Unstable evolution of pointwise trajectory solutions to chaotic maps

Simple chaotic maps are used to illustrate the inherent instability of trajectory solutions to the Frobenius-Perron equation. This is demonstrated by the difference in the behavior of delta-function solutions and of extended densities. Extended densities evolve asymptotically and irreversibly into invariant measures on stationary attractors. Pointwise trajectories chaotically roam over these attractors forever. Periodic Gaussian distributions on the unit interval are used to provide insight. Viewing evolving densities as ensembles of unstable pointwise trajectories gives densities a stochastic interpretation. (c) 1995 American Institute of Physics.