[47]

S. Kobayashi, Hyperbolic Complex Spaces,, Grundlehren der Mathematischen Wissenschaften, (1998).  doi: 10.1007/978-3-662-03582-5.  Google Scholar

[48]

P. Lelong, Fonctions Plurisousharmoniques Et Formes Différentielles Positives,, Gordon & Breach, (1968).   Google Scholar

[49]

C. T. McMullen, Dynamics on blowups of the projective plane,, Publ. Math. Inst. Hautes Études Sci., 105 (2007), 49.  doi: 10.1007/s10240-007-0004-x.  Google Scholar

[50]

M. McQuillan, Diophantine approximations and foliations,, Inst. Hautes Études Sci. Publ. Math., 87 (1998), 121.   Google Scholar

[51]

A. Moncet, Géométrie et dynamique sur les surfaces algébriques réelles,, Ph.D thesis, (2012).   Google Scholar

[52]

R. Nevanlinna, Analytic Functions,, Translated from the second German edition by P. Emig, (1970).   Google Scholar

[53]

K. Oguiso, A remark on dynamical degrees of automorphisms of hyperkähler manifolds,, Manuscripta Math., 130 (2009), 101.  doi: 10.1007/s00229-009-0271-6.  Google Scholar

[54]

I. P. Shestakov and U. U. Umirbaev, The tame and the wild automorphisms of polynomial rings in three variables,, J. Amer. Math. Soc., 17 (2004), 197.  doi: 10.1090/S0894-0347-03-00440-5.  Google Scholar

[55]

N. Sibony, Dynamique des applications rationnelles de $\mathbbP^k$,, in Dynamique et Géométrie Complexes (Lyon, (1997), 97.   Google Scholar

[56]

T. Uehara, Rational surface automorphisms with positive entropy,, , (2010).   Google Scholar

[57]

C. Voisin, Théorie de Hodge et Géométrie Algébrique Complexe,, Cours Spécialisés, (2002).  doi: 10.1017/CBO9780511615344.  Google Scholar

[58]

J. Taflin, Equidistribution speed towards the Green current for endomorphisms of $ \mathbbP^k$,, Adv. Math., 227 (2011), 2059.  doi: 10.1016/j.aim.2011.04.010.  Google Scholar

[59]

B. Teissier, Du théorème de l'index de Hodge aux inégalités isopérimétriques,, C. R. Acad. Sci. Paris Sér. A-B, 288 (1979).   Google Scholar

[60]

V. A. Timorin, Mixed Hodge-Riemann bilinear relations in a linear context,, Funktsional. Anal. i Prilozhen., 32 (1998), 63.  doi: 10.1007/BF02463209.  Google Scholar

[61]

S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I,, Comm. Pure Appl. Math., 31 (1978), 339.  doi: 10.1002/cpa.3160310304.  Google Scholar

[62]

Y. Yomdin, Volume growth and entropy,, Israel J. Math., 57 (1987), 285.  doi: 10.1007/BF02766215.  Google Scholar

show all references

References:
[1]

L. V. Ahlfors, Zur Theorie der Überlagerungsflächen,, Acta Math., 65 (1935), 157.  doi: 10.1007/BF02420945.  Google Scholar

[2]

L. V. Ahlfors and L. Sario, Riemann Surfaces,, Princeton Mathematical Series, (1960).   Google Scholar

[3]

E. Bedford and K. Kim, Dynamics of rational surface automorphisms: Rotation domains,, Amer. J. Math., 134 (2012), 379.  doi: 10.1353/ajm.2012.0015.  Google Scholar

[4]

E. Bedford, M. Lyubich and J. Smillie, Distribution of periodic points of polynomial diffeomorphisms of $C^2$,, Invent. Math., 114 (1993), 277.  doi: 10.1007/BF01232671.  Google Scholar

[5]

E. Bedford, M. Lyubich and J. Smillie, Polynomial diffeomorphisms of $C^2$. IV. The measure of maximal entropy and laminar currents,, Invent. Math., 112 (1993), 77.  doi: 10.1007/BF01232426.  Google Scholar

