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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.

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    $\begingroup$ I guess we are still struggling with the "correct" generalization of these concepts to infinite dimensional spaces. Do you know W. Desch, W. Schappacher and G.F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dynam. Systems, 17, 1997, 793–819? Now, roughly 20 years later, people (e.g A. Weber or T. Kalmes) are still thinking and publishing about chaotic C_0 semigroups acting on Banach spaces. You might want to consult them directly. $\endgroup$
    – Uwe Stroinski
    Commented Nov 13, 2013 at 6:40

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Jack Hale had important work in generalizing concepts of dynamical systems to the infinite dimensional setting, see his monographs:

Asymptotic behavior of dissipative systems

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

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

Linear Chaos

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.

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