Non-Archimedean Scale Invariance and Cantor Sets

TL;DR

This paper extends a non-archimedean scale invariant analysis on Cantor sets, introducing a valuation-based measure that aligns with Hausdorff measure and redefining differentiability in this context.

Contribution

It develops a non-archimedean framework for analyzing Cantor sets, linking valuation to measure and differentiability, and extends previous work with explicit constructions.

Findings

Valuation induces a non-archimedean structure on Cantor sets.

Valued measure corresponds to the Hausdorff measure of the set.

Differentiability is redefined in the non-archimedean setting.

Abstract

The framework of a new scale invariant analysis on a Cantor set $C\subset $ $% I=[0,1] $, presented originally in {\it S. Raut and D. P. Datta, Fractals, 17, 45-52, (2009)}, is clarified and extended further. For an arbitrarily small $\varepsilon >0$, elements $\tilde{x}$ in $I\backslash C$ satisfying $0<\tilde{x}<\varepsilon <x, x\in C $ together with an inversion rule are called relative infinitesimals relative to the scale $\varepsilon$. A non-archimedean absolute value $v(% \tilde{x})=\log_{\varepsilon ^{-1}}\frac{\varepsilon}{\tilde{x}}, \varepsilon \to 0$ 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 $\frac{1}{r}$ leaving out $p$ numbers of closed intervals so that $p+q=r.$