which we get the [*round metric*]{} – the only palindromic balanced metrics for any $k$. This can be explained by the fact that both the $T$ and $T_K$ maps respect the induced action of $\mathrm{SL}(2, \mathbb{C})$ on the space of metrics, while $T_{\nu}$ does not.

Starting with arbitrary $G = (a_0, a_1, \ldots, a_k)$ it is not entirely clear which balanced metric iterations of any of the operators $T, T_{\nu}, T_K$ will tend towards; all we can say is the coefficients will be of the form $B = (b_0, \ldots, b_k)$ where $b_q = \alpha c^q {k \choose q}$ for some $\alpha, c >0$, and if $G$ is palindromic or the operator is $T_{\nu}$ then $c=1.$ We also note that when $k=2$ we can calculate the value $c$ as $c=\sqrt{a_2/a_0}$ and thus the balanced metric will be of the form $\alpha ( a_0, 2\sqrt{a_0a_2} , a_2 )$ for some scalar $\alpha>0$.

Asymptotic behavior {#subsection: Asymptotic behavior}
-------------------

In the long run the behavior of the iterations of $F = T, T_{\nu}, T_K$ is predictable. For each function the limiting ratio $$\sigma_{F, k} := \lim_{r \rightarrow \infty}
        \frac{
            \mathrm{dist}(F^{(r+1)}(G),\ F^{(\infty)}(G))
        }{
            \mathrm{dist}(F^{(r)}(G),\ F^{(\infty)}(G))
        }$$ exists, and converges to a simple limit. 