be from the balanced metric.

Let $F = T, T_{\nu}, T_K$, let $G$ be any metric, and set $B = F^{(\infty)}(G)$ to be the balanced metric which the dynamical system $\{ F^{(r)}(G), r = 0, 1, 2, \ldots \}$ converges to. Recall that we define $$\mathrm{err}_{F, k} (G, r) = \mathrm{dist}\left( F^{(r)}(G),\ B \right).$$ Let $d$ denote the initial distance from $G$ to $B$ in the space of Hermitian metrics. Then we propose that in fact $$\label{bound}
    \mathrm{err}_{F, k} (G, r) < \log\left(1 + e^{kd} \sigma_{F, k}^r\right)$$ for every $k>1$. We do not expect this bound to be sharp.

Examples {#section: Examples}
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In this section we illustrate the findings from section \[section: Findings\] with some examples. We will always scale all metrics uniformly so that the limiting balanced metric begins with a one. Note that each of the operators $T, T_{\nu}, T_K$ respect scaling.

We begin with $k=2$ and a non-palindromic metric proportional to $G = (1, 17, 36),$ and consider the $T_K$ iterations. According to section \[subsection: The balanced metrics\] the limiting balanced metric will be, after scaling, $B=(1, 12, 36).$ Below is a table displaying the results: the first column gives the iteration $r$; the next three 