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December  2020, 40(12): 6767-6781. doi: 10.3934/dcds.2020163

On the regularity of the Green current for semi-extremal endomorphisms of $ \mathbb{P}^2 $

Christophe Dupont and  Axel Rogue

Université de Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

Received  June 2019 Revised  January 2020 Published  December 2020 Early access  March 2020

We study the regularity of the Green current for semi-extremal endomorphisms of $ \mathbb{P}^2 $. Under suitable assumptions, we show that the pointwise lower Radon-Nikodym derivative of stable slices with respect to the one dimensional Lebesgue measure is bounded at almost every point for the equilibrium measure. This provides a weak amount of metric regularity for the Green current along holomorphic discs.

Keywords: Holomorphic dynamics, Lyapunov exponents, Green current.
Mathematics Subject Classification: 37F10, 32U40, 28A15.
Citation: Christophe Dupont, Axel Rogue. On the regularity of the Green current for semi-extremal endomorphisms of $ \mathbb{P}^2 $. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6767-6781. doi: 10.3934/dcds.2020163
References:
[1] L. Barreira and Y. Pesin, Nonuniform Hyperbolicity, Encyclopedia of Mathematics and its Applications, 115, Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9781107326026.  Google Scholar
[2]

E. Bedford and M. Jonsson, Dynamics of regular polynomial endomorphisms of Ck, Amer. J. Math., 122 (2000), 153-212.  doi: 10.1353/ajm.2000.0001.  Google Scholar

[3]

F. Berteloot and J.-J. Loeb, Spherical hypersurfaces and Lattès rational maps, J. Math. Pures Appl. (9), 77 (1998), 655-666. doi: 10.1016/S0021-7824(98)80003-2.  Google Scholar

[4]

F. Berteloot and J.-J. Loeb, Une caractérisation géométrique des exemples de Lattès de ${ \mathbb P}^k$, Bull. Soc. Math. France, 129 (2001), 175-188.  doi: 10.24033/bsmf.2392.  Google Scholar

[5]

F. Berteloot, C. Dupont and L. Molino, Normalization of bundle holomorphic contractions and applications to dynamics, Ann. Inst. Fourier (Grenoble), 58 (2008), 2137-2168.  doi: 10.5802/aif.2409.  Google Scholar

[6]

F. Berteloot and C. Dupont, Une caractérisation des endomorphismes de Lattès par leur mesure de Green, Comment. Math. Helv., 80 (2005), 433-454.  doi: 10.4171/CMH/21.  Google Scholar

[7]

I. Binder and L. DeMarco, Dimension of pluriharmonic measure and polynomial endomorphisms of $ \mathbb C^n$, Int. Math. Res. Not., 2003 (2003), 613-625.  doi: 10.1155/S1073792803206048.  Google Scholar

[8]

J.-Y. Briend and J. Duval, Exposants de Liapounoff et distribution des points périodiques d'un endomorphisme de CPk, Acta Math., 182 (1999), 143-157.  doi: 10.1007/BF02392572.  Google Scholar

[9]

S. Cantat and S. Le Borgne, Théorème limite central pour les endomorphismes holomorphes et les correspondances modulaires, Int. Math. Res. Not., 2005 (2005), 3479-3510.  doi: 10.1155/IMRN.2005.3479.  Google Scholar

[10]

T.-C. Dinh and C. Dupont, Dimension de la mesure d'équilibre d'applications méromorphes, J. Geom. Anal., 14 (2004), 613-627.  doi: 10.1007/BF02922172.  Google Scholar

[11]

T.-C. Dinh, V.-A. Nguyên and N. Sibony, Exponential estimates for plurisubharmonic functions and stochastic dynamics, J. Differential Geom., 84 (2010), 465-488.  doi: 10.4310/jdg/1279114298.  Google Scholar

[12]

T.-C. Dinh and N. Sibony, Decay of correlations and the central limit theorem for meromorphic maps, Comm. Pure Appl. Math., 59 (2006), 754-768.  doi: 10.1002/cpa.20119.  Google Scholar

[13]

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

[14]

R. Dujardin, Fatou directions along the Julia set for endomorphisms of $ \mathbb{CP}^k$, J. Math. Pures Appl. (9), 98 (2012), 591-615. doi: 10.1016/j.matpur.2012.05.004.  Google Scholar

