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Regularity of dynamical Green's functions


Authors: Jeffrey Diller and Vincent Guedj
Journal: Trans. Amer. Math. Soc. 361 (2009), 4783-4805
MSC (2000): Primary 32H50, 37F10, 37D25
DOI: https://doi.org/10.1090/S0002-9947-09-04740-0
Published electronically: April 7, 2009
MathSciNet review: 2506427
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Abstract: For meromorphic maps of complex manifolds, ergodic theory and pluripotential theory are closely related. In nice enough situations, dynamically defined Green's functions give rise to invariant currents which intersect to yield measures of maximal entropy. `Nice enough' is often a condition on the regularity of the Green's function. In this paper we look at a variety of regularity properties that have been considered for dynamical Green's functions. We simplify and extend some known results and prove several others which are new. We also give some examples indicating the limits of what one can hope to achieve in complex dynamics by relying solely on the regularity of a dynamical Green's function.


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  • [BD] E. Bedford & J. Diller: Energy and invariant measures for birational surfaces maps. Duke Math. J. 128 (2005), no. 2, 331-368. MR 2140266 (2006c:32016)
  • [BK] E. Bedford & K. Kim: On the degree growth of birational mappings in higher dimension. J. Geom. Anal. 14 (2004), no. 4, 567-596. MR 2111418 (2006b:14018)
  • [BS] E. Bedford & J. Smillie: Polynomial diffeomorphisms of $ \mathbf{C}^2$. III. Ergodicity, exponents, and entropy of the equilibrium measure. Math. Ann. 294 (1992), 395-420. MR 1188127 (93k:32062)
  • [Bri] J.-Y. Briend: Exposants de Lyapunoff et points périodiques d'endomorphismes holomorphes de $ \mathbf{P}^k$. Thése, Université Paul Sabatier, Toulouse (France), 1997.
  • [Bro] H. Brolin: Invariant sets under iteration of rational functions. Ark. Mat. 6 (1965) 103-144. MR 0194595 (33:2805)
  • [Bu] X. Buff: On the Bieberbach conjecture and holomorphic dynamics. Proc. Amer. Math. Soc. 131 (2003), no. 3, 755-759. MR 1937413 (2003i:37041)
  • [CG] L. Carleson & T. Gamelin: Complex dynamics. Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993. MR 1230383 (94h:30033)
  • [Di 1] J. Diller: Dynamics of birational maps of $ \mathbf{P}^2$. Indiana Univ. Math. J. 45 (1996), no. 3, 721-772. MR 1422105 (97k:32044)
  • [Di 2] J. Diller: Invariant measure and Lyapunov exponents for birational maps of $ \mathbf{P}^2$. Comment. Math. Helv. 76 (2001), no. 4, 754-780. MR 1881705 (2003a:32029)
  • [DF] J. Diller & C. Favre: Dynamics of bimeromorphic maps of surfaces. Amer. J. Math. 123 (2001), no. 6, 1135-1169. MR 1867314 (2002k:32028)
  • [DS 1] T.-C. Dinh & N. Sibony: Regularization of currents and entropy. Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 6, 959-971. MR 2119243 (2006c:32045)
  • [DS 2] T.-C. Dinh & N. Sibony: Dynamique des applications d'allure polynomiale. J. Math. Pures Appl. 82 (2003), 367-423. MR 1992375 (2004e:37063)
  • [DS 3] T.-C. Dinh & N. Sibony: Green currents for holomorphic automorphisms of compact Kahler manifolds. J. Amer. Math. Soc. 18 (2004), no. 2, 291-312. MR 2137979 (2007d:32015)
  • [Du] R. Dujardin: Laminar currents and birational dynamics. Duke Math. J. 131 (2006), no. 2, 219-247. MR 2219241 (2007b:37097)
  • [F] C. Favre: Points périodiques d'applications birationnelles de $ \mathbf{P}^2$. Ann. Inst. Fourier (Grenoble) 48 (1998), no. 4, 999-1023. MR 1656005 (99i:32031)
  • [FG] C. Favre & V. Guedj: Dynamique des applications rationnelles des espaces multiprojectifs. Indiana Univ. Math. J. 50 (2001), no. 2, 881-934. MR 1871393 (2002m:32024)
