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

Abstract

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)∈ 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 gives an auxiliary map S_(ab) : S^1 → S^1 of the circle, which has a well-defined rotation number. This paper characterizes the possible dynamics under iteration of T_(ab) when the auxiliary map S_(ab) has rational rotation number. It characterizes cases where the map T_(ab) is a periodic map.