Evidently, "the natural measure associated with a chaotic attractor gives the fraction of the time that the long orbit on the attractor spends in any given region of state space." An illustrative way to explain the natural measure is thus to imagine a heat map of the attractor. Hot spots correspond to regions of the attractor in which a trajectory visits with "high" probability (or "high" frequency).
Now, I have recently read up on prime periodic orbits (a.k.a, prime cycles, primitive orbits, fundamental periodic orbits, etc) of chaotic systems. A prime orbit is a periodic orbit which is not the repeat of a shorter orbit. Many orbit expansions can be simplified, approximated, or expedited simply by calculating the expansion over prime orbits (e.g., see any paper of Cvitanovic's).
My question is thus: what is the relationship between the natural measure of a (for example low-dimensional, hyperbolic, etc) chaotic system and its prime orbits? Does the natural measure concentrate on the prime orbits? That is, is the heat map hottest on the prime orbits since they are more "important" than the other non-prime orbits? And what is the reference(s) for these results/answers?
Thanks!