Weierstrass fractal drums: II towards a fractal cohomology.

We explore the connections between the Complex Dimensions and the cohomological properties of a fractal object. In the case of the Weierstrass Curve, we define the corresponding fractal cohomology and show that the functions belonging to the cohomology groups associated to the Curve (and its prefractal approximations) are obtained, by induction, as (finite or countably infinite) sums indexed by the underlying Complex Dimensions. The functions that constitute the cohomology groups also satisfy the same discrete local Hölder conditions as the Weierstrass function W itself. More precisely, for any natural integer m, the m th cohomology group is obtained as the set of continuous functions f on the Curve such that, for any point M located in the m th finite graph of the prefractal approximation to the Curve, f(M) has an expansion which might be interpreted as a generalized Taylor expansion, with fractional derivatives of orders the underlying Complex Dimensions. Those expansions are then compared to the fractal expansion of the function which is the most naturally defined on the Curve, namely, the Weierstrass function W itself. An important new result comes from the fact that, contrary to fractal tube formulas, which are obtained for small values of a positive parameter ε , the aforementioned fractal expansions are only valid for the values of the (multi-scales) cohomology infinitesimal ε associated to the scaling relationship obeyed by the Weierstrass Curve. As a consequence, it makes it possible to express, in a very precise way, the relations satisfied by the functions which belong to the cohomology groups, as well as to completely characterize the elements of these groups. We also obtain a suitable counterpart of Poincaré Duality in this context. Our results shed new light on the theory and the interpretation of Complex Fractal Dimensions. [ABSTRACT FROM AUTHOR]