CANTORIAN DISTANCE, STATISTICAL-MECHANICS AND UNIVERSAL BEHAVIOR OF MULTIDIMENSIONAL TRIADIC SETS

We give different descriptions of an abstract n dimensional dynamical system. First, we use a Sierpinski space setting and subsequently we use a statistical cellular space setting. The results of the analysis elucidates certain universal behaviour which was observed in a wide category of cellular automata. The results further show that in four dimensions the phase space dynamics is Peano-like and resembles an Anosov diffeomorphism of a compact manifold which is dense and quasi ergodic. The fractal average distance dimension in this case is d(D)(4) = 4.00631 congruent-to 4, and we conjecture that fully developed turbulence is related to d(D)(5) = 6.36297. The corresponding Shannon information entropy of the second analysis are S(S)(4) = 3.68 and S(S)(5) = 6.12. In the case of more-than-seven-dimensional phase space, both descriptions lead to almost identical numerical results. Possible implications of these theoretical results to physical spatio-temporal chaos are discussed.