  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.
