1
$\begingroup$

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.

$\endgroup$

1 Answer 1

1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.