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

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dc.creatorLagarias, Jeffrey C.
dc.creatorRains, Eric
dc.date2005-05-06
dc.date2006-09-19
dc.date.accessioned2026-07-07T06:39:54Z
dc.date.available2026-07-07T06:39:54Z
dc.descriptionThis 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 the plane that map rays from the origin into 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 written in an eigenvalue form for a nonlinear difference operator of Schrodinger type, in which μ= 1/2(a-b) is viewed as fixed and the energy E=2- 1/2(a+b). The paper studies the set of parameter values where T_{ab} has at least one nonzero bounded orbit, which corresponds to an l_{\infty} eigenfunction of the difference operator. It shows that the for transcendental μthe set of allowed energy values E for which there is a bounded orbit is a Cantor set. Numerical simulations suggest that this Cantor set have positive one-dimensional measure for all real values of μ.
dc.descriptionv1 21 pages latex, 2 postscript figures; This was former part II in earlier version. Current part I is math.DS/0301294 and part II is math.DS/0303007; v2 20 pages latex- revised to reference prior work of Beardon, Bullett and Rippon
dc.identifierhttps://arxiv.org/abs/math/0505103
dc.identifierhttp://arxiv.org/abs/math/0505103
dc.identifierJournal of Difference Equations and Applications 11 (2005), No. 14, 1205-1224
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/101256
dc.subjectDynamical Systems
dc.subject37E30; 52C23, 82D30
dc.titleDynamics of a family of piecewise-linear area-preserving plane maps III. Cantor set spectra
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