Equivalence of “nonequilibrium” ensembles for simple maps

In this paper we discuss the equivalence of different approaches to the same transport problems, with particular attention to recent works by Breyman et al. These suggest a singular equivalence between the stationary states of models from Gaussian dynamics, and those from chaotic scattering theory. Although, the mentioned equivalence may concern but a limited set of the properties of those systems, it is very important from a theoretical standpoint. We argue that a special form of dissipativity, in the evolution equations of the phase space distributions, is to be invoked to understand those facts. We discuss one application of periodic orbit theory, to illustrate how the stationary states of thermostatted systems can be investigated and some qualitative features obtained.