Peano-Pólya motions, when time is intrinsic or binomial (uniform or multifractal)
type: Item
instance of: scholarly article
title: Peano-Pólya motions, when time is intrinsic or binomial (uniform or multifractal)
zbMATH Open document ID: 0922.28006
DOI: 10.1007/BF03024410
author: Stéphane Jaffard
author: Benoît B. Mandelbrot
published in: The Mathematical Intelligencer
publication date: 1999-09-19T00:00:00Z
review text: The authors argue that the classical term ``Peano curve'' is improper and ought to be replaced by the term ``Peano motion''. To support their view, the authors introduce an intrinsic time and show that in this intrinsic time \(t\), the Peano motion \(P(t)\) shares some local properties with Brownian motion. For instance, \(P(t)\) is nowhere differentiable, no matter what the generating angle \(\psi\) is. The authors also consider the properties of Peano motion \(\Pi(\tau)\) in an alternative time \(\tau\) introduced by Pólya and prove a strengthened form of Lax's theorem [\textit{P. D. Lax}, Adv. Math. 10, 456-464 (1973; Zbl 0262.26007)], which states that the differentiability of \(\Pi(\tau)\) depends on the generating angle \(\psi\). Their proof makes use of the multifractal nature of the singular function \(t(\tau)\).
reviewed by: Y. Xiao
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 37D45
zbMATH DE Number: 1233979
zbMATH Keywords: fractal
zbMATH Keywords: Peano-Pólya motion
zbMATH Keywords: intrinsic time
zbMATH Keywords: local Hölder exponent
zbMATH Keywords: multifractal
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03024410
OpenAlex ID: W2098614348
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description: scientific article; zbMATH DE number 1233979