  We present a simple one-dimensional Ising-type spin system on which we define
a completely asymmetric Markovian single spin-flip dynamics. We study the
system at a very low, yet non-zero, temperature and we show that for empty
boundary conditions the Gibbs measure is stationary for such dynamics, while
introducing in a single site a $+$ condition the stationary measure changes
drastically, with macroscopical effects. We achieve this result defining an
absolutely convergent series expansion of the stationary measure around the
zero temperature system. Interesting combinatorial identities are involved in
the proofs.
