to $c_j$. We consider the following two cases separately.

[*Case I: $j$ is free.*]{}  For free segments we use Lemma \[outside\], and for bound segments we use Lemma \[reclem1\](c). We obtain exponential growth of derivatives from time $i$ to $j$, which is much better than what is asserted by (G1$)_{n,c}$.

[*Case II: $j$ is bound.*]{} Let $\hat j$ denote the free return with a bound period $p$ such that $j\in[\hat j+1,\hat
j+p]$. We have $\hat j \leq n$, for otherwise $j>n+1$. Consequently, $$J^{j-i}(c_i)= \frac{J^{\hat
j}(c_0)}{J^i(c_0)} \cdot \frac{J^{\hat j+1}(c_0)}{J^{\hat
j}(c_0)} \cdot \frac{J^j(c_0)}{J^{\hat j +1}(c_0)} >
L^{\frac{1}{3} \lambda(\hat j-i)} \cdot K^{-1} Ld_C(v_{\hat j})
\cdot K^{-1} L^{\lambda(j-\hat j-1)}$$ where for the last inequality, we use Lemma \[reclem1\](c) combined with Lemma \[outside\] for the first factor. For the third factor we use bound distortion and (G2)$_{n-1}$ for the binding critical orbit. It then follows that $
J^{j-i}(c_0) \geq L^{\alpha(j-i)- \alpha \hat
j}\geq L^{-\alpha i}.
$ Hence (G1$)_{n,c}$ holds.

As for (G2$)_{n,c}$, we introduce the bound-free structure starting from $c_0$ to $c_{n+1}$. Observe that the sum of the lengths of all bound periods for the orbit of $c_0$ up to time $n$ is $\leq \frac{2
\alpha}{\lambda}n \ll n$. This follows from (\[prop-s3.1\]) and Lemma \[reclem1\](a). 