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Regularity of dynamical Green’s functions. (English) Zbl 1172.32004
Summary: For meromorphic maps of complex manifolds, ergodic theory and pluripotential theory are closely related. In nice enough situations, dynamically defined Green 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 function.
In this paper we look at a variety of regularity properties that have been considered for dynamical Green 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 function.

MSC:
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
32U40 Currents
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