Bypassing dynamical systems : A simple way to get the box-counting dimension of the graph of the Weierstrass function

TL;DR

This paper introduces a straightforward method to compute the box-counting dimension of the Weierstrass function's graph by using graph sequences, bypassing traditional dynamical systems techniques.

Contribution

It presents a simple approach to determine the box dimension of the Weierstrass function's graph without relying on complex dynamical systems tools.

Findings

Successfully computes the box dimension using graph sequences.

Provides a new, simpler method for fractal dimension calculation.

Applicable to classical Weierstrass functions with given parameters.

Abstract

In the following, bypassing dynamical systems tools, we propose a simple means of computing the box dimension of the graph of the classical Weierstrass function defined, for any real number~$x$, by~$ {\cal W}(x)=\displaystyle \sum_{n=0}^{+\infty} \lambda^n\,\cos \left ( 2\, \pi\,N_b^n\,x \right) $, where~$\lambda$ and~$N_b$ are two real numbers such that~\mbox{$0 <\lambda<1$},~\mbox{$ N_b\,\in\,\N$} and~$ \lambda\,N_b > 1 $, using a sequence a graphs that approximate the studied one.