## Notions of irreducibility

Let ${(X,f)}$ be a topological dynamical system. (Generally this means, for me at least, a continuous self-map of a compact metric space. However, sometimes one may be interested in examples that are not compact or that are only piecewise continuous.) We can study ${(X,f)}$ as an object in topological dynamics or as an object in ergodic theory (by studying the space of ${f}$-invariant Borel measures on ${X}$), and the interplay between the two turns out to be very rich.

An important principle in the study of any mathematical object is to understand those examples that are in some sense irreducible or indecomposable. From the topological point of view, irreducibility of ${(X,f)}$ corresponds to transitivity, the existence of a dense orbit. (This is very different from minimality, which demands that every orbit be dense, and which is not the setting I am interested in here; I want to study systems where both the topological and statistical asymptotic behaviour of orbits can be very diverse.)

From the point of view of ergodic theory, the picture is much more complex. For a fixed invariant measure ${\mu}$, irreducibility corresponds to ergodicity, the requirement that if ${E\subset X}$ is ${f}$-invariant, then either ${E}$ or its complement has zero measure. However, ${(X,f)}$ may have many invariant measures, some ergodic, some not. Indeed, writing ${{\mathcal{M}_f}}$ for the space of all Borel ${f}$-invariant probability measures on ${X}$ and ${{\mathcal{M}_f^e}}$ for the ergodic measures in ${{\mathcal{M}_f}}$, one finds that ${{\mathcal{M}_f}}$ is a simplex whose extreme points are precisely the elements of ${{\mathcal{M}_f^e}}$. For many examples of interest, this simplex is infinite-dimensional. What corresponds to “measure-theoretic irreducibility” for the whole system in such cases?

Since the ergodic measures are the indecomposable ones, and since moreover a simplex is determined by its extreme values, it makes sense to study the properties of ${{\mathcal{M}_f^e}}$ as a subset of ${{\mathcal{M}_f}}$. In the remainder of the post I’ll explore two fundamental facts about the space of ergodic measures for the full shift:

1. ${{\mathcal{M}_f^e}}$ is dense in ${{\mathcal{M}_f}}$;
2. ${{\mathcal{M}_f^e}}$ is arc-connected.

The second of these gives a concrete sense in which the full shift is measure-theoretically irreducible. Furthermore, both these properties fail for simple-minded topologically reducible examples, such as the system obtained by taking two disjoint copies of the full shift.

Invariant measures for the full shift

The full shift on two symbols is one of the canonical examples of a dynamical system with very rich behaviour. Here ${X = \Sigma_2^+ = \{ 0,1 \}^{\mathbb N}}$ and ${f=\sigma\colon x_1 x_2\dots \mapsto x_2 x_3\dots}$; this is topologically transitive.

First we address density, and show that in fact the class of periodic orbit measures is dense. If ${x}$ is periodic, write ${\delta_x^p}$ for the measure supported on the (finite) orbit of ${x}$ that gives equal weight to each point. Let ${{\mathcal{M}_f^p} = \{ \delta_x^p \mid x \text{ is periodic} \}}$; then ${{\mathcal{M}_f^p} \subset {\mathcal{M}_f^e}}$.

Proposition 1 ${{\mathcal{M}_f^p}}$ is dense in ${{\mathcal{M}_f}}$.

Proof: Let ${d}$ be a metric on ${{\mathcal{M}_f}}$ that induces the weak* topology. Given ${\mu\in {\mathcal{M}_f}}$ and ${\varepsilon>0}$, let ${\mu_1,\dots,\mu_k\in {\mathcal{M}_f^e}}$ and ${\alpha_i\geq 0}$ be such that ${\sum \alpha_i = 1}$ and ${d(\mu,\sum \alpha_i \mu_i) < \varepsilon}$. Let ${x^i = x^i_1 x^i_2 \dots \in X}$ be generic points for ${\mu_i}$. That is, if we write ${\delta_n(x) = \frac 1n \sum_{j=1}^n \delta_{f^j(x)}}$, then ${\delta_n(x^i) \rightarrow \mu_i}$. (The ergodic theorem states that every ergodic measure has a full-measure set of generic points.) Now let ${n_1,\dots,n_k}$ be such that ${\sum_j |\frac{n_j}{\sum n_i} - \alpha_j| < \varepsilon}$, and also ${d(\mu_j, \delta_{n_j}(y_j)) < \varepsilon}$ for every ${y\in [x^j_1\dots x^j_{n_j}]}$. Finally, let ${z}$ be the periodic point obtained by concatenating the words ${x^j_1 \dots x^j_{n_j}}$ in order and repeating. Then ${d(\delta_z^p, \mu) < 10\varepsilon}$, which suffices. $\Box$

Proposition 1 was proved by Sigmund in 1970, and has as a corollary that ${{\mathcal{M}_f^e}}$ is dense in ${{\mathcal{M}_f}}$. The second claim above, that ${{\mathcal{M}_f^e}}$ is arc-connected, was also proved by Sigmund (in 1977). (Thanks to Andrey Gogolev for the reference, via MathOverflow.)

I’ll say more in a future post about a dynamical proof of arc-connectedness based on Sigmund’s. For now, I’ll round out this post with the observation that in fact, arc-connectedness of ${{\mathcal{M}_f^e}}$ follows from density of ${{\mathcal{M}_f^e}}$ in ${{\mathcal{M}_f}}$.

For finite-dimensional simplices, the set of extreme points is closed, and so the phenomenon of extreme points (${{\mathcal{M}_f^e}}$) being dense is very far removed from what happens in more familiar simplices. The first example of a simplex with dense extreme points was given by Poulsen in 1961. (Thanks to Gerald Edgar for pointing this out to me, also via MathOverflow.) Later, it was shown by Lindenstrauss, Olsen, and Sternfeld in 1978 that the Poulsen simplex is unique in the following sense.

Theorem 2 Let ${S_1}$ and ${S_2}$ be metrisable simplices whose extreme points are dense. Then there is an affine homeomorphism from ${S_1}$ to ${S_2}$.

In the same paper, it was shown that the set of extreme points for the Poulsen simplex is arc-connected. Together, these results have the following consequences for the space of invariant measures of a topological dynamical system:

1. If ${(X,f)}$ is such that ${{\mathcal{M}_f^e}}$ is dense in ${{\mathcal{M}_f}}$, then ${{\mathcal{M}_f}}$ is the Poulsen simplex;
2. In this case the set of ergodic measures is arc-connected.

Proposition 1 shows that all of this holds for the full shift. Now it is natural to ask for which other systems the space of invariant measures is the Poulsen simplex, and this will lead us to discuss the specification property in the next post.