For $0\leq n\leq N$, let $$\Delta_n =\{a\in [0,1)\colon f_a^{i+1}(C)\cap(C_\sigma\cup
S_\sigma)=\emptyset\text{ for every }0\leq i\leq n\}.$$ Observe that $\Delta_n$ is a union of intervals. We start with the following statement, the proof of which is given in Appendix.

\[initial-1\] For any large integer $N$ there exists $L_0 = L_0(N)\gg1$ such that if $L\geq L_0$, then $
|\Delta_{N}|\geq 1  - L^{-\frac19}.
$

Lemma \[initial-1\] is sufficient for us to move forward. This approach for initial setups is easier, and leads to the desired asymptotic measure estimate on $\Delta$ as $L\to\infty$.

We move to the expanding property of the maps corresponding to parameters in $\Delta_{N}$. We frequently use the following notation: for $c\in C$ and $n\geq1$, $c_0=fc$ and $c_n=f^nc_0$: for $x\in S^1$ and $n\geq1$, $J(x)=|f'x|$ and $J^n(x)=J(x)J(fx)\cdots J(f^{n-1}x)$.

Let $\alpha=10^{-6}$ and $\delta = L^{-\alpha N}$. In what follows, we suppose $N$ to be a large integer for which $\delta\ll\sigma$, and the conclusion of Lemma \[initial-1\] holds. The value of $N$ will be replaced if necessary, but only a finite number of times. The letter $K$ will be used to denote generic constants which are independent of $N$ and $L$.

The next lemma, the proof of which is given in Appendix, ensures an exponential 