On the regularity of the Green current for semi-extremal endomorphisms of <inline-formula><tex-math id="M1">\begin{document} \mathbb{P}^2 \end{document}</tex-math></inline-formula>

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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 and 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.
[2]	E. Bedford and M. Jonsson, Dynamics of regular polynomial endomorphisms of C^k, Amer. J. Math., 122 (2000), 153-212. doi: 10.1353/ajm.2000.0001.
[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.
[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.
[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.
[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.
[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.
[8]	J.-Y. Briend and J. Duval, Exposants de Liapounoff et distribution des points périodiques d'un endomorphisme de CP^k, Acta Math., 182 (1999), 143-157. doi: 10.1007/BF02392572.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[19]	J. E. Fornæss and N. Sibony, Hyperbolic maps on $ \mathbf P^2$, Math. Ann., 311 (1998), 305-333. doi: 10.1007/s002080050189.
[20]	M. Jonsson, Dynamics of polynomial skew products on $ \mathbf C^2$, Math. Ann., 314 (1999), 403-447. doi: 10.1007/s002080050301.
[21]	M. Jonsson and D. Varolin, Stable manifolds of holomorphic diffeomorphisms, Invent. Math., 149 (2002), 409-430. doi: 10.1007/s002220200220.
[22]	M. Klimek, Pluripotential Theory, London Mathematical Society Monographs, New Series, 6, The Clarendon Press, Oxford University Press, New York, 1991.
[23]	F. Ledrappier, Quelques propriétés ergodiques des applications rationnelles, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 37-40.
[24]	V. Mayer, Comparing measures and invariant line fields, Ergodic Theory Dynam. Systems, 22 (2002), 555-570. doi: 10.1017/S0143385702000275.
[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.
[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.
[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.

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.
[2]	E. Bedford and M. Jonsson, Dynamics of regular polynomial endomorphisms of C^k, Amer. J. Math., 122 (2000), 153-212. doi: 10.1353/ajm.2000.0001.
[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.
[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.
[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.
[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.
[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.
[8]	J.-Y. Briend and J. Duval, Exposants de Liapounoff et distribution des points périodiques d'un endomorphisme de CP^k, Acta Math., 182 (1999), 143-157. doi: 10.1007/BF02392572.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[19]	J. E. Fornæss and N. Sibony, Hyperbolic maps on $ \mathbf P^2$, Math. Ann., 311 (1998), 305-333. doi: 10.1007/s002080050189.
[20]	M. Jonsson, Dynamics of polynomial skew products on $ \mathbf C^2$, Math. Ann., 314 (1999), 403-447. doi: 10.1007/s002080050301.
[21]	M. Jonsson and D. Varolin, Stable manifolds of holomorphic diffeomorphisms, Invent. Math., 149 (2002), 409-430. doi: 10.1007/s002220200220.
[22]	M. Klimek, Pluripotential Theory, London Mathematical Society Monographs, New Series, 6, The Clarendon Press, Oxford University Press, New York, 1991.
[23]	F. Ledrappier, Quelques propriétés ergodiques des applications rationnelles, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 37-40.
[24]	V. Mayer, Comparing measures and invariant line fields, Ergodic Theory Dynam. Systems, 22 (2002), 555-570. doi: 10.1017/S0143385702000275.
[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.
[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.
[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.

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