Consider probability measures $\mu$ on the space $\{0,1\}^\mathbb{N}$ that are shift-invariant with respect to the left-shift map.
Is there a nice characterization of the ergodic shift-invariant measures?
I know that the ergodic measures are the extremal points in the set of shift-invariant measures. However, I wouldn't call this a nice characterization.
As an example of a nice characterization, we know by de Finetti's theorem that the ergodic measures that are invariant under permutations (which switch bits around by permuting their positions) is exactly the set of Bernoulli measures.
I think all the examples of shift-invariant ergodic measures I know of are Markov chains. Hence this leads to the following sub-question?
Are there shift-invariant ergodic measures which are not Markov chains?
I am sure all this is standard and well-known, however I could not find an answer by Googling. Therefore, I am asking on Math.StackExchange (as opposed to MathOverflow).
Arguably, there is no nice characterization of ergodic shift-invariant measures.
The reason is that the (finite) orbit of any periodic sequence (x_i) in [0,1]^N supports a shift-invariant ergodic measure, and these measures are dense among all invariant measures (ergodic or not). So there is no closed condition which separates ergodicity from simple invariance, and moreover the ergodic measures are not very rigid -- they are as flexible as periodic sequences.
Smooth maps $f : S^1 \rightarrow S^1$ give lots of examples of invariant measures on $\{0,1\}^N$ that do not come from Markov chains. If $f$ preserves linear measure, is of degree 2, and expanding, meaning $f'(x)>1$ everywhere, then $f$ is ergodic. Also $f(x)$ is topologically conjugate to $F(x) = 2x \mod 1$, which is itself a quotient of the shift map on $\{0,1\}^N$ (use binary expansion). This quotient is a bijection outside a countable set.
Thus $f$ gives rise to an ergodic invariant measure on the shift. The Radon-Nikodym derivative of $f$ is given by $f'(x)$, which provides a strong invariant distinguishing $f$ from a (finite) Markov chain. (E.g. for a Markov chain, the derivative of the shift map takes on only finitely many values, the derivative only depends on finitely many coordinates, etc.)