## Dmitry Turaev

### Abstract:

We show that any smooth area-preserving diffeomorphism of a two-dimensional surface which has an elliptic fixed point can be perturbed to one exhibiting a chaotic island whose metric entropy is positive, the perturbation is small in the C-infinity topology. This proves a conjecture by Herman stating that the identity map of the disk can be perturbed to a conservative diffeomorphism with positive metric entropy. This implies also that the Chirikov standard map for large and small parameter values can be arbitrarily well approximated by a conservative diffeomorphism with a positive metric entropy (a weak version of Sinai’s positive metric entropy conjecture). We argue that these results question the relevance of the notions of metric entropy and Lyapunov exponents to the grand problem of understanding the Hamiltonian chaos. This is a joint work with Pierre Berger.

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