I am looking for examples of $C^{\infty}$ diffeomorphisms $f$ of a surface that have positive topological entropy($h(f) > 0$), but has Lebesgue measure entropy(metric entropy) 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!!