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), 157-194. doi: 10.1007/BF02420945.

[2]

L. V. Ahlfors and L. Sario, Riemann Surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J. 1960.

[3]

E. Bedford and K. Kim, Dynamics of rational surface automorphisms: Rotation domains, Amer. J. Math., 134 (2012), 379-405. doi: 10.1353/ajm.2012.0015.

[4]

E. Bedford, M. Lyubich and J. Smillie, Distribution of periodic points of polynomial diffeomorphisms of $C^2$, Invent. Math., 114 (1993), 277-288. doi: 10.1007/BF01232671.

[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), 77-125. doi: 10.1007/BF01232426.

[6]

E. Bedford and J. Smillie, Polynomial diffeomorphisms of $C^2$: Currents, equilibrium measure and hyperbolicity, Invent. Math., 103 (1991), 69-99. doi: 10.1007/BF01239509.

[7]

E. Bedford and J. Smillie, Polynomial diffeomorphisms of $C^2$. III. Ergodicity, exponents and entropy of the equilibrium measure, Math. Ann., 294 (1992), 395-420. doi: 10.1007/BF01934331.

[8]

J.-B. Bost, H. Gillet and C. Soulé, Heights of projective varieties and positive Green forms, J. Amer. Math. Soc., 7 (1994), 903-1027. doi: 10.1090/S0894-0347-1994-1260106-X.

[9]

D. Burns and N. Sibony, Limit currents and value distribution of holomorphic maps, Ann. Inst. Fourier (Grenoble), 62 (2012), 145-176. doi: 10.5802/aif.2703.

[10]

S. Cantat, Dynamique des automorphismes des surfaces K3, Acta Math., 187 (2001), 1-57. doi: 10.1007/BF02392831.

[11]

S. Cantat, Croissance des variétés instables, Ergodic Theory Dynam. Systems, 23 (2003), 1025-1042. doi: 10.1017/S0143385702001591.

[12]

D. Coman and V. Guedj, Invariant currents and dynamical Lelong numbers, J. Geom. Anal., 14 (2004), 199-213. doi: 10.1007/BF02922068.

[13]

J.-P. Demailly, Regularization of closed positive currents and intersection theory, J. Algebraic Geom., 1 (1992), 361-409.

[14]

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

[15]

J.-P. Demailly and M. Paun, Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. of Math. (2), 159 (2004), 1247-1274. doi: 10.4007/annals.2004.159.1247.

[16]

H. De Thélin, Sur la laminarité de certains courants, Ann. Sci. École Norm. Sup. (4), 37 (2004), 304-311. doi: 10.1016/j.ansens.2003.06.002.

[17]

H. De Thélin, Sur les automorphismes réguliers de $\mathbbC^k$, Publ. Mat., 54 (2010), 243-262. doi: 10.5565/PUBLMAT_54110_14.

[18]

H. De Thélin and T.-C. Dinh, Dynamics of automorphisms on compact Kähler manifolds, Adv. Math., 229 (2012), 2640-2655. doi: 10.1016/j.aim.2012.01.014.

[19]

T.-C. Dinh, Decay of correlations for Hénon maps, Acta Math., 195 (2005), 253-264. doi: 10.1007/BF02588081.

[20]

T.-C. Dinh, Suites d'applications méromorphes multivaluées et courants laminaires, J. Geom. Anal., 15 (2005), 207-227. doi: 10.1007/BF02922193.

[21]

T.-C. Dinh and V.-A. Nguyên, The mixed Hodge-Riemann bilinear relations for compact Kähler manifolds, Geom. Funct. Anal., 16 (2006), 838-849. doi: 10.1007/s00039-006-0572-9.

[22]

T.-C. Dinh, V.-A. Nguyên and N. Sibony, Heat equation and ergodic theorems for Riemann surface laminations, Math. Ann., 354 (2012), 331-376. doi: 10.1007/s00208-011-0730-8.

[23]

T.-C. Dinh, V.-A. Nguyên and N. Sibony, Dynamics of horizontal-like maps in higher dimension, Adv. Math., 219 (2008), 1689-1721. doi: 10.1016/j.aim.2008.07.006.

[24]

T.-C. Dinh and N. Sibony, Groupes commutatifs d'automorphismes d'une variété kählérienne compacte, Duke Math. J., 123 (2004), 311-328. doi: 10.1215/S0012-7094-04-12323-1.

[25]

T.-C. Dinh and N. Sibony, Regularization of currents and entropy, Ann. Sci. École Norm. Sup. (4), 37 (2004), 959-971. doi: 10.1016/j.ansens.2004.09.002.