[15]

C. Dupont, Exemples de Lattès et domaines faiblement sphériques de $ \mathbb C^n$, Manuscripta Math., 111 (2003), 357-378.  doi: 10.1007/s00229-003-0378-0.  Google Scholar

[16]

C. Dupont, Bernoulli coding map and almost sure invariance principle for endomorphisms of $\mathbb{P}^k$, Probab. Theory Related Fields, 146 (2010), 337-359.  doi: 10.1007/s00440-008-0192-4.  Google Scholar

[17]

C. Dupont, On the dimension of invariant measures of endomorphisms of $\mathbb{CP}$k, Math. Ann., 349 (2011), 509-528.  doi: 10.1007/s00208-010-0519-1.  Google Scholar

[18]

C. Dupont and J. Taflin, Dynamics of fibered endomorphisms of ${ \mathbb P}^k$, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci., arXiv: 1811.06909. Google Scholar

[19]

J. E. Fornæss and N. Sibony, Hyperbolic maps on $ \mathbf P^2$, Math. Ann., 311 (1998), 305-333.  doi: 10.1007/s002080050189.  Google Scholar

[20]

M. Jonsson, Dynamics of polynomial skew products on $ \mathbf C^2$, Math. Ann., 314 (1999), 403-447.  doi: 10.1007/s002080050301.  Google Scholar

[21]

M. Jonsson and D. Varolin, Stable manifolds of holomorphic diffeomorphisms, Invent. Math., 149 (2002), 409-430.  doi: 10.1007/s002220200220.  Google Scholar

[22] M. Klimek, Pluripotential Theory, London Mathematical Society Monographs, New Series, 6, The Clarendon Press, Oxford University Press, New York, 1991.   Google Scholar
[23]

F. Ledrappier, Quelques propriétés ergodiques des applications rationnelles, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 37-40.   Google Scholar

[24]

V. Mayer, Comparing measures and invariant line fields, Ergodic Theory Dynam. Systems, 22 (2002), 555-570.  doi: 10.1017/S0143385702000275.  Google Scholar

[25]

Y. B. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2004 doi: 10.4171/003.  Google Scholar

[26]

N. Sibony, Dynamique des applications rationnelles de $ \mathbf P^k$, in Dynamique et Géométrie Complexes (Lyon, 1997), Panor. Synthèses, 8, Soc. Math. France, Paris, 1999, 97–185.  Google Scholar

[27]

A. Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Math., 99 (1990), 627-649.  doi: 10.1007/BF01234434.  Google Scholar

show all references

References:
[1] L. Barreira and Y. Pesin, Nonuniform Hyperbolicity, Encyclopedia of Mathematics and its Applications, 115, Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9781107326026.  Google Scholar
[2]

E. Bedford and M. Jonsson, Dynamics of regular polynomial endomorphisms of Ck, Amer. J. Math., 122 (2000), 153-212.  doi: 10.1353/ajm.2000.0001.  Google Scholar

[3]

F. Berteloot and J.-J. Loeb, Spherical hypersurfaces and Lattès rational maps, J. Math. Pures Appl. (9), 77 (1998), 655-666. doi: 10.1016/S0021-7824(98)80003-2.  Google Scholar

[4]

F. Berteloot and J.-J. Loeb, Une caractérisation géométrique des exemples de Lattès de ${ \mathbb P}^k$, Bull. Soc. Math. France, 129 (2001), 175-188.  doi: 10.24033/bsmf.2392.  Google Scholar

[5]

F. Berteloot, C. Dupont and L. Molino, Normalization of bundle holomorphic contractions and applications to dynamics, Ann. Inst. Fourier (Grenoble), 58 (2008), 2137-2168.  doi: 10.5802/aif.2409.  Google Scholar

[6]

F. Berteloot and C. Dupont, Une caractérisation des endomorphismes de Lattès par leur mesure de Green, Comment. Math. Helv., 80 (2005), 433-454.  doi: 10.4171/CMH/21.  Google Scholar

[7]

