Ultrametric Cantor Sets and Growth of Measure

TL;DR

This paper introduces a novel ultrametric framework for Cantor sets using relative infinitesimals, revealing new properties such as measure growth and enabling the construction of Cantor sets with positive measure.

Contribution

It develops an ultrametric approach based on relative infinitesimals and inversion, leading to new insights into measure deformation and the introduction of a valuated exponent.

Findings

Ultrametric $d_u$ is scale and reparametrisation invariant.

Cantor functions become locally constant in the ultrametric space.

Deformation of measure can turn measure zero sets into positive measure sets.

Abstract

A class of ultrametric Cantor sets $(C, d_{u})$ introduced recently in literature (Raut, S and Datta, D P (2009), Fractals, 17, 45-52) is shown to enjoy some novel properties. The ultrametric $d_{u}$ is defined using the concept of {\em relative infinitesimals} and an {\em inversion} rule. The associated (infinitesimal) valuation which turns out to be both scale and reparametrisation invariant, is identified with the Cantor function associated with a Cantor set $\tilde C$ where the relative infinitesimals are supposed to live in. These ultrametrics are both metrically as well as topologically inequivalent compared to the topology induced by the usual metric. Every point of the original Cantor set $C$ is identified with the closure of the set of gaps of $\tilde C$. The increments on such an ultrametric space is accomplished by following the inversion rule. As a consequence, Cantor functions are reinterpretd as (every where) locally constant on these extended ultrametric spaces. An interesting phenomenon, called {\em growth of measure}, is studied on such an ultrametric space. Using the reparametrisation invariance of the valuation it is shown how the scale factors of a Lebesgue measure zero Cantor set might get {\em deformed} leading to a {\em deformed} Cantor set with a positive measure. The definition of a new {\em valuated exponent} is introduced which is shown to yield the fatness exponent in the case of a positive measure (fat) Cantor set. However, the valuated exponent can also be used to distinguish Cantor sets with identical Hausdorff dimension and thickness. A class of Cantor sets with Hausdorff dimension $\log_3 2$ and thickness 1 are constructed explicitly.