|\gamma_j^{(c)}(I_{n})|\leq 2|\tau_j(a_*)||I_{n}|\leq
4|(f_{a_*}^j)' c_0(a_*)||I_{n}| \leq
2L^{-\frac{1}{2}}d_C(c_j(a_*))d_S(c_j(a_*)).\end{aligned}$$ This implies $d_S(c_j(a))\geq \frac{1}{2} d_S(c_j(a_*))$ for all $a\in I_{n}$. Thus from Lemma \[derivative\](b), $$|f''|\leq \frac{KL}{d_S(c_j(a_*))^{2}}\quad\text{on $\gamma_j^{(c)}(I_{n})$.}$$ It then follows that $$\left|f_{a}'c_j(a)-f_{a_*}'c_j(a_*)\right| =
\left|f_{a_*}'c_j(a)-f_{a_*}'c_j(a_*)\right| \leq
\frac{KL}{d_S(c_j(a_*))^{2}}|c_j(I_{n})| \leq
\frac{KL}{d_S(c_j(a_*))^{2}}|\tau_j(a_*)| \cdot |I_{n}|,$$ where (\[samp-induct\]) is again used for the last inequality. We have then $$\left|f_{a}'c_j(a)-f_{a_*}'c_j(a_*)\right| \leq
\frac{KL^{\frac{1}{2}}d_C(c_j(a_*))}{d_S(c_j(a_*))}d_j^{-1}
\left(\sum_{i=0}^{n-1} d_i^{-1}\right)^{-1}\leq L^{-\frac{1}{3}}|(f_{a_*}')c_j(a_*)|d_j^{-1}\left(\sum_{i=0}^{n-1} d_i^{-1}\right)^{-1},$$ where for the last inequality we used Lemma \[derivative\](a). (\[twoa\]) follows directly from the last estimate.

As for $(I)_a$, we have from (\[f1-s3.2A\]), $$\label{onea}
(I)_a\leq \log \left(1+ \frac{1}{|(f_{a})'c_j(a)| \cdot
|\tau_j(a)|}\right) < \frac{1}{|(f_{a})'c_j(a)| \cdot|\tau_j(a)|}
< L^{-\frac{\lambda}{3}j}$$ where for the last estimates we use the inductive assumption (\[samp-induct\]) and Lemma \[trans\] for $|\tau_j(a_*)|$. We also use $$|(f_{a})'c_j(a)| \geq\frac{1}{2}|(f_{a_*})'c_j(a_*)| \geq \frac{
L}{2}\cdot\min\{\sigma,L^{-\alpha j} \}$$ where the second inequality follows from from (G1). Then $\frac{|\tau_k(a)|}{|\tau_k(a_*)|}\leq 2$ now follows from combining (\[onea\]) and (\[twoa\]) for all $j <
k$.

Exclusion on account of (R) {#R1}
---------------------------

To estimate the measure of parameters excluded due to (R$)_n$, we divide the critical orbit $\{ v_i, \ i \in [0, n] \}$ into free/bound segments for $a \in \Delta_{n-1}$, $c \in C$, and let $t_1 < t_2 < \cdots < t_{q} \leq n$ be the consecutive times for [*deep free returns*]{} to $C_{\delta}$. Let 