I. Binder and L. DeMarco, Dimension of pluriharmonic measure and polynomial endomorphisms of $ \mathbb C^n$, Int. Math. Res. Not., 2003 (2003), 613-625.  doi: 10.1155/S1073792803206048.  Google Scholar

[8]

J.-Y. Briend and J. Duval, Exposants de Liapounoff et distribution des points périodiques d'un endomorphisme de CPk, Acta Math., 182 (1999), 143-157.  doi: 10.1007/BF02392572.  Google Scholar

[9]

S. Cantat and S. Le Borgne, Théorème limite central pour les endomorphismes holomorphes et les correspondances modulaires, Int. Math. Res. Not., 2005 (2005), 3479-3510.  doi: 10.1155/IMRN.2005.3479.  Google Scholar

[10]

T.-C. Dinh and C. Dupont, Dimension de la mesure d'équilibre d'applications méromorphes, J. Geom. Anal., 14 (2004), 613-627.  doi: 10.1007/BF02922172.  Google Scholar

[11]

T.-C. Dinh, V.-A. Nguyên and N. Sibony, Exponential estimates for plurisubharmonic functions and stochastic dynamics, J. Differential Geom., 84 (2010), 465-488.  doi: 10.4310/jdg/1279114298.  Google Scholar

[12]

T.-C. Dinh and N. Sibony, Decay of correlations and the central limit theorem for meromorphic maps, Comm. Pure Appl. Math., 59 (2006), 754-768.  doi: 10.1002/cpa.20119.  Google Scholar

[13]

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

[14]

R. Dujardin, Fatou directions along the Julia set for endomorphisms of $ \mathbb{CP}^k$, J. Math. Pures Appl. (9), 98 (2012), 591-615. doi: 10.1016/j.matpur.2012.05.004.  Google Scholar

[15]

C. Dupont, Exemples de Lattès et domaines faiblement sphériques de $ \mathbb C^n$, Manuscripta Math., 111 (2003), 357-378.  doi: 10.1007/s00229-003-0378-0.  Google Scholar

[16]

C. Dupont, Bernoulli coding map and almost sure invariance principle for endomorphisms of $\mathbb{P}^k$, Probab. Theory Related Fields, 146 (2010), 337-359.  doi: 10.1007/s00440-008-0192-4.  Google Scholar

[17]

C. Dupont, On the dimension of invariant measures of endomorphisms of $\mathbb{CP}$k, Math. Ann., 349 (2011), 509-528.  doi: 10.1007/s00208-010-0519-1.  Google Scholar

[18]

C. Dupont and J. Taflin, Dynamics of fibered endomorphisms of ${ \mathbb P}^k$, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci., arXiv: 1811.06909. Google Scholar

[19]

J. E. Fornæss and N. Sibony, Hyperbolic maps on $ \mathbf P^2$, Math. Ann., 311 (1998), 305-333.  doi: 10.1007/s002080050189.  Google Scholar

[20]

M. Jonsson, Dynamics of polynomial skew products on $ \mathbf C^2$, Math. Ann., 314 (1999), 403-447.  doi: 10.1007/s002080050301.  Google Scholar

[21]

M. Jonsson and D. Varolin, Stable manifolds of holomorphic diffeomorphisms, Invent. Math., 149 (2002), 409-430.  doi: 10.1007/s002220200220.  Google Scholar

[22] M. Klimek, Pluripotential Theory, London Mathematical Society Monographs, New Series, 6, The Clarendon Press, Oxford University Press, New York, 1991.   Google Scholar
[23]

F. Ledrappier, Quelques propriétés ergodiques des applications rationnelles, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 37-40.   Google Scholar

[24]

V. Mayer, Comparing measures and invariant line fields, Ergodic Theory Dynam. Systems, 22 (2002), 555-570.  doi: 10.1017/S0143385702000275.  Google Scholar

[25]

Y. B. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2004 doi: 10.4171/003.  Google Scholar

[26]

N. Sibony, Dynamique des applications rationnelles de $ \mathbf P^k$, in Dynamique et Géométrie Complexes (Lyon, 1997), Panor. Synthèses, 8, Soc. Math. France, Paris, 1999, 97–185.  Google Scholar

[27]

A. Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Math., 99 (1990), 627-649.  doi: 10.1007/BF01234434.  Google Scholar

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