  We uncover the dynamics at the chaos threshold $\mu_{\infty}$ of the logistic
map and find it consists of trajectories made of intertwined power laws that
reproduce the entire period-doubling cascade that occurs for $\mu
<\mu_{\infty}$. We corroborate this structure analytically via the Feigenbaum
renormalization group (RG) transformation and find that the sensitivity to
initial conditions has precisely the form of a $q$-exponential, of which we
determine the $q$-index and the $q$-generalized Lyapunov coefficient $\lambda
_{q}$. Our results are an unequivocal validation of the applicability of the
non-extensive generalization of Boltzmann-Gibbs (BG) statistical mechanics to
critical points of nonlinear maps.
