  We recall that, at both the intermittency transitions and at the Feigenbaum
attractor in unimodal maps of non-linearity of order $\zeta >1$, the dynamics
rigorously obeys the Tsallis statistics. We account for the $q$-indices and the
generalized Lyapunov coefficients $\lambda_{q}$ that characterize the
universality classes of the pitchfork and tangent bifurcations. We identify the
Mori singularities in the Lyapunov spectrum at the edge of chaos with the
appearance of a special value for the entropic index $q$. The physical area of
the Tsallis statistics is further probed by considering the dynamics near
criticality and glass formation in thermal systems. In both cases a close
connection is made with states in unimodal maps with vanishing Lyapunov
coefficients.
