  Chaos thresholds of the $z$-logistic maps $x_{t+1}=1-a|x_t|^z$ $(z>1;
t=0,1,2,...)$ are numerically analysed at accumulation points of cycles 2, 3
and 5. We verify that the nonextensive $q$-generalization of a Pesin-like
identity is preserved through averaging over the entire phase space. More
precisely, we computationally verify $\lim_{t \to\infty}< S_{q_{sen}^{av}}
>(t)/t= \lim_{t \to\infty}< \ln_{q_{sen}^{av}} \xi >(t)/t \equiv
\lambda_{q_{sen}^{av}}^{av}$, where the entropy $S_{q} \equiv (1- \sum_i
p_i^q)/ (q-1)$ ($S_1=-\sum_ip_i \ln p_i$), the sensitivity to the initial
conditions $\xi \equiv \lim_{\Delta x(0) \to 0} \Delta x(t)/\Delta x(0)$, and
$\ln_q x \equiv (x^{1-q}-1)/ (1-q) $ ($\ln_1 x=\ln x$). The entropic index
$q_{sen}^{av}<1$, and the coefficient $\lambda_{q_{sen}^{av}}^{av}>0$ depend on
both $z$ and the cycle. We also study the relaxation that occurs if we start
with an ensemble of initial conditions homogeneously occupying the entire phase
space. The associated Lebesgue measure asymptotically decreases as
$1/t^{1/(q_{rel}-1)}$ ($q_{rel}>1$). These results led to (i) the first
illustration of the connection (conjectured by one of us) between sensitivity
and relaxation entropic indices, namely $q_{rel}-1 \simeq A
(1-q_{sen}^{av})^\alpha$, where the positive numbers $(A,\alpha)$ depend on the
cycle; (ii) an unexpected and new scaling, namely $q_{sen}^{av}(cycle n)=2.5
q_{sen}^{av}(cycle 2)+ \epsilon $ ($\epsilon=-0.03$ for $n=3$, and $\epsilon =
0.03$ for $n=5$).
