$c^{(i)}$ be the corresponding binding critical point at time $t_i$, $r_i$ is the unique integer such that $|c_i-f^{\nu_i}x_*|\in(L^{-r_i},L^{-r_i+1}].$ We call $${\bf i}: = (t_1, r^{(1)}, c^{(1)}; \ t_2, r^{(2)}, c^{(2)}; \
\cdots; \ t_q, r^{(q)}, c^{(q)})$$ the [*itinerary*]{} of $v_0 = f_a(c)$ up to time $n$.

Let $E_n(c, {\bf i})$ denote the set of all $a\in E_n$ for which (R$)_{n,c}$ fails at time $n$, and the itinerary for $c_0(a)=f_ac$ up to time $n$ is ${\bf i}$. To estimate the measure of $E_n$, we first estimate the measure of $E_n(c, {\bf i})$. We then combine it with a bound on the nubmer of all feasible itineraries.

\[pro-R1\] $|E_n(c, {\bf i})| \leq L^{-\frac{1}{3} R}$, where $R = r_1 + r_2
\cdots + r_q$.

First of all, for $a_* \in R_n(c, {\bf i})$ and $1\leq k\leq q$ we define a parameter interval $I_{t_k}(a_*)$ as follows. If $|\gamma_{t_k}^{(c)}(I_n(a_*,
c))|<\frac{1}{4}$, then we let $I_{t_k}(a_*)=I_{t_k}(a_*, c)$. Otherwise, take $I_{t_k}(a_*)$ to be the interval of length $\frac{1}{10|\gamma_{t_k}^{(c)} (I_{t_k}(a_*, c))|}|{I}_{t_k}(a_*, c)|$ centered at $a_*$.

The proof of Lemma \[pro-R1\] is outlined as follows. For a compact interval $I$ centered at $a$ and $r>0$, let $ r\cdot I$ denote the interval of length $r|I|$ centered at $a$. For each 