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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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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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: http://dx.doi.org/10.1090/S0002-9947-09-04740-0
PII: S 0002-9947(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.