[6]

E. Bedford and J. Smillie, Polynomial diffeomorphisms of $C^2$: Currents, equilibrium measure and hyperbolicity,, Invent. Math., 103 (1991), 69.  doi: 10.1007/BF01239509.  Google Scholar

[7]

E. Bedford and J. Smillie, Polynomial diffeomorphisms of $C^2$. III. Ergodicity, exponents and entropy of the equilibrium measure,, Math. Ann., 294 (1992), 395.  doi: 10.1007/BF01934331.  Google Scholar

[8]

J.-B. Bost, H. Gillet and C. Soulé, Heights of projective varieties and positive Green forms,, J. Amer. Math. Soc., 7 (1994), 903.  doi: 10.1090/S0894-0347-1994-1260106-X.  Google Scholar

[9]

D. Burns and N. Sibony, Limit currents and value distribution of holomorphic maps,, Ann. Inst. Fourier (Grenoble), 62 (2012), 145.  doi: 10.5802/aif.2703.  Google Scholar

[10]

S. Cantat, Dynamique des automorphismes des surfaces K3,, Acta Math., 187 (2001), 1.  doi: 10.1007/BF02392831.  Google Scholar

[11]

S. Cantat, Croissance des variétés instables,, Ergodic Theory Dynam. Systems, 23 (2003), 1025.  doi: 10.1017/S0143385702001591.  Google Scholar

[12]

D. Coman and V. Guedj, Invariant currents and dynamical Lelong numbers,, J. Geom. Anal., 14 (2004), 199.  doi: 10.1007/BF02922068.  Google Scholar

[13]

J.-P. Demailly, Regularization of closed positive currents and intersection theory,, J. Algebraic Geom., 1 (1992), 361.   Google Scholar

[14]

J.-P. Demailly, Complex Analytic and Differential Geometry., Available from: , ().   Google Scholar

[15]

J.-P. Demailly and M. Paun, Numerical characterization of the Kähler cone of a compact Kähler manifold,, Ann. of Math. (2), 159 (2004), 1247.  doi: 10.4007/annals.2004.159.1247.  Google Scholar

[16]

H. De Thélin, Sur la laminarité de certains courants,, Ann. Sci. École Norm. Sup. (4), 37 (2004), 304.  doi: 10.1016/j.ansens.2003.06.002.  Google Scholar

[17]

H. De Thélin, Sur les automorphismes réguliers de $\mathbbC^k$,, Publ. Mat., 54 (2010), 243.  doi: 10.5565/PUBLMAT_54110_14.  Google Scholar

[18]

H. De Thélin and T.-C. Dinh, Dynamics of automorphisms on compact Kähler manifolds,, Adv. Math., 229 (2012), 2640.  doi: 10.1016/j.aim.2012.01.014.  Google Scholar

[19]

T.-C. Dinh, Decay of correlations for Hénon maps,, Acta Math., 195 (2005), 253.  doi: 10.1007/BF02588081.  Google Scholar

[20]

T.-C. Dinh, Suites d'applications méromorphes multivaluées et courants laminaires,, J. Geom. Anal., 15 (2005), 207.  doi: 10.1007/BF02922193.  Google Scholar

[21]

T.-C. Dinh and V.-A. Nguyên, The mixed Hodge-Riemann bilinear relations for compact Kähler manifolds,, Geom. Funct. Anal., 16 (2006), 838.  doi: 10.1007/s00039-006-0572-9.  Google Scholar

[22]

T.-C. Dinh, V.-A. Nguyên and N. Sibony, Heat equation and ergodic theorems for Riemann surface laminations,, Math. Ann., 354 (2012), 331.  doi: 10.1007/s00208-011-0730-8.  Google Scholar

[23]

T.-C. Dinh, V.-A. Nguyên and N. Sibony, Dynamics of horizontal-like maps in higher dimension,, Adv. Math., 219 (2008), 1689.  doi: 10.1016/j.aim.2008.07.006.  Google Scholar

