  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 entirely 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.
