$i = p-1$ we obtain $$J^p(x)\geq
K^{-1} L^{-1-7\alpha p} |c-x|^{-1}.$$ Substituting into this the upper estimate of $p$ in Lemma \[reclem1\](a) we obtain $$J^p(x)\geq K^{-1}
L^{-1}|c-x|^{-1+\frac{15\alpha}{\lambda}} \geq
|c-x|^{-1+\frac{16\alpha}{\lambda}}.$$ We have used $|c-x|\leq\delta=L^{-\alpha N_0}$ for the last inequality.

It is left to prove the sublemma. (G1) implies $|f'c_i|\geq
L\min\{\sigma,L^{-\alpha i}\}$. Then (a) follows from Lemma \[derivative\](a). As for (b), let $j\in[0,i]$. By definition, $$J^i(c_0)d_j(c_0)=
\frac{J^{i}(c_0)}{J^j(c_0)}d_C(c_j)d_S(c_j).$$ We have: $\frac{J^i(c_0)}{J^j(c_0)}\geq
L^{-\alpha j}\sigma$ from (G1); $d_C(v_j)\geq K^{-1} \sigma
L^{-\alpha j}$ from (a); $d_S(v_j)\geq \sigma
L^{-4\alpha j}$ from (G3). Hence, $J^i(c_0)d_j\geq K^{-1}
\sigma^3L^{-6\alpha j}$, and thus $$\begin{aligned}
\sum_{j=0}^{i}J^i(c_0)^{-1}d_j^{-1}(c_0) &\leq \sigma^{-3}
L^{7\alpha i}.\end{aligned}$$ Taking reciprocals implies (b).

Global dynamical properties {#s2.4}
---------------------------

At step $n$ of induction, we wish to exclude all parameters for which one of (G1-3$)_n$ is violated for some $c\in C$, and to estimate the measure of the parameters deleted. Conditions (G1) (G2), however, can not be used directly as rules for exclusion, since they do not care about cummulative effects of “shallow returns". Hence we introduce a stronger condition, based on the notion of *deep returns*, and will use it as a rule for deletion in Section \[s3\].

[*Hypothesis in Sect.\[s2.4\]:*]{} $f =
f_{a}$ is such that $a \in \Delta_{N_0}$. $n\geq N_0$ 