Technical discussion of entropy in dynamical systems with no actionable allegations

of a related measure of the rapidity of dynamical expansion, the generation of new information seen as the rate of entering new boxes of the partition, a logarithmic rate of expansion of the possible. Counting the number of previously unoccupied squares entered by the dynamical systems orbit per unit time over the generating partition, for instance, yields an estimate of entropy that, as in the rat and computer mouse examples above, is called the topological entropy, Hr. Hr, is about how much new information is being generated by the system per unit time. Theorems have been proven that Hr is a maximal estimate of the global dynamical entropy with Hy proven to be a minimum estimate. Monitoring single or aggregate molecular motion in a system with the maximum randomness of a space filling gas, we find that, on the average, every box is entered and occupied uniformly such that H; = Hy or said another way, H7 — Hy = 0. As evidenced by the above described experiments in rats and people, the same entropic relations (but usually not with maximal or minimal measure) can be found in biological systems. We have previously described the manifold geometry of a generic (typical, idealized) nonlinear dynamical systems as hyperbolic defined by the presence of simultaneous but decomposable components of the motion including the straight ahead and round and round actions on the center manifold, the new possibility generating, expansive, away from the center manifold motions along unstable manifolds and the back to the center manifold, contracting motions, along the stable manifolds. Uniform expansive and contractive influences in the flow leads to mixing of the order of the initial sequence of the values inscribed by the orbits. This results in maximization of the entropies and satisfaction of a concomitant of the uniformly hyperbolic condition, H7 — Hy = 0. These clean and mathematically proven findings do not hold for the quasi- mess that is human neuropsychobiology. Enmeshed as most of us are in only intermittently random or nonuniformly hyperbolic systems with the in-between entropies of the only apparently real world of maya, Hr — Hy # 0. How the H;— Hy = 0 of uniform hyperbolicity fails, H7 — Hy # 0, and along with it the dispassionate detachment of entropic emptiness and fullness, becomes a problem not unrelated to the existence and quantitative qualities of personality styles and their dissolution 84 HOUSE_OVERSIGHT_013584