and (G1)$_{n-1}$-(G3)$_{n-1}$ hold for all $c\in C$.

For all $c\in C$ we have:

-   $f^{i+1}c\notin C\cup S$ for all $0\leq i\leq n$;

-   for the orbit of $c_0=fc$, the bound period initiated at all returns to $C_{\delta}$ before $n$ is $\ll n$.

([*Bound/free structure*]{}) We divide the orbit of $c_0$ into alternative bound/free segments as follows. Let $n_1$ be the smallest $j\geq0$ such that $c_j\in C_\delta$. For $k>1$, we define free return times $n_{k}$ inductively as follows. Let $p_{k-1}$ be the bound period of $c_{n_{k-1}}$, and let $n_{k}$ be the smallest $j\geq n_{k-1}+p_{k-1}$ such that $c_j\in C_\delta$. We decompose the orbit of $c_0$ into bound segments corresponding to time intervals $(n_k,n_k+p_k)$ and free segments corresponding to time intervals $[n_k+p_k,n_{k+1}]$. The times $n_k$ are the [*free return*]{} times. We have $$n_1<n_1+p_1 \leq n_2<n_2+p_2 \leq \cdots.$$

[Let $c\in C$ and assume that $c_0=fc$ makes a free return to $C_\delta$ at time $\nu\leq n$. We say $\nu$ is a [*deep return*]{} of $c_0$ if for every free return $i\in [0,\nu-1]$; $$\label{inessential}
\sum_{\stackrel{j\in[i+1,\nu]}{free \ return}}2\log
d_C(c_{j})\leq \log d_C(c_{i}).$$]{}

We say (R$)_{n,c}$ holds if $$\prod_{i\in[0, j]: \ \text{deep 
return}} d_C(c_i)  \geq L^{-\frac{1}{20} \lambda \alpha j}\ \ \text{for every } N_0\leq j\leq n.$$ We say (R$)_n$ 