Fractal dimension of chaotic dynamical spaces

In his paper [Chaos, Solitons & Fractals 4 (1994) 293-6], E1 Naschie has exposed the existence of sets with dimensions between 0 and −1, which he calls the cantorian sets. E1-Ghoul discussed some problems in fractal dimensions [Chaos, Solitons & Fractals, England 4 (2001) 77-80; Chaos, Solitons & Fractals, England 18 (2003) 187-92]. The present work is intended to extend their works for chaotic dynamical manifolds. This is obviously related to the cantorian geometry with a possible model for quantum mechanics. The folding of these spaces and their relations with dimensions are studied. Special emphasis on ε-dimension, 0 < ε ≪ 1 is given and some generalized theorems from classical dimension theory are introduced.