distance {#subsection: The effect on distance}
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Despite this simple long-term behavior of the $T, T_{\nu},$ and $T_K$ iterations, the early behavior is still somewhat mysterious. Perhaps one surprising fact along these lines is that [*in general the $T$ operator is not distance reducing on the space of Hermitian metrics on $\mathrm{H}^0\left(\mathbb{CP}^1, O(k)\right)$*]{}. An example when $k=6$ is given in section \[section: Examples\]. This is the smallest value of $k$ for which the author has found such an example.

In [@cc] Calabi and Chen show that the Calabi flow is, in a certain sense, distance reducing. Hence it might be surprising that $T$ is not given the expectation that it can be viewed as a discrete version of such a flow.

While it can happen that $T(G)$ is farther from the balanced metric than $G$ is, it does not appear to be the case that it can be [*arbitrarily*]{} farther. Indeed for each of the operators $T, T_{\nu}, T_K$ the amount it can “magnify” the distance from the balanced metric appears to be simply bounded by a slow function of $k$. This leads us to conjecture a bound for how far the $r$th iteration of any of the operators can 