The phase space dynamics of the discrete dynamical system is
just what you describe--- x(n+1) as a function of x(n). The
phase space itself is the range of values of the x(n), whatever
space they might live on, while the dynamics is the function
that specifies the evolution in one step in time. The connection
with mechanical phase space is provided by a Poincare section.
The Poincare section describes a dynamical continuous system by
its intersections with a given surface in the full phase space.
For a 1d motion, you can consider the half-line x=0,p>0, or in
canonical action-angle coordinates $\theta$ fixed, J arbitrary.
When you have a separable integrable motion, you take any one of
the $\theta$ variables and define a surface by setting it to
zero. Then the motion will intersect this surface once every
period. In mechanical phase space, the phase-space volume is
conserved, but this is not so for maps. The condition of
transversal intersection means that the map from the Poincare
surface to itself can get Th