Lagarias, Jeffrey C. and Rains, Eric
(2005)
*Dynamics of a family of piecewise-linear area-preserving plane maps III. Cantor set spectra.*
Journal of Difference Equations and Applications, 11
(14).
pp. 1205-1224.
ISSN 1023-6198.
doi:10.1080/10236190500273184.
https://resolver.caltech.edu/CaltechAUTHORS:20171002-100214237

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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. These maps are area-preserving homeomorphisms of ℝ^2 that map rays from the origin to rays from the origin. Orbits of the map 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 difference equation can be rewritten in an eigenvalue form for a nonlinear difference operator of Schrödinger type – x_(n+2)+2x_(n+1) – x_n +V_μ(x_(n+1))x_(n+1) = Ex_(n+1), in which μ = (1/2)(a − b) is fixed, and V_μ(x) = μ(sgn(x)) is an antisymmetric step function potential, and the energy E = 2 − 1/2(a+b). We study the set Ω_(SB) of parameter values where the map T_(ab) has at least one bounded orbit, which correspond to l∞-eigenfunctions of this difference operator. The paper shows that for transcendental μ the set Spec∞[μ] of energy values E having a bounded solution is a Cantor set. Numerical simulations suggest the possibility that these Cantor sets have positive (one-dimensional) measure for all real values of μ.

Item Type: | Article | ||||||||||||
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Additional Information: | © 2005 Taylor & Francis. Received 25 May 2005, Accepted 05 Jun 2005, Published online: 19 Aug 2006. | ||||||||||||

Subject Keywords: | Area preserving map, Discrete Schrödinger operator, Symbolic dynamics, Tight binding model | ||||||||||||

Issue or Number: | 14 | ||||||||||||

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

DOI: | 10.1080/10236190500273184 | ||||||||||||

Record Number: | CaltechAUTHORS:20171002-100214237 | ||||||||||||

Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20171002-100214237 | ||||||||||||

Official Citation: | Dynamics of a family of piecewise-linear area-preserving plane maps III. Cantor set spectra Jeffrey C. Lagarias & Eric Rains Journal of Difference Equations and Applications Vol. 11 , Iss. 14,2005 | ||||||||||||

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

ID Code: | 81940 | ||||||||||||

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

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

Deposited On: | 02 Oct 2017 17:13 | ||||||||||||

Last Modified: | 15 Nov 2021 19:47 |

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