F_\theta(Y)$, $b\in{{\mathbb R}}$,

-   $|\hat R_0(ib)v|_\infty\le|v|_\infty$,

-   $|\hat R_0(ib)^nv|_\theta \le C\{(1+|b|^\epsilon\int_Y\varphi^\epsilon\,d\mu)|v|_\infty+\theta^n|v|_\theta\}$.

-   $\|R^nv-\int_Yv\,d\mu\|_\theta\le C\tau^n\|v\|_\theta$.

Note that $\hat R_0(s)^nv=R^n(e^{-s\varphi_n}v)$. Since $|R|_\infty=1$, it follows that part (a) is valid. Part (c) follows from quasicompactness [@Aaronson Section 4.7].

It remains to prove (b). Our argument improves [@BHM05] where it is assumed that $\varphi\in L^1(Y)$. Let $y,y'\in Y$. Then $$(\hat R_0(ib)^n(1_av ))(y)-(\hat R_0(ib)^n(1_av ))(y')=D_1+D_2+D_3,$$ where $$\begin{aligned}
D_1 & = (e^{p_n(y_a)}-e^{p_n(y'_a)})e^{ib\varphi_n(y_a)}v(y_a), \quad
D_2  = e^{p_n(y'_a)}(e^{ib\varphi_n(y_a)}-e^{ib\varphi_n(y'_a)})v(y_a), \\
D_3 & = e^{p_n(y'_a)}e^{ib\varphi_n(y'_a)}(v(y_a)-v(y'_a)).\end{aligned}$$ By the estimates , $$\begin{aligned}
|D_1| & \le C_1 \mu(a)|v|_\infty d_\theta(y,y'), \quad
|D_3|  \le C_1 \mu(a)|v|_\theta d_\theta(y_a,y'_a) = C_1\theta^n\mu(a)|v|_\theta d_\theta(y,y').\end{aligned}$$ Summing over $a\in\alpha_n$, we obtain that the terms of type $D_1$ and $D_3$ contribute $C_1|v|_\infty$ and $C_1\theta^n|v|_\theta$ respectively to $|\hat R_0(ib)^nv|_\theta$.

Next, $$|D_2|\le C_1 \mu(a)\sum_{j=0}^{n-1}|e^{ib\varphi(F^jy_a)}-e^{ib\varphi(F^jy'_a)}||v|_\infty,$$ so using the inequality $|e^{ix}-1|\le\min\{1,|x|\}\le |x|^\epsilon$ for $x\in{{\mathbb R}}$, $\epsilon\in[0,1]$, $$|D_2|\le C_1 \mu(a)|b|^\epsilon\sum_{j=0}^{n-1}|1_{F^ja}\varphi|_{\theta_0}^\epsilon
d_{\theta_0}(F^jy_a,F^jy'_a)^\epsilon|v|_\infty.$$ Let $\theta=\theta_0^\epsilon$. Then $d_{\theta_0}(F^jy_a,F^jy'_a)^\epsilon
=d_\theta(F^jy_a,F^jy'_a)
=\theta^{n-j}d_\theta(y,y')$. By (A1), $$|D_2|\le C_1C_2^\epsilon \mu(a)|b|^\epsilon\sum_{j=0}^{n-1}\inf(1_{F^ja}\varphi)^\epsilon
\theta^{n-j}d_\theta(y,y')|v|_\infty.$$

Hence, summing over $a\in\alpha_n$, the $D_2$ terms contribute $C_1C_2^\epsilon|b|^\epsilon|v|_\infty S$, where $$\begin{aligned}
S & =\sum_{a\in\alpha_n}\mu(a)\sum_{j=0}^{n-1}\theta^{n-j}\inf(1_{F^ja}\varphi)^\epsilon 
= \sum_{d\in\alpha}\sum_{j=0}^{n-1}\sum_{a\in\alpha_n:F^ja=d}\mu(a)\theta^{n-j}\inf(1_d\varphi)^\epsilon \\ &
= \sum_{d\in\alpha}\sum_{j=0}^{n-1}\mu(d)\theta^{n-j}\inf(1_d\varphi)^\epsilon
\le \theta(1-\theta)^{-1}\sum_{d\in\alpha}\mu(d)\inf(1_d\varphi)^\epsilon 
\le \theta(1-\theta)^{-1}\int_Y\varphi^\epsilon\,d\mu.\end{aligned}$$ This completes the proof of part (b).

For $b\in{{\mathbb R}}$, define $M_b:L^\infty(Y)\to L^\infty(Y)$, $M_bv=e^{ib\varphi}v\circ F$.

\[defn-approx\] There are [*approximate eigenfunctions*]{} on a subset $Z\subset Y$ if there exist constants 