case there are exactly $\lceil k/2 \rceil$ real (positive) parameters, while in the non-palindromic case there are $k+1.$ However we note that for any of the operators $F = T, T_{\nu}, T_K$, and any starting metric $(a_0, \ldots, a_k)$, if we let $(\tilde{a}_0, \ldots, \tilde{a}_k)$ denote the metric after an application of $F$, then we have a relation $$\label{eqn:  metric relation}
    \sum_{i=0}^k \frac{a_i}{\tilde{a}_i} = k+1.$$ This is immediately verified by checking formulas (\[T map\]), (\[Tnu map\]), (\[TK map\]).

Findings {#section: Findings}
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In investigating the behavior of the convergence of a sequence of Hermitian metrics we need to decide what we mean when we say two metrics are close. Let $M = \mathrm{GL}(k+1, \mathbb{C})/U(k+1)$ be the space of Hermitian metrics on $\mathbb{CP}^k$. The $\mathrm{GL}(k+1,\mathbb{C})$-invariant Kähler metric is given by the form $g_H(U,V) = tr(H^{-2}UV)$ where $U, V$ are in the tangent space to $H$ on $M$. Geodesics on $M$ are given by the images of one-parameter subgroups, e.g. $$\left( \begin{array}{ccc} e^{\alpha_0 t} \\ & \ddots \\ & & e^{\alpha_k t} \end{array} \right).$$ Let $A = (a_0, \dots, a_k)$, and $B = (b_0, \ldots, b_k)$ be two metrics in $M$. Writing $a_i = e^{\alpha_i}$ and $b_i = e^{\beta_i}$ for $i=0, \ldots, 