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Apologies if this is a vague question.

It seems that a lot of the literature in smooth dynamics is focused on understanding systems that exhibit hyperbolic/non-uniformly hyperbolic behavior. In other words, these are systems which have nonzero Lyapunov exponents on a set of positive measure. There are a variety of tools that one can use in this setting, which from my understanding often comes from the existence of local stable/unstable manifolds (e.g. notions like accessibility).

Is there a corresponding general theory/set of tools for systems with everywhere zero Lyapunov spectrum? Even a pointer to an analogous monograph such as the book on nonuniform hyperbolicity by Barreira and Pesin would be very helpful.

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The best I could find regarding this are the following notes of Katok on "elliptic" dynamics, found here.

Katok poses that the main difficulty in forming a coherent overall view of such systems is that the local linear model is too restrictive: an "elliptic" matrix is just a rotation, which has no orbit growth at all. On the other hand, a system with zero topological entropy merely has slow orbit growth.

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