STATISTICAL-MECHANICS OF MULTIDIMENSIONAL CANTOR SETS, GODEL THEOREM AND QUANTUM SPACETIME

Two different descriptions of an abstract n-dimensional dynamical system are discussed: a Sierpinski space setting and a statistical cellular space setting. The results suggest 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 Hausdorff capacity dimension in this case is d(C)(4) = 3.981 is-approximately-equal-to 4 and we conjecture that the simplest fully developed turbulence is related to d(C)(5) is-approximately-equal-to 6.3. The corresponding Shannon information entropy of the second analysis are L(S)(4) = 3.68 and L(S)(5) = 6.12. The implications of the results for quantum spacetime are outlined and found to be consistent with Heisenberg uncertainty relationship and Bekenstein-Hawking entropy. Finally, the connection between strange nonchaotic behaviour and Godel theorem is discussed.