For an ordinal number $\alpha$, the epsilon number $\varepsilon\_\alpha$ is defined as the "$\alpha$-th" fixed point of the map $n \mapsto \omega^n$, i.e. $\omega^{\varepsilon\_\alpha} = \varepsilon\_\alpha$.


Question: What about fixed points of $n \mapsto \varepsilon\_n$ or $n \mapsto \omega\_n$? Examples for the former would be $\varepsilon\_{\varepsilon\_{\varepsilon\_{\dots}}}$ and (if I'm not mistaken) all $\omega\_\alpha$ for $\alpha\ge 1$, and an example for the latter would be $\omega\_{\omega\_{\omega\_{\dots}}}$. Do they have names? Are they studied? Do they have interesting properties?



Set theory: fixed points of $n \mapsto \varepsilon_n$ and $n \mapsto \omega_n$
The least fixed point of $\mu \mapsto \varepsilon\_\mu$ is called $\zeta\_0$ and is written as $\varphi\_2(0)$ using the Veblen $\varphi$ function. It is also equal to $\psi(\Omega)$ in Madore's psi function and $\psi\_0(\psi\_1(\psi\_1(0)))$ in Buchholz's psi function. You can read more about it here: [https://googology.fandom.com/wiki/Cantor's\_ordinal](https://googology.fandom.com/wiki/Cantor%27s_ordinal)


If by $\omega\_\mu$ you mean the enumeration of infinite cardinals, then the least fixed point of $\mu \mapsto \omega\_\mu$ doesn't have a specific name, but it is written as $\Phi\_1(0)$ using Rathjen's $\Phi$ function. The ordinal doesn't have that many specific properties, apart from $\psi\_{\chi\_0(0)}(\Phi\_1(0))$ in [Rathjen's psi function](https://googology.fandom.com/wiki/Rathjen%27s_psi_function) being the countable limit of the extended Buchholz's psi function.


