Let $\lambda$ be an ordinal, and let $f(\lambda)$ be the ordinal representing the order type of the limit ordinals strictly less than $\lambda$. Then $f$ is a weakly increasing function, $f(\lambda + 2) = f(\lambda+1)$ is always true, and $f(\lambda) \le \lambda$ is obvious.
- Is there a reference that describes the fixed points of $f$? I would expect at least some infinite cardinal numbers represented as their representative minimal ordinal should be a fixed point; obviously not $\aleph_0$, but I believe any unbounded set of ordinals smaller than $\aleph_1$ must be uncountable.
- Is there a name/reference for $f$, or perhaps for some canonical one-sided inverse of $f$, or in some other manner a closely related function?
This question defines the same function and asks about the same fixed points, but doesn't offer anything in the way of Google-fu assistance.