$D_n$ for $D_n(c_0)$, and let $I
=[c_0-D_n,c_0+D_n]$. Then $$\log \frac{J^n(x)}{J^n(y)}=\sum_{j=0}^{n-1}\log \frac{J(f^jx)}{J(f^jy)} \leq\sum_{j=0}^{
n-1}|f^jI| \sup_{\phi\in f^jI}\frac{|f''\phi|}{|f' \phi|}.$$ Lemma \[dist\] would hold if for all $j \leq n-1$ we have $f^jI\cap (S\cup C)=\emptyset$ and $$\label{disteq}
|f^jI|\sup_{\phi\in f^jI}\frac{|f''\phi|}{|f' \phi|}\leq
\log 2 \cdot d_j^{-1}(c_0) \left[\sum_{i=0}^{
n-1}d_i^{-1}(c_0)\right]^{-1}.$$ We prove (\[disteq\]) by induction on $j$. Assume (\[disteq\]) holds for all $j < k$. Summing (\[disteq\]) over $j=0,1,\cdots,
k-1$ implies $
\frac{1}{2} \leq \frac{J^k(\eta)}{J^k(c_0)}\leq 2$ for all $\eta \in I$. We have $$\label{derivative6}
|f^{k}I|\leq 2 J^{k}(c_0) D_n = 2d_{k}^{-1}
\cdot d_C(c_k) d_S(c_k) D_n \leq
\frac{2}{\sqrt{L}}d_C(c_k)d_S(c_k).$$ We have $f^k I \cap (C\cup S)
= \emptyset$ from (\[derivative6\]), and for $\phi\in f^k
I$, $$\begin{aligned}
|f^kI|\frac{|f''\phi|}{|f'\phi|} &\leq 2
d_k^{-1}  d_C(c_k) d_S(c_k) D_n \cdot \frac{K_0^2}{d_C(\phi) d_S(\phi)} \\
& = 2K_0^2 d_k^{-1}D_n  \cdot \frac{d_C(c_k)
d_S(c_k) }{d_C(\phi) d_S(\phi)}
\leq\frac{2K_0^2}{\sqrt{L}}\cdot d_{k}^{-1} \left[\sum_{i=0}^{n-1}
d_i^{-1}\right]^{-1},\end{aligned}$$ where we used Lemma \[derivative\](a) for $\frac{|f''\phi|}{|f'\phi|}$ for the first inequality. For the last inequality we observe that the second factor of the left-hand-side is $< 2$ by (\[derivative6\]).

Initial setup {#s2.2}
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In one-dimensional dynamics, a general strategy for constructing positive measure sets of “good" parameters is to start an inductive construction in small parameter intervals, in which orbits of critical points are kept out of bad sets for certain 