What is the proper terminology to describe the stability of a fixed point in a system with a discrete state-space? The states compose a high dimensional discrete torus, and the fixed point in question is stable in some directions and unstable in others. If it was a continuous system, I would use the term saddle point and stable/unstable manifolds. Can these terms be applied to a discrete system? What are my alternatives if not?
Edit: I am thinking about some absorbing state phase transition, e.g., a sandpile model without dissipation. The system can be in an absorbing state even in the supercritical state. Some perturbations will leave the system inactive (similar to displacing a continuous system in the indifferent manifold), some perturbations will relax back to the quiescent state (similar to a stable manifold), and some will trigger a neverending macroscopic cascade, pushing the system to an active stable branch of the phase transition (similar to unstable manifold).
Can you describe the map explicitly? I don't think it makes sense to talk about the behaviour "near" a fixed point in a finite discrete space. If you have a one-to-one discrete map, you can't have a stable manifold, because there can't be any other points whose forward orbits approach your fixed point. If your map is the discretisation of a discrete-time, continuous-space differentiable map, you could talk about the stable and unstable directions of the differentiable map.
Edit: if you're only interested in long-term behaviour, you could talk about basins of attraction. The basin of attraction of a point $p_0$ is the set of points whose forward orbit converges to $p_0$. A point $p$ near $p_0$ might lie in the basin of attraction of $p_0$, or it might be a new fixed point (I think that's what you mean by the 'indifferent manifold'), or it might leave the basin of attraction of $p_0$ (or it might have a more complicated orbit like a limit cycle).