By the way, a word of clarification about discrete dynamical systems.

The discreteness of a D.S. does not stipulate that manifold M is discrete, rather just that the time domain T is discrete.

\\(\phi^t\\) still maps a differentiable manifold into itself. The difference here is that we just have the discrete sequence of functions \\(\phi^0, \\phi^1, \\phi^2\\), and due to the requirement that we have an _action_ of T on the manifold, the nth function here, \\(\phi^n\\), turns out to be just the composition of \\(\phi^1\\) with itself n times, i.e., \\((\phi^1)^n\\).

In other words, the discrete dynamical system is characterized simply by an iterator \\(f = \phi^1: M \rightarrow M\\) that maps each state to its successor state.

The trajectory functions are sequences of states \\([s, f(s), f^2(s), ...]\\).

And the orbits are the corresponding sets of states \\(\lbrace s, f(s), f^2(s), ... \rbrace\\).