We consider the period-doubling and intermittency transitions in iterated one-dimensional maps to point out a clear connection between renormalization group (RG) fixed points and non-extensive entropy properties. The exact RG fixed-point map and perturbation expressions for the tangent bifurcation apply also to period-doublings and in both cases convey the physical meaning of universal non-extensive entropy maxima. The degrees of non-extensivity q and non-linearity z are equivalent and the generalized Lyapunov exponent is the leading map expansion coefficient. Also, we expose the dynamics at the chaos threshold of these maps and find it consists of self-similar trajectories that reproduce the entire period-doubling cascade that occurs outside the threshold. We corroborate this structure analytically via the Feigembaum RG transformation, find that the sensitivity to initial conditions has precisely the form of a q-exponential and determine the associated generalized Lyapunov exponent.