growth of derivatives for orbit segments lying outside of $C_{\delta}$.

\[outside\] There exists $L_1\geq L_0$ such that if $L\geq L_1$ and $f = f_{a}$ is such that $a \in \Delta_{N}(L)$, then the following holds:

-   if $n\geq1$ and $x$, $fx,\cdots, f^{n-1}x\notin C_{\delta}$, then $J^{n}(x)\geq \delta L^{2\lambda n}$;

-   if moreover $f^nx\in C_{\delta}$, then $J^{n}(x) \geq
    L^{2 \lambda n}$.

[*Standing assumption for the rest of this section:  $L\geq L_1$ and $a \in \Delta_{N}$.*]{}

Recovering expansion {#s2.3}
--------------------

For $f = f_a$, $c \in C$ and $n>N$ we introduce three conditions:

-   $J^{j-i}(c_i)\geq L \min\{\sigma, L^{-\alpha
    i}\}\ \ 0\leq \forall i<\forall j\leq n+1;$

-   $J^i(c_0)\geq L^{\lambda i}\ \ 0<
    \forall i\leq n+1;$

-   $d_S(c_i)\geq L^{-4\alpha i}\ \ N
    \leq \forall i\leq n$.

We say $f$ satisfies (G1$)_n$ if (G1$)_{n,c}$ holds for every $c\in C$. The definitions of (G2)$_n$, (G3)$_n$ are analogous. These conditions are taken as inductive assumptions in the construction of the parameter set $\Delta$.

We establish a recovery estimate of expansion. Let $c\in C$, $c_0=fc$ and assume that (G1)$_{n,c}$-(G3)$_{n,c}$. For $p\in[2,n]$, let $$I_p(c)=\begin{cases} &f^{-1}[c_0+D_{p-1}(c_0),c_0+D_p(c_0))\ \text{ if $c$ is a local minima of $x\to x+a+L\cdot\ln|\Phi(x)|$};\\
& f^{-1}(c_0-D_{p}(c_0),c_0-D_{p-1}(c_0)]\ \text{ if $c$ is a local maxima of $
x\to x+a+L\cdot\ln|\Phi(x)|$}.
\end{cases}$$ By the 