ANALYSIS ON A FRACTAL SET

Abstract

The formulation of a new analysis on a zero measure Cantor set C(⊂I = [0,1]) is presented. A non-Archimedean absolute value is introduced in C exploiting the concept of relative infinitesimals and a scale invariant ultrametric valuation of the form log ε-1 (ε/x) for a given scale ε > 0 and infinitesimals 0 < x < ε, x ∈ I\C. Using this new absolute value, a valued (metric) measure is defined on C and is shown to be equal to the finite Hausdorff measure of the set, if it exists. The formulation of a scale invariant real analysis is also outlined, when the singleton {0} of the real line R is replaced by a zero measure Cantor set. The Cantor function is realized as a locally constant function in this setting. The ordinary derivative dx/dt in R is replaced by the scale invariant logarithmic derivative d log x/d log t on the set of valued infinitesimals. As a result, the ordinary real valued functions are expected to enjoy some novel asymptotic properties, which might have important applications in number theory and in other areas of mathematics.