Lagarias, Jeffrey C. and Rains, Eric
(2005)
*Dynamics of a family of piecewise-linear area-preserving plane maps I. Rational rotation numbers.*
Journal of Difference Equations and Applications, 11
(12).
pp. 1089-1108.
ISSN 1023-6198.
doi:10.1080/10236190500273069.
https://resolver.caltech.edu/CaltechAUTHORS:20170927-154918316

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## Abstract

This paper studies the behavior under iteration of the maps T_(ab)(x,y) = (F_(ab)(x)−y,x) of the plane ℝ^2 in which F_(ab)(x) = ax if x ≥ 0 and bx if x < 0. The orbits under iteration correspond to solutions of the nonlinear difference equation x_(n+2) = 1/2(a−b)|x_(n+1)|+1/2(a+b)x_(n+1)–x_n. This family of piecewise-linear maps has the parameter space(a,b)∈ ℝ^2. These maps are area-preserving homeomorphisms of ℝ^2 that map rays from the origin into rays from the origin. The action on rays gives an auxiliary map S_(ab) : S^1 → S^1 of the circle, which has a well-defined rotation number. This paper characterizes the possible dynamics under iteration of T_(ab) when the auxiliary map S_(ab) has rational rotation number. It characterizes cases where the map T_(ab) is a periodic map.

Item Type: | Article | ||||||||||||
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Additional Information: | © 2005 Taylor & Francis. We did most of the work on this paper while employed at AT&T Labs-Research, whom we thank for support; most of the results in parts I and II were obtained in the summer of 1993. We thank T. Spencer for helpful comments on the relation of (1.7) to nonlinear Schrödinger operators, and M. Kontsevich for bringing the work of Bedford, Bullett and Rippon [2] to our attention. | ||||||||||||

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Subject Keywords: | area preserving map, iterated map, discrete Schrödinger operator | ||||||||||||

Issue or Number: | 12 | ||||||||||||

Classification Code: | AMS Subject Classification: Primary: 37E30 Secondary: 52C23, 82D30 | ||||||||||||

DOI: | 10.1080/10236190500273069 | ||||||||||||

Record Number: | CaltechAUTHORS:20170927-154918316 | ||||||||||||

Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20170927-154918316 | ||||||||||||

Official Citation: | Dynamics of a family of piecewise-linear area-preserving plane maps I. Rational rotation numbers Jeffrey C. Lagarias & Eric Rains Journal of Difference Equations and Applications Vol. 11 , Iss. 12,2005 | ||||||||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||||

ID Code: | 81890 | ||||||||||||

Collection: | CaltechAUTHORS | ||||||||||||

Deposited By: | Tony Diaz | ||||||||||||

Deposited On: | 27 Sep 2017 23:02 | ||||||||||||

Last Modified: | 15 Nov 2021 19:46 |

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