Box-counting dimension and differentiability of box-like statistically self-affine functions

TL;DR

This paper investigates the fractal dimension and differentiability properties of a class of self-affine functions, revealing conditions under which they are almost surely differentiable or non-differentiable almost everywhere.

Contribution

It introduces a class of box-like statistically self-affine functions and determines their almost-sure box-counting dimension and differentiability behavior.

Findings

Computed the almost-sure box-counting dimension of the functions' graphs.

Established conditions for almost sure differentiability or non-differentiability.

Linked the differentiability properties to an explicit functional of the model.

Abstract

We consider a class of "box-like" statistically self-affine functions, and compute the almost-sure box-counting dimension of their graphs. Furthermore, we consider the differentiability of our functions, and prove that, depending on an explicitly computable functional of the model, they are almost surely either differentiable almost everywhere or non-differentiable almost everywhere.