On the Asymptotics of the Hopf Characteristic Function
Zachary Guralnik, Cengiz Pehlevan, Gerald Guralnik

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…
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