[26]

T.-C. Dinh and N. Sibony, Green currents for holomorphic automorphisms of compact Kähler manifolds, J. Am. Math. Soc., 18 (2005), 291-312. doi: 10.1090/S0894-0347-04-00474-6.

[27]

T.-C. Dinh and N. Sibony, Dynamics of regular birational maps in $P^k$, J. Funct. Anal., 222 (2005), 202-216. doi: 10.1016/j.jfa.2004.07.018.

[28]

T.-C. Dinh and N. Sibony, Geometry of currents, intersection theory and dynamics of horizontal-like maps, Ann. Inst. Fourier (Grenoble), 56 (2006), 423-457. doi: 10.5802/aif.2188.

[29]

T.-C. Dinh and N. Sibony, Super-potentials of positive closed currents, intersection theory and dynamics, Acta Math., 203 (2009), 1-82. doi: 10.1007/s11511-009-0038-7.

[30]

T.-C. Dinh and N. Sibony, Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms, J. Algebraic Geom., 19 (2010), 473-529. doi: 10.1090/S1056-3911-10-00549-7.

[31]

T.-C. Dinh and N. Sibony, Dynamics in several complex variables: Endomorphisms of projective spaces and polynomial-like mappings, in Holomorphic Dynamical Systems, Lecture Notes in Math., 1998, Springer, Berlin, 2010, 165-294. doi: 10.1007/978-3-642-13171-4_4.

[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, Contemp. Math., 532, Amer. Math. Soc., Providence, RI, 2010, 107-114. doi: 10.1090/conm/532/10486.

[33]

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

[34]

R. Dujardin, Hénon-like mappings in $\mathbbC^2$, Amer. J. Math., 126 (2004), 439-472.

[35]

J. E. Fornæss and N. Sibony, Complex Hénon mappings in $C^2$ and Fatou-Bieberbach domains, Duke Math. J., 65 (1992), 345-380. doi: 10.1215/S0012-7094-92-06515-X.

[36]

J. E. Fornæss and N. Sibony, Complex dynamics in higher dimensions. Notes partially written by Estela A. Gavosto, in Complex Potential Theory (Montreal, PQ, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 439, Kluwer Acad. Publ., Dordrecht, 1994, 131-186.

[37]

J. E. Fornæss and N. Sibony, Harmonic currents of finite energy and laminations, Geom. Funct. Anal., 15 (2005), 962-1003. doi: 10.1007/s00039-005-0531-x.

[38]

J. E. Fornæss and N. Sibony, Unique ergodicity of harmonic currents on singular foliations of $P^2$, Geom. Funct. Anal., 19 (2010), 1334-1377. doi: 10.1007/s00039-009-0043-1.

[39]

S. Friedland and J. Milnor, Dynamical properties of plane polynomial automorphisms, Ergodic Theory Dynam. Systems, 9 (1989), 67-99. doi: 10.1017/S014338570000482X.

[40]

M. Gromov, Convex sets and Kähler manifolds, in Advances in Differential Geometry and Topology, World Sci. Publ., Teaneck, NJ, 1990, 1-38.

[41]

M. Gromov, On the entropy of holomorphic maps, Enseign. Math. (2), 49 (2003), 217-235.

[42]

R. C. Gunning, Introduction to Holomorphic Functions of Several Variables. Vol. I, II. Function Theory, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990.

[43]

L. Hörmander, The Analysis of Linear Partial Differential Operators. II. Differential Operators with Constant Coefficients, Reprint of the 1983 original, Classics in Mathematics, Springer-Verlag, Berlin, 2005.

[44]

H. W. E. Jung, Über ganze birationale Transformationen der Ebene, J. Reine Angew. Math., 184 (1942), 161-174.

[45]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, With a supplementary chapter by Katok and L. Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.

[46]

A. G. Khovanskii, The geometry of convex polyhedra and algebraic geometry, Uspehi Mat. Nauk., 34 (1979), 160-161.

[47]

S. Kobayashi, Hyperbolic Complex Spaces, Grundlehren der Mathematischen Wissenschaften, 318, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03582-5.

[48]

P. Lelong, Fonctions Plurisousharmoniques Et Formes Différentielles Positives, Gordon & Breach, Paris-London-New York (Distributed by Dunod éditeur, Paris), 1968.

[49]

C. T. McMullen, Dynamics on blowups of the projective plane, Publ. Math. Inst. Hautes Études Sci., 105 (2007), 49-89. doi: 10.1007/s10240-007-0004-x.

