Dynamics of a family of piecewise-linear area-preserving plane maps III. Cantor set spectra

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. These maps are area-preserving homeomorphisms of the plane that map rays from the origin into rays from the origin. Orbits of the map 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 difference equation can be written in an eigenvalue form for a nonlinear difference operator of Schrodinger type, in which \mu= 1/2(a-b) is viewed as fixed and the energy E=2- 1/2(a+b). The paper studies the set of parameter values where T_{ab} has at least one nonzero bounded orbit, which corresponds to an l_{\infty} eigenfunction of the difference operator. It shows that the for transcendental \mu the set of allowed energy values E for which there is a bounded orbit is a Cantor set. Numerical simulations suggest that this Cantor set have positive one-dimensional measure for all real values of \mu.