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 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 These maps are area-preserving homeomorphisms of 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.