Equilibrium measures for a strongly irreducible TMS (topological Markov shift) are Gibbs measures. In the one dimensional case, they are unique and weak Bernoulli. There are two dimensional, strongly irreducible TMS's with many equilibrium measures. We show that these give rise to three dimensional , strongly irreducible TMS's with (i) Gibbs measures which are not shift invariant, and (ii) equilibrium measures having a variety of spectral properties.