  Deterministic dynamical systems such as the baker maps are useful to shed
light on some of the conditions verified by deterministic models in
non-equilibrium statistical physics. We investigate a 2D dynamical system,
enjoying a weak form of reversibility, with peculiar basins of attraction and
steady states. In particular, we test the conditions required for the validity
of the Transient Fluctuation Relation. Our analysis illustrates by means of
concrete examples why ergodicity of the equilibrium dynamics seems to be a
necessary condition for the Transient Fluctuation Relation to hold. This
investigation then leads to the numerical verification of a kind of transient
relation which, differently from the usual Transient Fluctuation Relation (FR),
holds only asymptotically. At the same time, it is not a steady state
fluctuation relation, because no fluctuations are present in the steady state.
