Lebesgue entropy zero and positive topological entropy

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!!

• When you refering to the Lebesgue measure entropy, are you implying tht the Lebesgue measure of the surface is $f$-invariant?
– Asaf
Feb 7, 2014 at 11:55
• Regarding the second question, assuming $f$ is "nice enough" (the magic words here are Markov partitions), by the Adler-Weiss theorem, the dynamical system will be isomorphic to a Bernoulli system, hence $\mu$ can be thought of as a measure on a Bernoulli system, and there are plenty of such measures, and in this case the entropy of the measure is related to the dimension of its support, in particular, by using only $1$ parameter dynamics, you cannot rule out $0$-dimensional support.
– Asaf
Feb 7, 2014 at 11:59
• Yes, I'm assuming $f$ is preserving volume, I'll fix that. Thanks. Feb 7, 2014 at 15:40
• Thanks for the comment, do you know about references that might help me to understand what you mean by "nice enough"? and also what do you mean by "1 parameter" dynamics and "0 dimensional support" Feb 8, 2014 at 7:12
• "nice enough" - meaning $C^{1+alpha}$ say, with proper algebraic description, the usual examples are hyperbolic toral automorphisms, geodesic flow over homogeneous spaces, and others as well. I think the most recent work in this subject is by Omri Sarig - wisdom.weizmann.ac.il/~sarigo/MP22.pdf . $0$ dimensional support meaning the Hausdorff dimension of the support is $0$. $1$-parameter dynamics meaning that you have here a $\mathbb{N}$ action and not $\mathbb{N}^d$ action for $d>1$.
– Asaf
Feb 8, 2014 at 20:11

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).