Multi-dimensional cantor sets in classical and quantum mechanics

We give two 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 elucidate 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 diffcomorphism of a compact manifold which is dense and quasi-ergodic. The fractal dimension in this case is dc(4) = 3.981 ≅ 4 and we conjecture that fully developed turbulence is related to dc(5) = 6.3. The corresponding Shannon information entropy of the second analysis are δs(4) = 3.68 and δs(5) = 6.12. In the case of eight dimensional phase space both descriptions lead to almost identical numerical results. Possible implications of these theoretical results to physical spatio-temporal chaos and the reduction of complexity are discussed. In conclusion, the relevance of Cantor-like space-time for the Copenhagen intepretation of quantum mechanics and the connection to non-standard analysis and Boscovich covariance are touched upon.