[24]

T.-C. Dinh and N. Sibony, Groupes commutatifs d'automorphismes d'une variété kählérienne compacte,, Duke Math. J., 123 (2004), 311.  doi: 10.1215/S0012-7094-04-12323-1.  Google Scholar

[25]

T.-C. Dinh and N. Sibony, Regularization of currents and entropy,, Ann. Sci. École Norm. Sup. (4), 37 (2004), 959.  doi: 10.1016/j.ansens.2004.09.002.  Google Scholar

[26]

T.-C. Dinh and N. Sibony, Green currents for holomorphic automorphisms of compact Kähler manifolds,, J. Am. Math. Soc., 18 (2005), 291.  doi: 10.1090/S0894-0347-04-00474-6.  Google Scholar

[27]

T.-C. Dinh and N. Sibony, Dynamics of regular birational maps in $P^k$,, J. Funct. Anal., 222 (2005), 202.  doi: 10.1016/j.jfa.2004.07.018.  Google Scholar

[28]

T.-C. Dinh and N. Sibony, Geometry of currents, intersection theory and dynamics of horizontal-like maps,, Ann. Inst. Fourier (Grenoble), 56 (2006), 423.  doi: 10.5802/aif.2188.  Google Scholar

[29]

T.-C. Dinh and N. Sibony, Super-potentials of positive closed currents, intersection theory and dynamics,, Acta Math., 203 (2009), 1.  doi: 10.1007/s11511-009-0038-7.  Google Scholar

[30]

T.-C. Dinh and N. Sibony, Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms,, J. Algebraic Geom., 19 (2010), 473.  doi: 10.1090/S1056-3911-10-00549-7.  Google Scholar

[31]

T.-C. Dinh and N. Sibony, Dynamics in several complex variables: Endomorphisms of projective spaces and polynomial-like mappings,, in Holomorphic Dynamical Systems, (1998), 165.  doi: 10.1007/978-3-642-13171-4_4.  Google Scholar

[32]

T.-C. Dinh and N. Sibony, Exponential mixing for automorphisms on compact Kähler manifolds,, in Dynamical Numbers-Interplay Between Dynamical Systems and Number Theory, (2010), 107.  doi: 10.1090/conm/532/10486.  Google Scholar

[33]

T.-C. Dinh and N. Sibony, Equidistribution of saddle periodic points for Hénon-type automorphisms of $\mathbbC^k$,, , ().   Google Scholar

[34]

R. Dujardin, Hénon-like mappings in $\mathbbC^2$,, Amer. J. Math., 126 (2004), 439.   Google Scholar

[35]

J. E. Fornæss and N. Sibony, Complex Hénon mappings in $C^2$ and Fatou-Bieberbach domains,, Duke Math. J., 65 (1992), 345.  doi: 10.1215/S0012-7094-92-06515-X.  Google Scholar

[36]

J. E. Fornæss and N. Sibony, Complex dynamics in higher dimensions. Notes partially written by Estela A. Gavosto,, in Complex Potential Theory (Montreal, (1993), 131.   Google Scholar

[37]

J. E. Fornæss and N. Sibony, Harmonic currents of finite energy and laminations,, Geom. Funct. Anal., 15 (2005), 962.  doi: 10.1007/s00039-005-0531-x.  Google Scholar

[38]

J. E. Fornæss and N. Sibony, Unique ergodicity of harmonic currents on singular foliations of $P^2$,, Geom. Funct. Anal., 19 (2010), 1334.  doi: 10.1007/s00039-009-0043-1.  Google Scholar

[39]

S. Friedland and J. Milnor, Dynamical properties of plane polynomial automorphisms,, Ergodic Theory Dynam. Systems, 9 (1989), 67.  doi: 10.1017/S014338570000482X.  Google Scholar

[40]

M. Gromov, Convex sets and Kähler manifolds,, in Advances in Differential Geometry and Topology, (1990), 1.   Google Scholar

[41]

