How do I know if a discrete time-invariant homogeneous dynamic system will reach, at some point, an equilibrium point? Is this even possible? Given a time-invariant homogeneous dynamic system:
$$x(k+1) = Ax(k)$$
My textbook defines an equilibrium point of the system as:

A vector $\bar x$ is an equilibrium point if it has the
  property that once the system state vector is equal to  $\bar x$ it remains
  equal to  $\bar x$ for all future time.

Obvious consequence is that the equilibrium point must satisfy $\bar x = A \bar x$, hence should be an eigenvector of $A$ with the corresponding eigenvalue equal to $1$.
Given the initial state $x(0)$ of the system and the equilibrium point $\bar x$, how do I know if the system will reach this equilibrium point in the future?
Without knowing of the initial state, can I determine the initial state that will cause the system to reach an equilibrium point in the future?
Does the following make sense and can I solve it?
$$
\bar x = A^kx(0)\\
A^{-k} \bar x = A^{-k}A^kx(0)\\
A^{-k} \bar x = x(0)
$$
 A: To write $A^{-k}$ you need to know that $A$ is invertible, which needs not always be the case in general. If this is true, then what you wrote is correct but not so interesting : if $\overline{x}=A\overline{x}$ then for all $k \in \mathbb{Z}$, $\overline{x}=A^k\overline{x}$.
In other words, if your dynamical system is invertible, then the only initial conditions that lead to an equilibrium (in finite time !) is this equilibrium.
If however the system is not invertible, then it may happen that the equation $\overline{x}=A^kx_0$ has other solutions, and so that you get non-trivial initial conditions leading to this equilibrium in finite time. 
Think of an invertible dynamical system as one where you can "go back in time", i.e. where given an initial condition you can find uniquely what previous states the system must have been in in the past to lead to this initial state. A non-invertible system is one where there are multiple "past histories" possible.
Note that these questions also make sense (and are much more complex and interesting!) if $A$ is not a linear map.
A: 
How do I know if a discrete time-invariant homogeneous dynamic system will reach, at some point, an equilibrium point?

This is a rare case.  Really one should speak of convergence toward equilibrium, not attaining equilibrium.

Is this even possible?

For any finite value of $k$ it is, in some sense, possible to describe the solutions of $A^k(x)=x$, which are the points that reach exact equilibrium within $k$ steps. It is much more complicated, and impossible for some specific dynamical systems and starting states, to determine whether for a given starting point $x$ there exists a number of steps $k$ that is enough to drive $x$ to exact equilibrium.  Most $x$ either are driven closer and closer to equilibrium without reaching it, or display some divergent or oscillatory behavior.
For continuous state spaces, usually the equilibrium points with $A(x)=x$ are a small subset of the possible starting points, and there is a slightly larger but still very small subset of points that reach equilibria in a finite number of steps.  These states are exceptions and they usually occupy zero volume in the state space.   Points that are driven toward one or another equilibrium point (or along a particular escape route "to infinity", which is often treated as another equilibrium point), but never reach the target in a finite number of steps, will generally be a large subset of the space of starting states, such as all of the states, or an open subset of states, or all states except some zero volume exceptional set.   
For a nonlinear dynamical system there can be other kinds of complicated behavior but as far as fixed-point $f(x)=x$ type of equilibria are concerned, convergence toward a fixed point is much more typical than exactly attaining the equilibrium. 
