The ergodic theory of chaotic feedback systems

The notion of an invariant density for a dynamical system is defined, and it is shown that by the Birkhoff ergodic theorem such densities may be used to compute various interesting statistics for the trajectories. It is also shown how, for one class of systems, such densities satisfy a certain functional equation. Existence and uniqueness results for these functional equations are presented. It is shown that for some chaotic systems no continuous invariant densities exist whereas for a related class of systems there exist unique invariant uniform densities. The principal difference between these two classes of systems is that for the former there do not exist positively invariant sets with nonzero content.