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
The topological properties of maps on
The properties of maps on spaces are as complicated as you like. The question is then which topological properties are you interested in?
The simplest topological theorems on maps is the Brouwer fixed point theorem, which can be restated as follows:
- Link the points x and f(x) by a path. If you draw a contractible sphere, and you find that as you go around the boundary, this x-f(x) map has a nonzero winding, then there is a fixed point inside this sphere.
The winding of a sphere around another sphere is the index of the map--- it is how many times the sphere covers the other sphere in the map. The Brouwer theorem is classical.
Another classical theorem of this sort is Sharkovskii's theorem:
- There is a linear order on periods of periodic cycles in 1d maps, such that each periodic orbit of length l implies that there is a periodic orbit of length l' whenever l' is greater than l.
Some other results are given by symbolic dynamics, the coarse grained position as a function of time. The notions of the entropy of a dynamical system is related to this. These results are not really topological in character, but they are general, and give qualitative insight, so they are similar.
Many further results can be found here, http://elib.tu-darmstadt.de/tocs/35981431.pdf