non-degeneracy of $c$, $I_p(c)$ is the union of two intervals, one at the right of $c$ and the other at the left. According to Lemma \[dist\], if $x\in I_p(c)$ then the derivatives along the orbit of $fx$ shadow that of the orbit of $c_0$ for $p-1$ iterates. We regard the orbit of $x$ as been bound to the critical orbit of $c$ up to time $p$; and we call $p$ the [*bound period*]{} of $x$ to $c$.

\[reclem1\] If (G1)$_{n,c}$-(G3)$_{n,c}$ holds, then for $p\in[2,n]$ and $x\in
I_{p}(c)$ we have:

-   $p\leq\log |c-x|^{-\frac{2}{\lambda}}$;

-   if $x\in C_\delta$, then $J^p(x)|\geq
    |c-x|^{-1+\frac{16\alpha}{\lambda}}\geq L^{\frac{\lambda}{3}p}$.

By definition we have $$|c-x|^2\leq D_{p-1}(c_0)\leq L^{-\frac{1}{2}}d_{p-2}(c_0) <
L^{-\frac{1}{2}} J^{p-2}(c_0)^{-1}.$$ Then by (G2), $$\label{low}
|c-x|^2\leq L^{-\frac{1}{2}-\lambda(p-2)}\leq L^{-\lambda p}
,$$ from which (a) follows. The second inequality of (b) follows from (\[low\]).

\[add-lem1-s2.3B\] For $0\leq i \leq n$ we have:

-   $d_C(c_i)\geq
    K^{-1} \sigma L^{-\alpha i}$;

-   $J^i(c_0)D_{i+1}(c_0)\geq
    L^{-1-7\alpha i}$.

We finish the proof of Lemma \[reclem1\] assuming the conclusion of this sublemma. We have $$\begin{aligned}
 J^{p}(x)=
J^{p-1}(fx)J(x)\geq K^{-1}J^{p-1}(c_0)
\cdot L|c-x|\geq K^{-1}J^{p-1}(c_0) \cdot |c-x|^{-1}
D_{p}(c_0),\end{aligned}$$ where for the first inequality we use Lemma \[dist\] and Lemma \[derivative\](c), and for the last inequality we use $x\in I_{-p}(c)\cup I_p(c)$. Using Sublemma \[add-lem1-s2.3B\](b) for 