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.

Ultrametric Cantor sets and growth of measure

“…Choosing the full set C for E and noting that s is the Hausdorff dimension it thus follows that µ v (C) = 1. We remark that the metric properties of the present ultrametric are indeed distinct from the natural ultrametric (c.f., [4,5]), since the Lebesgue measure of C in the natural ultrametric is zero, but in the present case, the corresponding valued measure equals the Hausdorff measure. More importantly, topologies induced by the two ultrametrics are also different, as seen in the following example.…”

“…Some of the results presented here are new (Theorem 1 and Proposition 2): (i) The relation of the nontrivial valuation with the Cantor function, considered in Refs. [4,5], is made precise by proving that the valuation is indeed given by an appropriate Cantor function. (ii) The multiplicative representation that exists because of the nontrivial infinitesimals and the scale invariant ultrametric for every element of the Cantor set is verified explicitly, leading to a proof that the non-archimedean absolute value ||x|| = 3 −ns , x ∈ C, s = log 3 2 precisely corresponds to the ultrametric of [2] in the context of noncommutative geometry and (iii) the analysis of convergence of sequences of the form ǫ n−nl , 0 < ǫ, l < 1.…”

Nonlinear Scale invariant Formalism and its Application to Some Differential Equations

“…An ordinary real x is extended over to the fattened variables x of a Cantor set is replaced by a connected segment over which the real variable x (with a slight abuse of notation, we are here using the same symbol which denoted infinitesimals in 0 ) is supposed to live in [8,9].…”

“…Let us consider the simplest differential equation 1 = dt dx (8) This may be assumed to represent the uniform motion (of the centre of mass) of a rigid ball. Accordingly, the ball will roll at uniform rate 1 along the x -axis when the position at any instance may, for To summarize, the extension of the singleton set of the form {0} of R over to a deleted set of the form {0} \ 1,1) ( is realized explicitly in the context of the linear equation (8).…”

“…Accordingly, the ball will roll at uniform rate 1 along the x -axis when the position at any instance may, for To summarize, the extension of the singleton set of the form {0} of R over to a deleted set of the form {0} \ 1,1) ( is realized explicitly in the context of the linear equation (8). Plugging in all the steps together we can also write down a generalized class of solutions of this equation in the form…”