viewed as discrete approximations to the Ricci and Calabi flows.

Instead of pursuing general questions of existence, in this paper we pick a simple compact complex manifold – the Riemann sphere – and investigate the convergence properties of each of $T$, $T_{\nu}$, and $T_K$ on the space of Hermitian metrics induced from Veronese embedings into $\mathbb{CP}^n.$ In section \[section: Higher dimensional projective space\] we briefly investigate the case $\mathbb{CP}^n$ when $n>1$.

There is a natural notion of distance on the space of Hermitian metrics $\mathrm{GL}(n+1, \mathbb{C})/\mathrm{U}(n+1)$, and indeed as $k$ increases this distance function is expected [@ps2] to approximate that on the infinite dimensional space of Kähler metrics [@mabuchi1], [@semmes], [@donaldson1], [@chen]. A natural question one might ask is: Are the $T, T_{\nu},$ or $T_K$ iterations distance reducing on the space of metrics? In section \[subsection: The effect on distance\] we show that the $T$ operator does *not* satisfy this property.

One goal of this study was to find an effective bound on the distance between the $r$th iteration of a metric under $T$, $T_{\nu}$, or $T_K$ and the limiting balanced metric. One is proposed in section \[subsection: The effect on distance\]. In section \[subsection: Asymptotic behavior\] we list the 