  We consider both the dynamics within and towards the supercycle attractors
along the period-doubling route to chaos to analyze the development of a
statistical-mechanical structure. In this structure the partition function
consists of the sum of the attractor position distances known as supercycle
diameters and the associated thermodynamic potential measures the rate of
approach of trajectories to the attractor. The configurational weights for
finite $2^{N}$, and infinite $N \rightarrow \infty $, periods can be expressed
as power laws or deformed exponentials. For finite period the structure is
undeveloped in the sense that there is no true configurational degeneracy, but
in the limit $N\rightarrow \infty $ this is realized together with the analog
property of a Legendre transform linking entropies of two ensembles. We also
study the partition functions for all $N$ and the action of the Central Limit
Theorem via a binomial approximation.
