Physical Ergodicity and Exact Response Relations for Low-dimensional Maps

Recently, novel ergodic notions have been introduced in order to find physically relevant formulations and derivations of fluctuation relations. These notions have been subsequently used in the development of a general theory of response, for time continuous deterministic dynamics. The key ingredient of this theory is the Dissipation Function Ω, that in nonequilibrium systems of physical interest can be identified with the energy dissipation rate, and that is used to determine exactly the evolution of ensembles in phase space. This constitutes an advance compared to the standard solution of the (generalized) Liouville Equation, that is based on the physically elusive phase space variation rate. The response theory arising in this framework focuses on observables, rather than on details of the dynamics and of the stationary probability distributions on phase space. In particular, this theory does not rest on metric transitivity, which amounts to standard ergodicity. It rests on the properties of the initial equilibrium, in which a system is found before being perturbed away from that state. This theory is exact, not restricted to linear response, and it applies to all dynamical systems. Moreover, it yields necessary and sufficient conditions for relaxation of ensembles (as in usual response theory), as well as for relaxation of single systems. We extend the continuous time theory to time discrete systems, we illustrate our results with simple maps and we compare them with other recent theories.