M. Gromov, On the entropy of holomorphic maps,, Enseign. Math. (2), 49 (2003), 217.   Google Scholar

[42]

R. C. Gunning, Introduction to Holomorphic Functions of Several Variables. Vol. I, II. Function Theory,, The Wadsworth & Brooks/Cole Mathematics Series, (1990).   Google Scholar

[43]

L. Hörmander, The Analysis of Linear Partial Differential Operators. II. Differential Operators with Constant Coefficients,, Reprint of the 1983 original, (1983).   Google Scholar

[44]

H. W. E. Jung, Über ganze birationale Transformationen der Ebene,, J. Reine Angew. Math., 184 (1942), 161.   Google Scholar

[45]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, With a supplementary chapter by Katok and L. Mendoza, (1995).  doi: 10.1017/CBO9780511809187.  Google Scholar

[46]

A. G. Khovanskii, The geometry of convex polyhedra and algebraic geometry,, Uspehi Mat. Nauk., 34 (1979), 160.   Google Scholar

[47]

S. Kobayashi, Hyperbolic Complex Spaces,, Grundlehren der Mathematischen Wissenschaften, (1998).  doi: 10.1007/978-3-662-03582-5.  Google Scholar

[48]

P. Lelong, Fonctions Plurisousharmoniques Et Formes Différentielles Positives,, Gordon & Breach, (1968).   Google Scholar

[49]

C. T. McMullen, Dynamics on blowups of the projective plane,, Publ. Math. Inst. Hautes Études Sci., 105 (2007), 49.  doi: 10.1007/s10240-007-0004-x.  Google Scholar

[50]

M. McQuillan, Diophantine approximations and foliations,, Inst. Hautes Études Sci. Publ. Math., 87 (1998), 121.   Google Scholar

[51]

A. Moncet, Géométrie et dynamique sur les surfaces algébriques réelles,, Ph.D thesis, (2012).   Google Scholar

[52]

R. Nevanlinna, Analytic Functions,, Translated from the second German edition by P. Emig, (1970).   Google Scholar

[53]

K. Oguiso, A remark on dynamical degrees of automorphisms of hyperkähler manifolds,, Manuscripta Math., 130 (2009), 101.  doi: 10.1007/s00229-009-0271-6.  Google Scholar

[54]

I. P. Shestakov and U. U. Umirbaev, The tame and the wild automorphisms of polynomial rings in three variables,, J. Amer. Math. Soc., 17 (2004), 197.  doi: 10.1090/S0894-0347-03-00440-5.  Google Scholar

[55]

N. Sibony, Dynamique des applications rationnelles de $\mathbbP^k$,, in Dynamique et Géométrie Complexes (Lyon, (1997), 97.   Google Scholar

[56]

T. Uehara, Rational surface automorphisms with positive entropy,, , (2010).   Google Scholar

[57]

C. Voisin, Théorie de Hodge et Géométrie Algébrique Complexe,, Cours Spécialisés, (2002).  doi: 10.1017/CBO9780511615344.  Google Scholar

[58]

J. Taflin, Equidistribution speed towards the Green current for endomorphisms of $ \mathbbP^k$,, Adv. Math., 227 (2011), 2059.  doi: 10.1016/j.aim.2011.04.010.  Google Scholar

[59]

B. Teissier, Du théorème de l'index de Hodge aux inégalités isopérimétriques,, C. R. Acad. Sci. Paris Sér. A-B, 288 (1979).   Google Scholar

[60]

V. A. Timorin, Mixed Hodge-Riemann bilinear relations in a linear context,, Funktsional. Anal. i Prilozhen., 32 (1998), 63.  doi: 10.1007/BF02463209.  Google Scholar

[61]

S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I,, Comm. Pure Appl. Math., 31 (1978), 339.  doi: 10.1002/cpa.3160310304.  Google Scholar

[62]

Y. Yomdin, Volume growth and entropy,, Israel J. Math., 57 (1987), 285.  doi: 10.1007/BF02766215.  Google Scholar

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