This paper studies the behavior under iteration of the maps T ab (x,y) ¼ (F ab (x) 2 y, x) of the plane R 2 , in which F ab (x) ¼ 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 equationThis difference equation can be rewritten in an eigenvalue form for a nonlinear difference operator of (x)) is an antisymmetric step function potential, and the energy E ¼ 2 2 1/2(a þ b). We study the set V SB of parameter values where the map T ab has at least one bounded orbit, which correspond to l 1 -eigenfunctions of this difference operator. The paper shows that for transcendental m the set Spec 1 [m ] 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 m.