We now study the linearization problem in the case $|\lambda| = 1$, with $\lambda$ not a root of unity. We set, for $m \geqslant 1$, $$\alpha_m := \min(1/5, \omega_{m+1}, \omega_{m+2}, \ldots, \omega_{2m}), \quad \gamma_m := \alpha_m^{2/m},$$ where $\omega_k := |\lambda^k - \lambda|, k \geqslant 2$, and $G$ as defined in question (26). Show that $$\hat{G}(r) \leqslant \left(\alpha_m + (1 + \alpha_m)\alpha_m^2 + \frac{\alpha_m(1 + \alpha_m)(1 + \alpha_m^2)}{1 - \alpha_m}\right) r \leqslant r$$ for all $r$ such that $$0 \leqslant r \leqslant \frac{1 - \alpha_m}{(1 + \alpha_m)(1 + \alpha_m^2)} \gamma_m r_0$$
We now study the linearization problem in the case $|\lambda| = 1$, with $\lambda$ not a root of unity. We set, for $m \geqslant 1$,
$$\alpha_m := \min(1/5, \omega_{m+1}, \omega_{m+2}, \ldots, \omega_{2m}), \quad \gamma_m := \alpha_m^{2/m},$$
where $\omega_k := |\lambda^k - \lambda|, k \geqslant 2$, and $G$ as defined in question (26).
Show that
$$\hat{G}(r) \leqslant \left(\alpha_m + (1 + \alpha_m)\alpha_m^2 + \frac{\alpha_m(1 + \alpha_m)(1 + \alpha_m^2)}{1 - \alpha_m}\right) r \leqslant r$$
for all $r$ such that
$$0 \leqslant r \leqslant \frac{1 - \alpha_m}{(1 + \alpha_m)(1 + \alpha_m^2)} \gamma_m r_0$$