On the Asymptotics of the Hopf Characteristic Function

TL;DR

This paper investigates the large argument asymptotic behavior of the Hopf characteristic function for fractals and chaotic systems, linking it to fractal dimensions through explicit calculations for specific examples.

Contribution

It provides explicit calculations of the Hopf characteristic function for the generalized Cantor set and Lorenz attractor, clarifying its relation to fractal dimensions.

Findings

Asymptotic behavior corresponds to fractal dimension

Explicit calculation for Cantor set matches known dimension

Numerical results for Lorenz attractor are consistent with fractal dimensions

Abstract

We study the asymptotic behavior of the Hopf characteristic function of fractals and chaotic dynamical systems in the limit of large argument. The small argument behavior is determined by the moments, since the characteristic function is defined as their generating function. Less well known is that the large argument behavior is related to the fractal dimension. While this relation has been discussed in the literature, there has been very little in the way of explicit calculation. We attempt to fill this gap, with explicit calculations for the generalized Cantor set and the Lorenz attractor. In the case of the generalized Cantor set, we define a parameter characterizing the asymptotics which we show corresponds exactly to the known fractal dimension. The Hopf characteristic function of the Lorenz attractor is computed numerically, obtaining results which are consistent with Hausdorff or correlation dimension, albeit too crude to distinguish between them.