Multi-dimensional Cantor sets in classical and quantum mechanics
The author's summary: ``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 diffeomorphism of a compact manifold which is dense and quasi- ergodic. The fractal dimension in this case is \(d_ c^{(4)}=3.981\cong 4\) and we conjecture that fully developed turbulence is related to \(d_ c^{(5)}=6.3\). 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 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 interpretation of quantum mechanics and the connection to non-standard analysis and Boscovich covariance are touched upon.
abstract \(n\)-dimensional dynamical system
scientific article; zbMATH DE number 64962