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On globally hypoelliptic abelian actions and their existence on homogeneous spaces
On the regularity of the Green current for semi-extremal endomorphisms of $ \mathbb{P}^2 $
Université de Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France |
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.
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 Ck, 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 CPk, 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 Ck, 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 CPk, 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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