$$R_K = \frac{\mathrm{dim}\,  \mathrm{H}^0(X, L^k)}{\mathrm{Vol}(X, \omega_{G,K})}.$$ As before set $T_K(G) = \mathrm{Hilb}_K(\mathrm{FS}(G)).$

A Hermitian metric $G$ is [*balanced with respect to* ]{} $T$ (resp. $T_{\nu}, T_K$) if $T(G) = G$ (resp. $T_{\nu}(G) = G, T_K(G)=G$). The basic philosophy is that if $F = T, T_{\nu}, T_K$ and if there exists some balanced metric, then starting with any Hermitian metric $G$ the iterations $F^{(r)}(G)$ should tend to a balanced metric as $r$ tends to infinity (see [@donaldson5] and also [@sano]). In this paper we will concern ourselves only with a very simple case and study in some detail the properties of this convergence.

Specifically we take as our manifold the Riemann sphere $X = \mathbb{CP}^1$ and line bundle $L = O_X(1)$. We note that the presence of the automorphism group $\mathrm{SL}(2, \mathbb{C})$ means that, strictly speaking, some aspects of the theory may need to be developed further, in the manner of [@mabuchi2] for example, but since we are focusing on numerical results here we will not dwell on this issue. Fix a holomorphic coordinate $z \in \mathbb{C}.$ Then $\mathrm{H}^0(X,L^k) = \mathrm{H}^0(\mathbb{CP}^1, O(k)) \cong \mathbb{C}^{k+1}$ has basis $1, z, z^2, \ldots, z^k$. Hermitian metrics can now be associated with $(k+1)\times(k+1)$ positive definite 