Given some $f: \Bbb N \to \Bbb N$, we have various ways to talk about how fast it's growing: big-O notation, little-o notation, and so on. These are a good way to compare growth rates, so we can talk about some $f$ growing faster than some $g$, and so on.
Does a similar notion exist for functions from $\textbf{Ord} \to \textbf{Ord} $, or at least a well-behaved subclass like the "normal" functions (strictly increasing and continuous)?
The main issue would be that for any such function $f$, there is always a proper class of fixed points, where $f(x) = x$. As a result, no matter where you are in the function, there is some point later where the identity function $\text{id}(x)$ "catches up" with $f(x)$. For example, the function $f(x) = \omega\cdot x = \text{id}(x)$ whenever $x = \omega^\omega \cdot a$ for some ordinal $a$. It then periodically skips ahead after that and catches up with it again at the next $\omega^\omega \cdot (a+1)$.
Regardless, it seems possible that some notion of the growth rate of ordinal functions could exist. Has one been developed and does anyone have any references?