  Steady state fluctuation relations for dynamical systems are commonly derived
under the assumption of some form of time-reversibility and of chaos. There
are, however, cases in which they are observed to hold even if the usual notion
of time reversal invariance is violated, e.g. for local fluctuations of
Navier-Stokes systems. Here we construct and study analytically a simple
non-smooth map in which the standard steady state fluctuation relation is
valid, although the model violates the Anosov property of chaotic dynamical
systems. Particularly, the time reversal operation is performed by a
discontinuous involution, and the invariant measure is also discontinuous along
the unstable manifolds. This further indicates that the validity of fluctuation
relations for dynamical systems does not rely on particularly elaborate
conditions, usually violated by systems of interest in physics. Indeed, even an
irreversible map is proved to verify the steady state fluctuation relation.
