Continuous versions of some discrete maps or what goes on when the lights go out
type: Item
instance of: scholarly article
title: Continuous versions of some discrete maps or what goes on when the lights go out
zbMATH Open document ID: 0809.58020
DOI: 10.1007/BF02790360
published in: Journal d'Analyse Mathématique
publication date: 1994-10-18T00:00:00Z
review text: The author considers the problem of realizing a diffeomorphism \(\phi : M \to M\) as the time-one map for a vector field \(V\) on \(TM\). He describes four examples of discrete maps on manifolds. Each of them is realized as the time-one map for a completely integrable Hamiltonian system. The \(k\)- iterate of the map corresponds to the time \(k\) evaluation for the Hamiltonian system. He states a sort of stroboscope theorem for these examples: what goes on when the lights go out is described by completely integrable Hamiltonian flow. The relevant ingredient from integrable mechanics is the classical \(R\)- matrix. Each example has a corresponding theorem to be found in the literature. Example \(1\) is the \(QR\) algorithm for computing the eigenvalues of a matrix \(M_ 0\). This is a sequence of isospectral matrices that relates to Toda flows [the author, \textit{L. C. Li} and \textit{C. Tomei}, Commun. Pure Appl. Math. 42, No. 4, 443-521 (1989; Zbl 0689.70006)]. Example 2 is the singular value decomposition algorithm for finding the singular values of a matrix \(M_ 0\) (i.e. the spectrum of \(\sqrt {M_ 0^ T M_ 0}\)). The theorem can be found in [the author, \textit{J. Demmel, L. C. Li} and \textit{C. Tomei}, SIAM J. Numer. Anal. 28, No. 5, 1463-1516 (1991; Zbl 0737.65035)]. The 3rd example is an algorithm for computing an elliptic integral by performing successive arithmetic and geometric means on the parameters. For this there is a theorem by the author, Li, Previato, Tomei. The 4th example is a billiard ball problem on an elliptical billiard table considered in [the author, \textit{L. C. Li} and \textit{C. Tomei}, Mem. Am. Math. Soc. 479, 101 p. (1992; Zbl 0770.34005)].
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 37B99
zbMATH DE Number: 640334
zbMATH Keywords: discrete maps
zbMATH Keywords: completely integrable Hamiltonian system
zbMATH Keywords: \(R\)-matrix
zbMATH Keywords: singular value decomposition
zbMATH Keywords: elliptic integral
zbMATH Keywords: billiard ball problem
author: Percy A. Deift
reviewed by: Luis A. Piovan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 640334