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\$$.