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) ∈ 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 : S 1 → 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.