  We show that the two complementary parts of the dynamics associated to the
Feigenbaum attractor, inside and towards the attractor, form together a q
-deformed statistical-mechanical structure. A time-dependent partition function
produced by summing distances between neighboring positions of the attractor
leads to a q-entropy that measures the fraction of ensemble trajectories still
away at a given time from the attractor (and the repellor). The values of the
q-indexes are given by the attractor's universal constants, while the
thermodynamic framework is closely related to that first developed for
multifractals.
