Cantorian distance, statistical mechanics and universal behaviour of multi-dimensional triadic sets
Mohamed Saladin El Naschie · Mathematical and Computer Modelling · 1993
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 dD(4) = 4.00631 ≅ 4, and we conjecture that fully developed turbulence is related to dD(5) = 6.36297. The corresponding Shannon information entropy of the second analysis are SS(4) = 3.68 and SS(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.