Non-Archimedean Scale Invariance and Cantor Sets

Abstract

The framework of a new scale invariant analysis on a Cantor set C ⊂ I = [0, 1] , presented originally in S. Raut and D. P. Datta, Fractals, 17, 45-52, (2009), is clarified and extended further. For an arbitrarily small ε > 0, elementsx in I\C satisfying 0 <x < ε < x, x ∈ C together with an inversion rule are called relative infinitesimals relative to the scale ε. A non-archimedean absolute value v(x) = log ε −1 ε x , ε → 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 nonarchimedean 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 1 r leaving out p numbers of closed intervals so that p + q = r.

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].…”

“…Nontrivial inversion induced variations are revealed only under double logarithmic scales of an ordianry linear variable, when there is a transition from one gap to another (that is to say, between two points of the underlying Cantor set , for more details see [9,10]). So far we have discussed about the scale free analysis.…”

Excitation of flow instabilities due to nonlinear scale invariance

“…Here, we present in brief the mathematical arguments [17,19,20] leading to the emergence of multifractal scalings from standard differential measures in a laminar flow as the original laminar flow tends to become turbulent. Recall that the traditional (differential) Lebesgue measure is well suited for simple systems, for instance, the uniform rolling of a billiard ball along a straight line, say, or in a laminar flow.…”

“…Finally, to justify the anomalous scaling for t as t → ∞ in the present formalism let us remark that as η = t −1 → 0 respecting 0 <η n < n < t −1 , relatively invisible smaller scalesη n residing in (0, n ) might have a coherent, cooperative effect on the visible variable η in the form η −α( ) where the slowly varying, locally constant effective exponent α( ) = lim n→∞ log −n ( n /η n ) > 0 is interpreted as an ultrametric valuation living in a multifractal set of microscopically small and macroscopically large scales [18,19,20]. Clearly, cascades of infinitesimally small scale invisible elementsη n are related dually (i.e.…”

“…More generally, α(η) = α 0 η s(η) is locally constant over countable open subintervals of (0,1) (i.e. η s(η) = constant), with nontrivial variability at the points of (classical) discontinuity, as represented by the variable factor η s(η) with s(η) = s remaining constant on an infinitesimally small subinterval of the set of variability of α(η) [19,20], reflecting the cooperative effects of smaller scale cascades. The effective renormalized exponent α(η) thus has an intermittent behaviour as claimed.…”