Asymptotic periodicity and banded chaos

Treating one dimensional maps as dynamical systems, we examine their evolution in terms of density flow. In particular, a type of density evolution known as asymptotic periodicity is studied. Unlike statistically stable (or exact) systems, asymptotically periodic systems do not in general evolve to an invariant density, even though they possess one. Consequently, a nonequilibrium formalism, using several metastable states rather than one invariant density, is examined as an alternative to study the physical properties of asymptotically periodic systems. Asymptotic periodicity is demonstrated for the hat map and the quadratic map at the parameters where these maps generate banded chaos or quasiperiodicity. Using the ergodic properties of asymptotic periodicity, a compact expression for the time correlation function is obtained that does not make any assumptions about decoupling the periodic and stochastic components of C(t). Finally a generalization to the Boltzmann-Gibbs entropy, known as the conditional entropy, is studied as an index characterizing the asymptotic density sequence that emerges in asymptotically periodic systems.