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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.

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    $\begingroup$ 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". $\endgroup$
    – Ian Morris
    Commented Dec 1, 2015 at 17:07
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    $\begingroup$ 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). $\endgroup$
    – Ian Morris
    Commented Dec 9, 2016 at 15:41

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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, JMD 2011).

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. 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 (he also did ACIPs for interval maps without any Markov assumption).

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  • $\begingroup$ About the Sarig results -- he proves that the system itself is Bernoulli (with countable alphabet), or that the corresponding induced system is Bernoulli? Infinite alphabet shifts should have infinite entropy, if I understand right and it puzzles me. For the flows there are also Sinai billiards and Lorenz flow, but seemingly Bernoullicity for a flow does not necessarily imply Bernoullicity for its time-1 map (or a Poincare map), although I don't see an immediate example showing why it is so. $\endgroup$
    – demitau
    Commented Dec 1, 2015 at 16:42
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    $\begingroup$ 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. $\endgroup$
    – Vaughn Climenhaga
    Commented Dec 1, 2015 at 16:54
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    $\begingroup$ @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 $p_n \ll n^{-1-\varepsilon}$. $\endgroup$
    – Ian Morris
    Commented Dec 1, 2015 at 17:04
  • $\begingroup$ Thanks for the clarification, Ian - I should have mentioned this. A full shift on an infinite alphabet has infinite topological entropy, but as Ian's comment makes clear, even in this case a fully-supported Bernoulli measure can have finite measure-theoretic entropy. $\endgroup$
    – Vaughn Climenhaga
    Commented Dec 1, 2015 at 18:02
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On the other side of the word...Bernoulli systems can be builded in the theory of Cellular Automata. Indeed, the canonical factor of the class of positively expansive one-sided cellular automata are Bernoulli (see [Dynamical properties of expansive one-sided cellular automata]).

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