  • [FJ] C. Favre & M. Jonsson: Eigenvaluations. Ann. Sci. École Norm. Sup. 40 (2007), no. 2, 309-349. MR 2339287
  • [FS 1] J.-E. Fornaess & N. Sibony: Complex Hénon mappings in $ \mathbf{C}^2$ and Fatou-Bieberbach domains. Duke Math. J. 65 (1992), no. 2, 345-380. MR 1150591 (93d:32040)
  • [FS 2] J.-E. Fornaess & N. Sibony: Complex dynamics in higher dimensions. Notes partially written by Estela A. Gavosto. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 439, Complex potential theory (Montreal, PQ, 1993), 131-186, Kluwer Acad. Publ., Dordrecht, 1994. MR 1332961 (96k:32057)
  • [FS 3] J.-E. Fornaess & N. Sibony: Complex dynamics in higher dimension II. Modern methods in complex analysis (Princeton, NJ, 1992), 135-182, Ann. of Math. Stud., 137, Princeton Univ. Press, Princeton, NJ, 1995. MR 1369137 (97g:32033)
  • [GH] P. Griffiths & H. Harris: Principles of algebraic geometry. Reprint of the 1978 original. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1994. xiv+813 pp. MR 1288523 (95d:14001)
  • [Gr] M. Gromov: Entropy, homology and semialgebraic geometry. Séminaire Bourbaki, Vol. 1985/86. Astérisque No. 145-146 (1987), 5, 225-240. MR 880035 (89f:58082)
  • [G 1] V. Guedj: Dynamics of polynomial mappings of $ \mathbf{C}^2$. Amer. J. Math. 124 (2002), no. 1, 75-106. MR 1879000 (2003b:32021)
  • [G 2] V. Guedj: Decay of volumes under iteration of meromorphic mappings. Ann. Inst. Fourier 54, no. 7 (2004), 2369-2386. MR 2139697 (2005m:32035)
  • [G 3] V. Guedj: Ergodic properties of rational mappings with large topological degree. Annals of Math. 161 no. 3 (2005), 1589-1607. MR 2179389 (2006f:32027)
  • [GS] V. Guedj & N. Sibony: Dynamics of polynomial automorphisms of $ \mathbf{C}^k$. Ark. Mat. 40 (2002), no. 2, 207-243. MR 1948064 (2004b:32029)
  • [HP] J.-H. Hubbard & P. Papadopol: Superattractive fixed points in $ \mathbf{C}^n$. Indiana Univ. Math. J. 43 (1994), 321-365. MR 1275463 (95e:32025)
  • [K] M. Kosek: Hölder continuity property of filled-in Julia sets in $ \mathbf{C}^n$. Proc. Amer. Math. Soc. 125 (1997), no. 7, 2029-2032. MR 1376994 (97i:32013)
  • [Ly] M. Ju. Lyubich: Entropy properties of rational endomorphisms of the Riemann sphere. Ergodic Theory Dyn. Syst. 3 (1983), 351-385. MR 741393 (85k:58049)
  • [Meo] M. Meo. Image inverse d'un courant positif fermé par une application analytique surjective. C. R. Acad. Sci. Paris Ser. I Math. 322 (1996), no. 12, 1141-1144. MR 1396655 (97d:32013)
  • [Sm] J. Smillie: The entropy of polynomial diffeomorphisms of $ \mathbf{C}^2$. Ergodic Theory Dynam. Systems 10 (1990), no. 4, 823-827. MR 1091429 (92b:58131)
  • [S] N. Sibony: Dynamique des applications rationnelles de $ \mathbb{P}^k$. In Dynamique et géométrie complexes, Panoramas et Synthèses (1999). MR 1760844 (2001e:32026)
  • [Y] Y. Yomdin: Volume growth and entropy. Israel J. Math. 57 (1987), no. 3, 285-300. MR 889979 (90g:58008)

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Additional Information

Jeffrey Diller
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: diller.1@nd.edu

Vincent Guedj
Affiliation: Centre de Mathématique et Informatique, Université Aix-Marseille 1, Latp, 13453 Marseille Cedex 13, France
Email: guedj@cmi.univ-mrs.fr

DOI: https://doi.org/10.1090/S0002-9947-09-04740-0
Keywords: Complex dynamics, meromorphic maps, pluripotential theory, Green's function
Received by editor(s): November 10, 2006
Received by editor(s) in revised form: August 28, 2007
Published electronically: April 7, 2009
Additional Notes: The first author gratefully acknowledges support from National Science Foundation grant DMS06-53678 during the preparation of this article.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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