  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.
