NON-ARCHIMEDEAN SCALE INVARIANCE AND CANTOR SETS

The framework of a new scale invariant analysis on a Cantor set C subset of I = [0, 1], presented recently(1) is clarified and extended further. For an arbitrarily small epsilon > 0, elements (x) over tilde in I/C satisfying 0 < <(x)over tilde> < epsilon < x, x is an element of C together with an inversion rule are called relative infinitesimals relative to the scale e. A non-archimedean absolute value v((x) over tilde) = log(epsilon)-1 epsilon/(x) over tilde, epsilon -> 0 is assigned to each such infinitesimal which is then shown to induce a non-archimedean structure in the full Cantor set C. A valued measure constructed using the new absolute value is shown to give rise to the finite Hausdorff measure of the set. The definition of differentiability on C in the non-archimedean sense is introduced. The associated Cantor function is shown to relate to the valuation on C which is then reinterpreated as a locally constant function in the extended non-archimedean space. The definitions and the constructions are verified explicitly on a Cantor set which is defined recursively from I deleting q number of open intervals each of length 1/r leaving out p numbers of closed intervals so that p + q = r.