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A global statistical context for these qualitative differential systems was inspired by the Russian mathematician, Andrei Nikolaevic Kolmogorov. In his now famous foundational talk about the stability of classical mechanical systems in the final session of the 1954 International Congress of Mathematics, he gave public birth to, among other ideas, what has come to be called the ergodic or statistical, measure theory of dynamical systems. Here, ergodic means the existence of an invariant statistical measure on the phase space attractor of the system that can be obtained using a variety of equivalent methods and beginning the count at any of its points. Two phase space objects generated by a dynamical system may look different in phase space but their statistical measures may all be the same, i.e. invariant. These qualitative orbits in a box-partitioned space can be visualized as Paulus and Geyer’s rats exploring a space and Selz’s path sequences of computer screen dot quenches produced by clicking on them with a computer mouse.

A precursor of Kolmogorov’s ergodicity was the earlier ergodicity of Ludwig Boltzmann. This describes a suitably partitioned system such that equivalent values come from quantitating the behavior of one single orbit exploring the space of the lattice of boxes over very long times time as those obtained from a single aggregate photograph of all orbits run from all possible starting places simultaneously. The ergodicity of gas-like molecular randomness implicates systems being in one of only two possible equilibrium statistical states: measure zero (at most occupying a single point, zero, minimal entropy) or its “complement,” full measure one (occupying all available space in a state of maximal entropy). Joseph Goldstein, a well known teacher of meditation, giving advice recorded in Daniel Goleman’s 1977 book on the subject said that all methods of nirvana directed meditation amounted to “...simple mathematics ...all systems aiming for One or Zero—union with God or emptiness.” In place of the maximal or minimal values for the HT and HM entropies of these states of transcendence, we in the world of samsara are stuck in states of in-between entropy which invariant statistical measures of on phase space shapes help quantify.

To generalize measures made on rat and computer mouse paths to more general and idealized systems, after plotting an orbital path in a phase space, we

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