Dynamics of a family of piecewise-linear area-preserving plane maps I. Rational rotation numbers

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dc.creatorLagarias, Jeffrey C.
dc.creatorRains, Eric
dc.date2003-01-25
dc.date2006-09-19
dc.date.accessioned2026-07-07T06:35:35Z
dc.date.available2026-07-07T06:35:35Z
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. 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)\in 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 defines a map S_{ab} of the circle, which has a well-defined rotation number. This paper characterizes the possible behaviors of T_{ab} under iteration when the rotation number is rational. It characterizes cases where the map T_{ab} is a periodic map.
dc.descriptionv2. original v1 (52 pages) divided in two parts, this is first part; 18 pages latex. Current part II is math.DS/0303007; part III is math.DS/0505103 v3. revised to reflect prior work by Beardon, Bullett and Rippon; v3 corrects misprints
dc.identifierhttps://arxiv.org/abs/math/0301294
dc.identifierhttp://arxiv.org/abs/math/0301294
dc.identifierJournal of Difference Equations and Applications 11 (2005), No. 12, 1089--1108
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/99834
dc.subjectDynamical Systems
dc.subject37E30; 52C23; 82D30
dc.titleDynamics of a family of piecewise-linear area-preserving plane maps I. Rational rotation numbers
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