  We analyze the fluctuating dynamics at the golden-mean transition to chaos in
the critical circle map and find that trajectories within the critical
attractor consist of infinite sets of power laws mixed together. We elucidate
this structure assisted by known renormalization group (RG) results. Next we
proceed to weigh the new findings against Tsallis' entropic and Mori's
thermodynamic theoretical schemes and observe behavior to a large extent richer
than previously reported. We find that the sensitivity to initial conditions
has the form of families of intertwined q-exponentials, of which we determine
the q-indexes and the generalized Lyapunov coefficient spectra. Further, the
dynamics within the critical attractor is found to consist of not one but a
collection of Mori's q-phase transitions with a hierarchical structure. The
value of Mori's `thermodynamic field' variable q at each transition corresponds
to the same special value for the entropic index q. We discuss the relationship
between the two formalisms and indicate the usefulness of the methods involved
to determine the universal trajectory scaling function and/or the ocurrence and
characterization of dynamical phase transitions.
