  We show that any area-preserving $C^r$-diffeomorphism of a two-dimensional
surface displaying an elliptic fixed point can be $C^r$-perturbed to one
exhibiting a chaotic island whose metric entropy is positive, for every $1\le
r\le \infty$. This proves a conjecture of Herman stating that the identity map
of the disk can be $C^\infty$-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 $C^\infty$-approximated by a
conservative diffeomorphisms displaying a positive metric entropy (a weak
version of Sinai's positive metric entropy conjecture). Finally, this sheds
light onto a Herman's question on the density of $C^r$-conservative
diffeomorphisms displaying a positive metric entropy: we show the existence of
a dense set formed by conservative diffeomorphism which either are weakly
stable (so, conjecturally, uniformly hyperbolic) or display a chaotic island of
positive metric entropy.
