# List of Bernoulli chaotic systems

Which discrete chaotic systems are known to be Bernoulli (i.e. measure theoretically isomorphic to a Bernoulli shift, one-sided or two-sided)?

I am aware that it is known for some uniformly hyperbolic systems (Axiom A, Markov maps of the interval), but I don't know much other examples.

• An interesting converse question would be "Which (natural invariant measures on) discrete-time smooth dynamical systems are known to be mixing, of positive metric entropy, and not Bernoulli?". I think the answer might be "none". – Ian Morris Dec 1 '15 at 17:07
• I have since learned that the answer to my question above is not "none". In 1980 Katok constructed a discrete-time smooth diffeomorphism of an $8$-manifold which is Kolmogorov but not Bernoulli. Lower-dimensional examples were subsequently constructed by Rudolph (dimension 5) and Kanigowski, Rodriguez-Hertz and Vinhage (dimension 4). – Ian Morris Dec 9 '16 at 15:41

The most well-understood examples are the ones you mention: Axiom A diffeomorphisms and Markov maps of the interval, since these can be modeled by SFTs. Note that "Bernoulli" refers to a particular choice of invariant measure; the SRB measure for an Axiom A attractor (or the ACIP for a Markov interval map) is Bernoulli, but other invariant measures, such as periodic orbit measures, need not be. More generally, given a system modeled by an SFT, equilibrium states for Hölder continuous potentials are always Bernoulli. (Recall that these are invariant measures $\mu$ that maximize the quantity $h(\mu) + \int\phi\,d\mu$, where $h(\mu)$ is Kolmogorov-Sinai entropy and $\phi\colon X\to \mathbb{R}$ is the potential function.)
• Sarig's JMD paper considers a diffeomorphism $f$ of a surface $M$ with a positive entropy equilibrium state $\mu$ and shows that $(M,f,\mu)$ is Bernoulli; that is, the system itself is Bernoulli. He does this by obtaining a semi-conjugacy to $(M,f,\mu)$ from a countable-state shift. An infinite alphabet shift has infinite entropy if it is the full shift, but if it is just a Markov shift then it can have finite entropy (e.g., if not many transitions are allowed). Even if it is the full shift, it may have finite Gurevich pressure for certain potentials, which is what Sarig uses. – Vaughn Climenhaga Dec 1 '15 at 16:54
• @demitau: there's no issue with a fully-supported Bernoulli measure on the countable full shift having finite entropy: the $(p_1,p_2,\ldots)$-Bernoulli measure on the full shift with countable alphabet has entropy $\sum_{n=1}^\infty -p_n\log p_n$, which is clearly finite if $p_n \ll n^{-1-\varepsilon}$. – Ian Morris Dec 1 '15 at 17:04