It is quite easy to construct a dynamical system which has a physical measure with a positive Lyapunov exponent and zero entropy, just a figure $\infty$ system. By Pesin's entropy formula such a measure can not be a Sinai-Ruelle-Bowen measure. Now my question is, if there exists a system with a physical measure with positive entropy that is not a SRB-measure. I think the answer is Yes, but I did not manage the construction of such a system. (Perhaps I have to add that I am interested in ergodic and hyperbolic physical measures that are not SRB)


1 Answer 1


Yes. The simplest construction is to let $f$ be the figure-eight system so that $\delta_p$ is a physical non-SRB measure (where $p$ is the saddle point) and let $g$ be an Anosov diffeomorphism with SRB measure $\mu$ (a hyperbolic toral automorphism with Lebesgue measure will do the job); then consider the product system $f\times g$. The product measure $\delta_p\times \mu$ is physical, not SRB, and has positive entropy.

Of course that example feels like cheating and is almost certainly not the sort of thing you had in mind. A more informative example comes from

Hofbauer, Franz; Keller, Gerhard, Quadratic maps without asymptotic measure, Commun. Math. Phys. 127, No. 2, 319-337 (1990). ZBL0702.58034.

Theorem 2 in that paper says that if one considers a full, continuous family of S-unimodal interval maps (for example, the family of quadratic maps $\{f_a\colon [0,1]\to [0,1] : a\in [0,4]\}$ given by $f_a(x) =ax(1-x)$), then for every $0 < h < \log(\frac{1+\sqrt{5}}2)$ there are uncountably many parameter values with an ergodic measure $\nu$ that is singular to Lebesgue, has entropy $h$, and is the limit of the empirical measures for Lebesgue-a.e. $x$ (in other words, it is physical).

I have not read the paper carefully enough to have any insight into how those parameters are chosen, beyond knowing that it has something to do with kneading sequences.

It is then reasonable to ask about diffeomorphisms (as opposed to non-invertible interval maps) and conjecture that similar examples exist in the Hénon family of maps, but I do not know of any results in this direction and did not find any in a quick search.

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    $\begingroup$ Many thanks for this detailed answer! $\endgroup$ May 3, 2020 at 13:29

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