[50]

M. McQuillan, Diophantine approximations and foliations, Inst. Hautes Études Sci. Publ. Math., 87 (1998), 121-174.

[51]

A. Moncet, Géométrie et dynamique sur les surfaces algébriques réelles, Ph.D thesis, arXiv:1207.0390, 2012.

[52]

R. Nevanlinna, Analytic Functions, Translated from the second German edition by P. Emig, {Die Grundlehren der mathematischen Wissenschaften}, Band 162 Springer-Verlag, New York-Berlin, 1970.

[53]

K. Oguiso, A remark on dynamical degrees of automorphisms of hyperkähler manifolds, Manuscripta Math., 130 (2009), 101-111. doi: 10.1007/s00229-009-0271-6.

[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), 197-227. doi: 10.1090/S0894-0347-03-00440-5.

[55]

N. Sibony, Dynamique des applications rationnelles de $\mathbbP^k$, in Dynamique et Géométrie Complexes (Lyon, 1997), Panoramas et Synthèses, 8, Soc. Math. France, Paris, 1999, ix-x, xi-xii, 97-185.

[56]

T. Uehara, Rational surface automorphisms with positive entropy, arXiv:1009.2143, 2010.

[57]

C. Voisin, Théorie de Hodge et Géométrie Algébrique Complexe, Cours Spécialisés, 10, Société Mathématique de France, Paris, 2002. doi: 10.1017/CBO9780511615344.

[58]

J. Taflin, Equidistribution speed towards the Green current for endomorphisms of $ \mathbbP^k$, Adv. Math., 227 (2011), 2059-2081. doi: 10.1016/j.aim.2011.04.010.

[59]

B. Teissier, Du théorème de l'index de Hodge aux inégalités isopérimétriques, C. R. Acad. Sci. Paris Sér. A-B, 288 (1979), A287-A289.

[60]

V. A. Timorin, Mixed Hodge-Riemann bilinear relations in a linear context, Funktsional. Anal. i Prilozhen., 32 (1998), 63-68, 96; translation in Funct. Anal. Appl., 32 (1998), 268-272. doi: 10.1007/BF02463209.

[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), 339-411. doi: 10.1002/cpa.3160310304.

[62]

Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300. doi: 10.1007/BF02766215.

show all references

References:
[1]

L. V. Ahlfors, Zur Theorie der Überlagerungsflächen, Acta Math., 65 (1935), 157-194. doi: 10.1007/BF02420945.

[2]

L. V. Ahlfors and L. Sario, Riemann Surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J. 1960.

[3]

E. Bedford and K. Kim, Dynamics of rational surface automorphisms: Rotation domains, Amer. J. Math., 134 (2012), 379-405. doi: 10.1353/ajm.2012.0015.

[4]

E. Bedford, M. Lyubich and J. Smillie, Distribution of periodic points of polynomial diffeomorphisms of $C^2$, Invent. Math., 114 (1993), 277-288. doi: 10.1007/BF01232671.

[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), 77-125. doi: 10.1007/BF01232426.

[6]

E. Bedford and J. Smillie, Polynomial diffeomorphisms of $C^2$: Currents, equilibrium measure and hyperbolicity, Invent. Math., 103 (1991), 69-99. doi: 10.1007/BF01239509.

[7]

E. Bedford and J. Smillie, Polynomial diffeomorphisms of $C^2$. III. Ergodicity, exponents and entropy of the equilibrium measure, Math. Ann., 294 (1992), 395-420. doi: 10.1007/BF01934331.

[8]

J.-B. Bost, H. Gillet and C. Soulé, Heights of projective varieties and positive Green forms, J. Amer. Math. Soc., 7 (1994), 903-1027. doi: 10.1090/S0894-0347-1994-1260106-X.

[9]

D. Burns and N. Sibony, Limit currents and value distribution of holomorphic maps, Ann. Inst. Fourier (Grenoble), 62 (2012), 145-176. doi: 10.5802/aif.2703.

[10]

S. Cantat, Dynamique des automorphismes des surfaces K3, Acta Math., 187 (2001), 1-57. doi: 10.1007/BF02392831.

[11]

S. Cantat, Croissance des variétés instables, Ergodic Theory Dynam. Systems, 23 (2003), 1025-1042. doi: 10.1017/S0143385702001591.

[12]

D. Coman and V. Guedj, Invariant currents and dynamical Lelong numbers, J. Geom. Anal., 14 (2004), 199-213. doi: 10.1007/BF02922068.

[13]

J.-P. Demailly, Regularization of closed positive currents and intersection theory, J. Algebraic Geom., 1 (1992), 361-409.

