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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!!

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    $\begingroup$ When you refering to the Lebesgue measure entropy, are you implying tht the Lebesgue measure of the surface is $f$-invariant? $\endgroup$
    – Asaf
    Commented Feb 7, 2014 at 11:55
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    $\begingroup$ 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. $\endgroup$
    – Asaf
    Commented Feb 7, 2014 at 11:59
  • $\begingroup$ Yes, I'm assuming $f$ is preserving volume, I'll fix that. Thanks. $\endgroup$
    – shurtados
    Commented Feb 7, 2014 at 15:40
  • $\begingroup$ 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" $\endgroup$
    – shurtados
    Commented Feb 8, 2014 at 7:12
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    $\begingroup$ "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 $d>1$. $\endgroup$
    – Asaf
    Commented Feb 8, 2014 at 20:11

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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).

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  • $\begingroup$ Thanks a lot for the comment, the references are very helpful. I'm actually interested in the real analytic case, do you know anything special about how relevant could be such assumption? $\endgroup$
    – shurtados
    Commented Feb 8, 2014 at 7:15
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    $\begingroup$ I guess the real analytic case does not help much. Either there is a known example, or it is not known. The standard map (and others which appear in the "coexistence" link above) are real analytic families where the topological entropy is known to be positive, but the metric entropy (or Lyapunov exponents) is not known (as far as I know). $\endgroup$
    – rpotrie
    Commented Feb 9, 2014 at 0:50

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