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Let $X=2^\omega = \lbrace 0,1 \rbrace^{\mathbb{N}}$ and $T\colon X \to X$ the (left) shift map. The space $\mathcal{M}$ of $T$-invariant Borel probability measures is convex with $\mathcal{M}^e$, the set of all ergodic measures, exactly the extremal points. Moreover, the set $\mathrm{C}(\mathcal{M}^e)$ of convex combinations of elements of $\mathcal{M}^e$ is dense in $\mathcal{M}$ (in the weak topology).

What is known about which measures in $\mathcal{M}$ are (and are not) in $\mathrm{C}(\mathcal{M}^e)$? In particular, what is an example of a measure in $\mathcal{M} \setminus \mathrm{C}(\mathcal{M}^e)$? What about just a measure in $\mathcal{M} \setminus \mathcal{M}^e$?

I've tagged with 'reference-request' because, besides just answers to the above questions, I'd like some references to where this stuff has been studied.

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Related question:… – Ian Agol Dec 28 '11 at 21:06
up vote 7 down vote accepted

The ergodic decomposition theorem states that any shift-invariant Borel probability measure, $\mu$, can be uniquely expressed in the form $$ \mu=\int_{\mathcal M^e(X)}\nu\,d\eta(\nu). $$

This means that $C(\mathcal M^e)$ is the set of measures that are (finite) convex combinations of ergodic measures and $\mathcal M\setminus C(\mathcal M^e)$ consists exactly of those measures, for which the measure $\nu$ appearing in the decomposition is not finitely supported.

Any non-trivial combination of ergodic measures is non-ergodic.

For a measure that is not a convex combination of ergodic measures, you need to take a measure that is a combination of an infinite number of measures.

For simple examples, let $\mu_\beta$ be the Bernoulli measure on $\lbrace 0,1\rbrace^{\mathbb Z}$ where in each coordinate the symbol 1 appears with probability $\beta$ independently of all other coordinates.

Then $\frac12(\mu_{1/2}+\mu_{1/17})$ is an element of $C(\mathcal M^e)\setminus \mathcal M^e$. A "typical point" from this measure is either (with probability .5 a string of 0's and 1's with a 50% density of 1's'; or with probability .5 a string of 0's and 1's with a 1/17 density of 1's).

The measure $\mu=\int_0^1 \mu_\beta\,d\beta$ (the uniform mixture of all Bernoulli measures) belongs to $\mathcal M\setminus C(\mathcal M^e)$. If you pick a typical point from this measure, there is some density $\beta$ (which itself is uniformly chosen from 0 to 1) and then the sequence of 0's and 1's has density $\beta$ of 1's.

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Thanks for the answer. I'm still trying to understand everything. In particular, I've asked a related question on math StackExchange:… – Quinn Culver Dec 29 '11 at 16:51
Do you know where I can read about such things? – Quinn Culver Jan 4 '12 at 20:24

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