[14]

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

[15]

J.-P. Demailly and M. Paun, Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. of Math. (2), 159 (2004), 1247-1274. doi: 10.4007/annals.2004.159.1247.

[16]

H. De Thélin, Sur la laminarité de certains courants, Ann. Sci. École Norm. Sup. (4), 37 (2004), 304-311. doi: 10.1016/j.ansens.2003.06.002.

[17]

H. De Thélin, Sur les automorphismes réguliers de $\mathbbC^k$, Publ. Mat., 54 (2010), 243-262. doi: 10.5565/PUBLMAT_54110_14.

[18]

H. De Thélin and T.-C. Dinh, Dynamics of automorphisms on compact Kähler manifolds, Adv. Math., 229 (2012), 2640-2655. doi: 10.1016/j.aim.2012.01.014.

[19]

T.-C. Dinh, Decay of correlations for Hénon maps, Acta Math., 195 (2005), 253-264. doi: 10.1007/BF02588081.

[20]

T.-C. Dinh, Suites d'applications méromorphes multivaluées et courants laminaires, J. Geom. Anal., 15 (2005), 207-227. doi: 10.1007/BF02922193.

[21]

T.-C. Dinh and V.-A. Nguyên, The mixed Hodge-Riemann bilinear relations for compact Kähler manifolds, Geom. Funct. Anal., 16 (2006), 838-849. doi: 10.1007/s00039-006-0572-9.

[22]

T.-C. Dinh, V.-A. Nguyên and N. Sibony, Heat equation and ergodic theorems for Riemann surface laminations, Math. Ann., 354 (2012), 331-376. doi: 10.1007/s00208-011-0730-8.

[23]

T.-C. Dinh, V.-A. Nguyên and N. Sibony, Dynamics of horizontal-like maps in higher dimension, Adv. Math., 219 (2008), 1689-1721. doi: 10.1016/j.aim.2008.07.006.

[24]

T.-C. Dinh and N. Sibony, Groupes commutatifs d'automorphismes d'une variété kählérienne compacte, Duke Math. J., 123 (2004), 311-328. doi: 10.1215/S0012-7094-04-12323-1.

[25]

T.-C. Dinh and N. Sibony, Regularization of currents and entropy, Ann. Sci. École Norm. Sup. (4), 37 (2004), 959-971. doi: 10.1016/j.ansens.2004.09.002.

[26]

T.-C. Dinh and N. Sibony, Green currents for holomorphic automorphisms of compact Kähler manifolds, J. Am. Math. Soc., 18 (2005), 291-312. doi: 10.1090/S0894-0347-04-00474-6.

[27]

T.-C. Dinh and N. Sibony, Dynamics of regular birational maps in $P^k$, J. Funct. Anal., 222 (2005), 202-216. doi: 10.1016/j.jfa.2004.07.018.

[28]

T.-C. Dinh and N. Sibony, Geometry of currents, intersection theory and dynamics of horizontal-like maps, Ann. Inst. Fourier (Grenoble), 56 (2006), 423-457. doi: 10.5802/aif.2188.

[29]

T.-C. Dinh and N. Sibony, Super-potentials of positive closed currents, intersection theory and dynamics, Acta Math., 203 (2009), 1-82. doi: 10.1007/s11511-009-0038-7.

[30]

T.-C. Dinh and N. Sibony, Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms, J. Algebraic Geom., 19 (2010), 473-529. doi: 10.1090/S1056-3911-10-00549-7.

[31]

T.-C. Dinh and N. Sibony, Dynamics in several complex variables: Endomorphisms of projective spaces and polynomial-like mappings, in Holomorphic Dynamical Systems, Lecture Notes in Math., 1998, Springer, Berlin, 2010, 165-294. doi: 10.1007/978-3-642-13171-4_4.

[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, Contemp. Math., 532, Amer. Math. Soc., Providence, RI, 2010, 107-114. doi: 10.1090/conm/532/10486.

[33]

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

[34]

R. Dujardin, Hénon-like mappings in $\mathbbC^2$, Amer. J. Math., 126 (2004), 439-472.

[35]

J. E. Fornæss and N. Sibony, Complex Hénon mappings in $C^2$ and Fatou-Bieberbach domains, Duke Math. J., 65 (1992), 345-380. doi: 10.1215/S0012-7094-92-06515-X.

[36]

J. E. Fornæss and N. Sibony, Complex dynamics in higher dimensions. Notes partially written by Estela A. Gavosto, in Complex Potential Theory (Montreal, PQ, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 439, Kluwer Acad. Publ., Dordrecht, 1994, 131-186.

