Symplectic reversible maps, tiles and chaos
type: Item
instance of: scholarly article
title: Symplectic reversible maps, tiles and chaos
zbMATH Open document ID: 0786.58015
DOI: 10.1016/0960-0779(92)90049-S
author: Igor Hoveijn
published in: Chaos, Solitons and Fractals
publication date: 1992-09-27T00:00:00Z
review text: This nice paper extends studies by G. M. Zaslavskij and coworkers [\textit{A. A. Chernikov, R. Z. Sagdeev}, and \textit{G. M. Zaslavskij}, Physica D 33, No. 1-3, 65-76 (1988; Zbl 0696.58034)] on stochastic webs. The author first establishes the reversibility properties of the symplectic map \(F(z)= R_ \alpha z+\varepsilon (x)(-\sin \alpha,\cos\alpha)\). For periodic \(f\), some values of \(\alpha\) produce a kind of tiling of the plane which resembles some of Escher's pictures. Those correspond to the famous ``crystallographic condition''. However, for \(k\neq 1,2,3,4,6\), \(\alpha = {2\pi\over k}\) yields a quasicrystal (or an approximate) in the sense of \textit{V. I. Arnol'd} [Physica D 33, No. 1-3, 21-25 (1988; Zbl 0684.52008)].
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 37B99
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 82D25
zbMATH DE Number: 64949
zbMATH Keywords: symplectic maps
zbMATH Keywords: quasicrystals
zbMATH Keywords: stochastic webs
reviewed by: Jair Koiller
MaRDI profile type: Publication
cites work: Reversible systems
cites work: Remarks on quasicrystallic symmetries
cites work: Number of critical points for a quasiperiodic potential
cites work: Stochastic webs
full work available at URL: 0960-0779(92)90049-s
OpenAlex ID: W1987258267
description: scientific article; zbMATH DE number 64949