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 ℝ^2 that map rays from the origin to 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 rewritten in an eigenvalue form for a nonlinear difference operator of Schrödinger type – x_(n+2)+2x_(n+1) – x_n +V_μ(x_(n+1))x_(n+1) = Ex_(n+1), in which μ = (1/2)(a − b) is fixed, and V_μ(x) = μ(sgn(x)) is an antisymmetric step function potential, and the energy E = 2 − 1/2(a+b). We study the set Ω_(SB) of parameter values where the map T_(ab) has at least one bounded orbit, which correspond to l∞-eigenfunctions of this difference operator. The paper shows that for transcendental μ 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.