[37]

J. E. Fornæss and N. Sibony, Harmonic currents of finite energy and laminations, Geom. Funct. Anal., 15 (2005), 962-1003. doi: 10.1007/s00039-005-0531-x.

[38]

J. E. Fornæss and N. Sibony, Unique ergodicity of harmonic currents on singular foliations of $P^2$, Geom. Funct. Anal., 19 (2010), 1334-1377. doi: 10.1007/s00039-009-0043-1.

[39]

S. Friedland and J. Milnor, Dynamical properties of plane polynomial automorphisms, Ergodic Theory Dynam. Systems, 9 (1989), 67-99. doi: 10.1017/S014338570000482X.

[40]

M. Gromov, Convex sets and Kähler manifolds, in Advances in Differential Geometry and Topology, World Sci. Publ., Teaneck, NJ, 1990, 1-38.

[41]

M. Gromov, On the entropy of holomorphic maps, Enseign. Math. (2), 49 (2003), 217-235.

[42]

R. C. Gunning, Introduction to Holomorphic Functions of Several Variables. Vol. I, II. Function Theory, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990.

[43]

L. Hörmander, The Analysis of Linear Partial Differential Operators. II. Differential Operators with Constant Coefficients, Reprint of the 1983 original, Classics in Mathematics, Springer-Verlag, Berlin, 2005.

[44]

H. W. E. Jung, Über ganze birationale Transformationen der Ebene, J. Reine Angew. Math., 184 (1942), 161-174.

[45]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, With a supplementary chapter by Katok and L. Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.

[46]

A. G. Khovanskii, The geometry of convex polyhedra and algebraic geometry, Uspehi Mat. Nauk., 34 (1979), 160-161.

[47]

S. Kobayashi, Hyperbolic Complex Spaces, Grundlehren der Mathematischen Wissenschaften, 318, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03582-5.

[48]

P. Lelong, Fonctions Plurisousharmoniques Et Formes Différentielles Positives, Gordon & Breach, Paris-London-New York (Distributed by Dunod éditeur, Paris), 1968.

[49]

C. T. McMullen, Dynamics on blowups of the projective plane, Publ. Math. Inst. Hautes Études Sci., 105 (2007), 49-89. doi: 10.1007/s10240-007-0004-x.

[50]

M. McQuillan, Diophantine approximations and foliations, Inst. Hautes Études Sci. Publ. Math., 87 (1998), 121-174.

[51]

A. Moncet, Géométrie et dynamique sur les surfaces algébriques réelles, Ph.D thesis, arXiv:1207.0390, 2012.

[52]

R. Nevanlinna, Analytic Functions, Translated from the second German edition by P. Emig, {Die Grundlehren der mathematischen Wissenschaften}, Band 162 Springer-Verlag, New York-Berlin, 1970.

[53]

K. Oguiso, A remark on dynamical degrees of automorphisms of hyperkähler manifolds, Manuscripta Math., 130 (2009), 101-111. doi: 10.1007/s00229-009-0271-6.

[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), 197-227. doi: 10.1090/S0894-0347-03-00440-5.

[55]

N. Sibony, Dynamique des applications rationnelles de $\mathbbP^k$, in Dynamique et Géométrie Complexes (Lyon, 1997), Panoramas et Synthèses, 8, Soc. Math. France, Paris, 1999, ix-x, xi-xii, 97-185.

[56]

T. Uehara, Rational surface automorphisms with positive entropy, arXiv:1009.2143, 2010.

[57]

C. Voisin, Théorie de Hodge et Géométrie Algébrique Complexe, Cours Spécialisés, 10, Société Mathématique de France, Paris, 2002. doi: 10.1017/CBO9780511615344.

[58]

J. Taflin, Equidistribution speed towards the Green current for endomorphisms of $ \mathbbP^k$, Adv. Math., 227 (2011), 2059-2081. doi: 10.1016/j.aim.2011.04.010.

[59]

B. Teissier, Du théorème de l'index de Hodge aux inégalités isopérimétriques, C. R. Acad. Sci. Paris Sér. A-B, 288 (1979), A287-A289.

[60]

V. A. Timorin, Mixed Hodge-Riemann bilinear relations in a linear context, Funktsional. Anal. i Prilozhen., 32 (1998), 63-68, 96; translation in Funct. Anal. Appl., 32 (1998), 268-272. doi: 10.1007/BF02463209.

[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), 339-411. doi: 10.1002/cpa.3160310304.

[62]

Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300. doi: 10.1007/BF02766215.

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