We uncover the basis for the validity of the Tsallis statistics at the onset of chaos in logistic maps. The dynamics within the critical attractor is found to consist of an infinite family of Mori's q-phase transitions of rapidly decreasing strength, each associated to a discontinuity in Feigenbaum's trajectory scaling function σ . The value of q at each transition corresponds to the same special value for the entropic index q, such that the resultant sets of q-Lyapunov coefficients are equal to the Tsallis rates of entropy evolution.