Mathematics – Dynamical Systems
Scientific paper
2003-01-25
Journal of Difference Equations and Applications 11 (2005), No. 12, 1089--1108
Mathematics
Dynamical Systems
v2. original v1 (52 pages) divided in two parts, this is first part; 18 pages latex. Current part II is math.DS/0303007; part
Scientific paper
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)\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.
Lagarias Jeffrey C.
Rains Eric
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