Dynamics of a Family of Piecewise-linear Area-preserving Plane Maps Ii. Cantor Set Spectra

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. This family of 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 orbits of the map are solutions of the nonlinear difference operator of Schrödinger type −xn+2+2xn+1−xn+Vμ(xn+1)xn+1 = Exn+1, with energy parameter E = 2 − 12(a + b) and with an antisymmetric step-function potential Vμ(x) specified by the parameter μ = 1 2(a − b). We study the set Ωbdd of parameter values where the map Tab has at least one bounded orbit, which correspond to l∞-eigenfunctions of the difference operator. For transcendental μ we prove that the set Spec∞[μ] of energy values E having a bounded solution is a Cantor set. Numerical simulations suggest the possibility that these Cantor sets have positive (one-dimensional) measure for all real values of μ.