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. 



List of Bernoulli chaotic systems
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.)


Beyond this, I know of basically two classes of examples that are known to be Bernoulli; both are "non-uniformly hyperbolic" in some sense. The first class contains systems that can be modeled by a Young tower or a countable-state Markov shift, where the measure is an equilibrium state for some sufficiently regular potential function; this includes the case where the measure is SRB, in particular when the measure is smooth. For example, this includes any positive entropy equilibrium state for a Hölder continuous potential on a surface diffeomorphism, thanks to recent work of Omri Sarig ([JAMS 2013](http://www.ams.org/mathscinet-getitem?mr=3011417), [JMD 2011](http://www.ams.org/mathscinet-getitem?mr=2854097)).


In fact, if the measure is smooth and has non-zero Lyapunov exponents, then you don't need to build a Young tower or a countable-state Markov shift; for measures like this ("hyperbolic" measures), Bernoullicity was proved by [Yakov Pesin in 1977](http://www.ams.org/mathscinet-getitem?mr=466791). In particular, this includes Liouville measure for geodesic flow of a compact surface of genus at least two without focal points (of course this is continuous-time rather than discrete-time as you asked for). The result for smooth invariant measures was extended to SRB measures by [Ledrappier in 1984](http://www.ams.org/mathscinet-getitem?mr=743818) (he also did [ACIPs for interval maps](http://www.ams.org/mathscinet-getitem?mr=627788) without any Markov assumption). 


