In Ordinal $\alpha$ such that $\alpha=\omega_\alpha$? the question is asked if fixed points exist for $\alpha=\omega_\alpha$.
I'm aware that $\varepsilon_0=\sup\{\omega, \omega^\omega, \omega^{\omega^\omega},\ldots\}$.
I'm not sure what the first fixed point of $\alpha=\omega_\alpha$ is called.
$$\sup\{\omega,\omega_\omega,\omega_{\omega_\omega},\ldots\}=\text{?}$$
I don't think it has a standard notation - it's generally just called "the first fixed point of the $\aleph$-function" (for some reason I haven't heard the term "$\omega$-function" nearly as frequently).
Some texts will introduce notation like "$\mathit{lfp}(G)$" for least fixed points of a continuous unbounded ordinal function $G$, at which point this can be written as e.g. "$\mathit{lfp}(\omega_-)$," but I don't think any of these are standard.