Apparently chaotic orbits embedded in closed curves (with S. Bahar). SIAM Journal on Applied Mathematics, 60(5), 1824-1840(2000)

In Bahar (Chaos, Solitons and Fractals, 5(6), (1995) and Chaos, Solitons and Fractals, 7(1), (1996)), pictures were presented of the forward orbit {z_n}_n in the plane, where z_n satisfies z_n+1 = z_n A_n+1 for certain affine transformations A_n. These orbits (a) appear to lie on closed curves, and (b) assume forms reminiscent of strange attractors. We answer the questions raised in Bahar (1995,1996) and prove that indeed in the aperiodic case the invariant orbits of this "iterated function system" lie on closed curves and that in some sense the maps act "chaotically" on the set of closed curves. A key role is being played by almost periodic functions. Furthermore, we provide an expression for the top Lyapunov exponent, that is, a sum in the periodic case and an integral in the aperiodic case. Our results have important implications in applications of computer visualization and imaging, and, shed some light onto nonlinear dynamical systems.