  We introduce functions associated to polarized dynamical systems that
generalize averages of the dynamical Arakelov-Green's functions for rational
functions due to Baker and Rumely. For a polarized dynamical system $X\to X$
over a product formula field, we prove an Elkies-style lower bound for these
functions evaluated on the adelic points of $X$. As an application, we prove a
Lehmer-type lower bound on the canonical height of a non-torsion point $P$ on
an abelian variety $A/K$, where $K$ is a product formula field having perfect
residue fields at its completions (for instance, $K$ may be a number field or
the function field of a curve over $\mathbb{C}$ or $\mathbb{F}_p$). For $A$ of
dimension $g$, the lower bound has the form
\[\widehat{h}(P)\ge\frac{C}{D^{2g+3}(\log D)^{2g}},\] where
$C=C(A,K,\widehat{h})>0$, $D=[K(P):K]\ge2$, and $P\in A(\bar{K})\setminus
A(\bar{K})_{\text{tors}}$ is not contained in a torsion translate of an abelian
subvariety of $A$ having everywhere potential good reduction.
