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Dynamics of a family of piecewise-linear area-preserving plane maps III. Cantor set spectra

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
Related URLs:
URLURL TypeDescription
https://doi.org/10.1080/10236190500273184DOIArticle
http://www.tandfonline.com/doi/abs/10.1080/10236190500273184PublisherArticle
https://arxiv.org/abs/math/0505103arXivDiscussion Paper
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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