Analysis on a Fractal Set

Abstract

The formulation of a new analysis on a zero measure Cantor set $C(\subset I=[0,1])$ is presented. A non-archimedean absolute value is introduced in $C$ exploiting the concept of {\em relative} infinitesimals and a scale invariant ultrametric valuation of the form ${\log }_{{\epsilon }^{-1}}(\epsilon /x)$ for a given scale $\epsilon >0$ and infinitesimals $0<x<\epsilon ,x\in I\backslash C$. 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 realised 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.

arXiv commentAMS-Latex 2e, 13 pages, no figures