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Traditionally, the 'thermodynamic averages' are calculated from the 'canonical' partition function (eqs. (13), (16)), and little use is made of periodic orbits. However, a more recent approach put forward by Cvitanovic and collaborators  suggested the use of a 'grand-canonical-like' partition, which explicitly takes account of the unstable periodic orbits of the system. Generalizing our model, we consider a particular restriction imposed on a three-scale Cantor set by simple, but nontrivial, grammars.
This object of observation is directly related to the instability properties of the system. The basic quantity to be considered is then the slope of the dynamical map or, more conveniently, the logarithm of the slope. To provide a simple example for illustration we consider the supercritical tent map (Fig. Ia). As the result of the first observation process, the Cantor set shown in Fig. ). As the result of an observation from the second point of view, the result shown in Fig Ic is obtained. Now, each element of the asymptotic set can be labelled by the sequence of the unstable elements which led to its appearance-examples are indicated in Fig.
470, Springer, Berlin 1975 9. S. Ellis: Entropy, large deviation and statistical mechanics. Springer, New York 1985 10. Y. Oono: Large deviation and statistical physics. Progr. Theor. Phys. Suppl. 99, 165 (1989) 11. O. Lopez: Entropy and large deviation. Nonlinearity 3, 527 (1990) 12. A. Muenster: Statistical thermodynamics. Springer, Berlin 1969 13. C. H. P. Kadanoff, I. Procaccia, B. Shraiman: Fractal measures and their singularities - the characterization of strange sets. Phys. Rev. A 33, 1141 (1986) 14.