TITLE: Lebesgue entropy zero and positive topological entropy
QUESTION [8 upvotes]: I am looking for examples of volume preserving $C^{\infty}$ diffeomorphisms $f$ of a surface, which have positive topological entropy ($h(f) > 0$), but that the Lebesgue measure entropy (metric entropy) is equal to zero. It was pointed out to me that such horrible things exist. 
These examples should be very weird in nature as positive topological entropy implies the existence of a homoclinic periodic point (by a theorem of A. Katok), which implies the existence of some horseshoe for a power of $f$.
I am also looking for conditions on a diffeomorphism $f$ of a surface or a measure $\mu$ invariant under $f$ (For example $f$ being real analytic or $\mu$ being SRB and $h_{\mu}(f) >0$) that would imply it has positive lebesgue entropy.
Any reference on these questions on anything somewhat related to this would be greatly appreciated.
Thanks!!

REPLY [3 votes]: I do not have an direct answer but since I fear the answer might not be known and you also ask for related stuff let me mention a couple of results. 
In the $C^1$ case, Mañe-Bochi's result asserts that far from Anosov systems, this is the ``typical'' phenomena. However, $C^1$-generic sets of diffeomorphisms may contain no smooth ones. 
In the $C^\infty$ case, one could imagine that it is possible to start with some diffeomorphism which has positive entropy (a linear Anosov in $T^2$ for example) and opening elliptic islands in the periodic orbits until they fill up a full lebesgue measure subset, but I am not sure if this can be done (nor if the existence of such an example is known). 
I guess a good place to look for this type of phenomena is looking at the problem of coexistence (for example, in section 1.1.2 of that paper a way of constructing the above map is proposed but it claims not to be known if the measure of the complement of the islands is positive).