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

This paper studies the behavior under iteration of the maps Tab(x, y) = (Fab(x) - y, x) of the plane R², in which Fab(x) = ax if x ≥ 0 and bx if x « 0. The orbits under iteration correspond to solutions of the nonlinear difference equation xn+2 = 1/2(a - b)[xn+1] + 1/2(a + b)xn+1 - xn. This family of piecewise-linear maps has the parameter space (a, b] ϵ R². These maps are area-preserving homeomorphisms of R² that map rays from the origin into rays from the origin. The action on rays gives an auxiliary map Sab: S¹ ↵ S¹ of the circle, which has a well-defined rotation number This paper characterizes the possible dynamics under iteration of Tab when the auxiliary map Sab has rational rotation number, It characterizes cases where the map Tab is a periodic map. [ABSTRACT FROM AUTHOR]