$\alpha>0$ arbitrarily large, $\beta>0$ and $C\ge1$, and sequences $|b_k|\to\infty$, $\psi_k\in[0,2\pi)$, $u_k\in F_\theta(Y)$ with $|u_k|\equiv1$, such that setting $n_k=[\beta\ln|b_k|]$, $$|M_{b_k}^{n_k}u_k(y)-e^{i\psi_k}u_k(y)|\le C|b_k|^{-\alpha},$$ for all $y\in Z$ and all $k\ge1$.

A subset $Z_0\subset Y$ is called a [*finite subsystem*]{} if $Z_0=\bigcap_{n\ge0} F^{-n}Z$ where $Z$ is a finite union of partition elements $a\in\alpha$.

\[prop-A2\] There exists a finite subsystem $Z_0$ such that there are no approximate eigenfunctions on $Z_0$.

By (A2), we can fix two periodic orbits with periods $\tau_1$ and $\tau_2$ such that $\tau_1/\tau_2$ is Diophantine. Let $Z$ be the union of the partition elements $a\in\alpha$ intersected by the periodic orbits and define $Z_0 =\bigcap_{n\ge0}F^{-n}Z$. It follows from [@Dolgopyat98b Section 13] that there are no approximate eigenfunctions on $Z_0$.

\[lem-approx\] There exists $\alpha>0$ and $C\ge1$ such that $\|(I-\hat R(ib))^{-1}\|_\theta\le C|b|^\alpha$ for all $b\in{{\mathbb R}}$ with $|b|\ge1$.

By , it suffices to prove this with $\hat R(ib)$ replaced by $R_0(ib):F_\theta(Y)\to F_\theta(Y)$. By Proposition \[prop-A2\], there is a finite subsystem on which there are no approximate eigenfunctions. The estimate for $(I-\hat R_0)^{-1}$ follows from this by exactly the argument used in [@M07 Lemmas 3.12 and 3.13]. Lemma \[lem-Rb\] plays the role of [@M07 Proposition 3.7].

\[rmk-norm\] We shall consider various families of linear operators acting 