a fundamental paper, Donaldson [@donaldson2] showed that, under an assumption on the space of automorphisms, the metrics induced from *balanced* embeddings [@zhang] of $X$ into projective space by sections of $L^k$ converge to the constant scalar curvature metric as $k\rightarrow \infty$. The balanced condition means that $$\int_X \frac{Z_i \overline{Z_j}}{|Z|^2}\, d\mu \ = \ c\delta_{ij},$$ (where $d\mu$ is the volume form on $X$ induced by the Fubini-Study metric and $c$ is a constant depending on the data $(X, L^k)$ and not on the particular embedding) and this is equivalent to the Chow stability of the embedding [@zhang], [@luo], [@ps1].

Recently [@donaldson4; @donaldson5], Donaldson has devised iterative procedures on the space of Hermitian metrics on $\mathrm{H}^0(X, L^k)$ to find approximations to these balanced metrics. For sufficiently large $k$, these approximations are close to a constant scalar curvature metric. Explicit numerical computations, focused in the Calabi-Yau case where there are possible applications to string theory, have been obtained in [@donaldson5], [@dklr1], [@dklr2]. See also [@hw], [@keller2], and [@rubinstein] where different methods are used.

Donaldson’s three iterative maps $T$, $T_{\nu}$, and $T_K$, described below, are interesting in their own right. Indeed, as pointed out in [@donaldson5], it is likely that these maps can be 