Peano-Pólya motions, when time is intrinsic or binomial (uniform or multifractal)
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 \(t(\tau)\).
scientific article; zbMATH DE number 1233979