NON-ARCHIMEDEAN SCALE INVARIANCE AND CANTOR SETS

Abstract

The framework of a new scale invariant analysis on a Cantor set C ⊂ I = [0,1], presented recently1 is clarified and extended further. For an arbitrarily small ε > 0, elements [Formula: see text] in I\C satisfying [Formula: see text], x ∈ C together with an inversion rule are called relative infinitesimals relative to the scale ε. A non-archimedean absolute value [Formula: see text], ε → 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 [Formula: see text] leaving out p numbers of closed intervals so that p + q = r.