  We consider stochastic dynamics of lattice systems with finite local state
space, possibly at low temperature, and possibly non-reversible. We assume the
additional regularity properties on the dynamics: a) There is at least one
stationary measure which is a Gibbs measure for an absolutely summable
potential Phi. b) Zero loss of relative entropy density under dynamics implies
the Gibbs property with the same Phi.
  We prove results on the attractor property of the set of Gibbs measures for
Phi: 1. The set of weak limit points of any trajectory of translation-invariant
measures contains at least one Gibbs state for Phi. 2. We show that if all
elements of a weakly convergent sequence of measures are Gibbs measures for a
sequence of some translation-invariant summable potentials with uniform bound,
then the limiting measure must be a Gibbs measure for Phi. 3. We give an
extension of the second result to trajectories which are allowed to be
non-Gibbs, but have a property of asymptotic smallness of discontinuities. An
example for this situation is the time evolution from a low temperature Ising
measure by weakly dependent spin flips.
