“Nowhere” differentiable horizons

Existence of Nowhere Differentiable Boundaries in a Realistic Map

If the world is a differential equation, as Boltzmann thought, the problem of the origin of the "nowhere differentiable nature of reality" (cf. Mandelbrot [1]) poses itself [2]. To our knowledge, only two differentiable maps exist in the literature which produce nowhere differentiable behavior, the 3-D diffeomorphisms indicated by Grebogi et al. [3] and by Kaneko [4], The first map [3] which co...

Fractional differentiability of nowhere differentiable functions and dimensions.

Weierstrass's everywhere continuous but nowhere differentiable function is shown to be locally continuously fractionally differentiable everywhere for all orders below the "critical order" 2-s and not so for orders between 2-s and 1, where s, 1<s<2 is the box dimension of the graph of the function. This observation is consolidated in the general result showing a direct connection between local ...

An Analysis of Katsuura’s Continuous Nowhere Differentiable Function

The Prevalence of Continuous Nowhere Differentiable Functions

In the space of continuous functions of a real variable, the set of nowhere dilferentiable functions has long been known to be topologically "generic". In this paper it is shown further that in a measure theoretic sense (which is different from Wiener measure), "almost every" continuous function is nowhere dilferentiable. Similar results concerning other types of regularity, such as Holder cont...

Simple Proofs of Nowhere-differentiability for Weierstrass’s Function and Cases of Slow Growth

Using a few basics from integration theory, a short proof of nowhere-differentiability of Weierstrass functions is given. Restated in terms of the Fourier transformation, the method consists in principle of a second microlocalisation, which is used to derive two general results on existence of nowhere differentiable functions. Examples are given in which the frequencies are of polynomial growth...