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.



Why aren't operator semigroups studied from a dynamical perspective?
Jack Hale had important work in generalizing concepts of dynamical systems to the infinite dimensional setting, see his monographs:


[Asymptotic behavior of dissipative systems](http://books.google.hu/books?id=fSzscCu37ygC&lpg=PP1&hl=de&pg=PP1#v=onepage&q&f=false)


It turns out that, under compactness assumtions, it is possible to generalize a lot of the finite dimensional concepts.
The work of Temam, Foiaș and their group connected to the Navier-Stokes equations is also extremely important, see


[Infinite dymensional dynamical systems](http://books.google.hu/books?id=0VZMYKSgkqMC&lpg=PP1&hl=de&pg=PP1#v=onepage&q&f=false)


On more recent work of the chaotic behavior (as mentioned b Uwe in his comment) you should consult the excellent monograph


[Linear Chaos](http://books.google.hu/books?id=UwsBihOIDkoC&lpg=PP1&hl=de&pg=PP1#v=onepage&q&f=false)


where there is an account on recent results.


But of course the main problem is, as remarked by Uwe, that we are still struggling with the proper notions of generalization of some of the concepts mentioned.


