Question: Are the following statements 1-7 correct ?
Let $\Lambda$ be the set of countable limit ordinals (0 added), $\psi:\omega_1\to\Lambda$ maps $\alpha\mapsto\omega\cdot\alpha$. Then:
- $\psi$ is normal (i.e. strictly increasing and continuous) bijection and $\psi(x)\geq x$ for all $x$.
- $S=\{\omega^{\omega^\alpha}\cdot\beta\}_{\alpha,\beta<\omega_1}$ is exactly the set of fixed points of $\psi$, i.e. $\psi(x)=x\iff x\in S$ (not used further)
- $\{\varepsilon_\alpha\}_{\alpha<\omega_1}\subset S$, where $\varepsilon_\alpha$ are fixed points of $\beta\mapsto\omega^\beta$.
If 1,3 are correct let $\varphi=\psi^{-1}:\Lambda\to\omega_1$. Obviously, $S$ is the set of all fixed points of $\varphi$ and $\varphi(x)\leq x$. Let $S_\alpha=(S\cap[0;\varepsilon_\alpha])\setminus 0$ be the set of all nonzero fixed points of $\varphi$ less than $\varepsilon_\alpha+1$, $~B_\alpha=([0;\varepsilon_\alpha]\setminus S_\alpha)\cap\Lambda$ be the set of all points of $\Lambda_\alpha=\Lambda\cap[0;\varepsilon_\alpha]$ which are not fixed. So, $\Lambda_\alpha=S_\alpha\cup B_\alpha$. Denote $\sigma_\alpha,\beta_\alpha$ the order types of $S_\alpha,B_\alpha$ respectively.
- $\beta_\alpha\geq\sigma_\alpha$, ($\beta_\alpha=\sigma_\alpha$ ?)
Let $f_\alpha:S_\alpha\to B_\alpha$ be monomorphism on initial segment (isomorphism?) of well-ordered sets.
- $f_\alpha(x)<x$ for all $x\neq 0$
Finally, we define the following family of functions $g_\alpha:\Lambda\to\omega_1$: $$ g_\alpha(x)= \begin{cases} \varphi(x),~ \text{ if } ~x\in B_\alpha \\ \varphi(f_\alpha(x)),~ \text{ if } ~x\in S_\alpha \\ \varepsilon_\alpha,~\text{ otherwise, i.e.} ~x>\varepsilon_\alpha \end{cases} $$
- $g_\alpha(x)<x$ for all $x\neq 0,~\alpha$ ($g_\alpha$ is regressive)
- $\omega_1$-sequence $g_\alpha(\varepsilon_\alpha)$ is strictly increasing