Often times one talks about iterating a continuous map to get discrete topological dynamics, or having a 1-parameter family of continuous maps to get continuous topological dynamics.
When studying Feller processes, or in general semigroups of operators defined on a Banach space, it seems the only notion of dynamical systems that appears is "ergodicity" and it is really about the asymptotic behavior of the family. Why aren't things like fractals in Banach spaces, and non-asymptotic objects such as the orbit and Hausdorf dimension of the orbit and so on studied? For instance, in the Feller case "ergodicity" means that there is exactly one stationary distribution for the semigroup and it is attracting in some sense. All other properties studied (keep in mind I study it from the perspective of a probabilist) seem to be functional analytic.