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may partition the space of values taken by the journey of the orbital action generated by the equation over time with rectangular grids of increasing fineness. The result is an equipartition of phase space such that there is at most one orbital point in each rectangle of the grid, with, of course, many rectangles in the finer grids being empty. This final grid partition is called a generating partition. The proportion of the available boxes of the partition occupied by points is called its area or volume measure. This measure has been given a variety of names including Liouville, Haar and Lesbegue measures. If every box is occupied, it has measure one. If at most one box, it has measure zero. If we allow partitions to be non-uniform and/or not fine enough to be generating and apply probability weightings for how many points fall into each particular box of the grid, the method is called the Sinai-Ruelle-Bowen or SRB measure after Kolmogorov’s students and followers, the Russian, Ya Sinai, the Belgian Frenchmen, David Ruelle and the American, Rufus Bowen.

Similar to the SRB measure, the distribution of box occupancy probabilities multiplied by their logarithms and summed over all cells of the partition yields a statistical measure that is close to the informational entropy of Claude Shannon as described above. It is called the metric entropy ( HM = -Σ(pi ln(pi)), where H means entropy and pi is the proportion of the total observations that occupy cell i of the phase space or state space partition. It was the above noted Russian father of modern dynamical systems, Kolmogorov, who in 1956 proved that the Shannon metric entropy is a quantifiable invariant of systems even in very complicated motion. Stanford University's Donald Ornstein won a Field’s Medal (the under forty year old mathematician’s Nobel Prize) for his late 1960’s work proving that the Shannon metric entropy, HM, was the only invariant for a large class of appropriately defined, expansive (near by points separating in time) dynamical systems. Recall that we refer to metric entropy reflecting the relative occupancy as probability among the possible boxes (or states) as HM. HM is maximal when the percentage occupancy of all occupied boxes is uniform.

IBM’s Roy Adler in New York and Brian Marcus in California, Hebrew University’s Benjamin Weiss, Warwick University’s English mathematicians, William Parry, Peter Walters, Mark Pollicott and others developed and proved the relevance

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