Count of non-trivial ergodic measures of a topological dynamical system

Question

Given a compact Hausdorff space $X$ and a continuous mapping $\varphi: X \to X$. We denote by $C(X)$ the space of continuous functions $f: X \to \mathbb{C}$. A probability measure $\mu$ on the Borel-$\sigma$-algebra of $X$ is said to be ergodic for $\varphi$ if it is $\varphi$-invariant, i.e.,

$$\int_X f \, d\mu = \int_X f \circ \varphi \, d\mu$$

for all $f \in C(X)$, and if all Borel sets $A \subseteq X$ with $\phi(A) \subseteq A$ satisfy $\mu(A) \in \{0,1\}$.

It would be of interest to know how large the set of all ergodic measures $\mu$ that have a support $\mathrm{supp}(\mu)$ that contains at least 2 elements, let's denote it by $M^\mathrm{erg}_{\varphi, \geq 2}(X)$, can become. More precisely, is $M^\mathrm{erg}_{\varphi, \geq 2}(X)$ always countable? And if not in general, are there conditions that assure that $M^\mathrm{erg}_{\varphi, \geq 2}(X)$ is countable, e.g., $\varphi$ is a homeomorphism, $X$ is metrizable or both.

Answer 1

Suppose $X$ is the unit circle and $\varphi$ is the doubling map (multiplicatively, $X = \{ z\in \mathbb{C} : |z| = 1\}$ and $\varphi(z) = z^2$, or additively, $X = \mathbb{R}/\mathbb{Z}$ and $\varphi(x) = 2x \mod \mathbb{Z}$). Then the set of ergodic measures is uncountable and in fact is path-connected and dense in the set of all invariant measures (with the weak* topology). See the answers at this other question for more details and references. That's written in terms of the full shift (and more generally, shifts of finite type) but the simplex of invariant measures is the same for the doubling map and the full two-shift.

Note that invertibility (or lack thereof) plays no role here; a non-invertible system has the same simplex of invariant measures as its natural extension, which is invertible, so the story is the same if you consider homeomorphisms.

I'll mention that all those examples are "high complexity" systems in the sense of having positive topological entropy, exponential growth of periodic orbits, etc. One gets rather different behavior for various classes of "low complexity" systems such as interval exchange transformations, translation surfaces, and subshifts of linear growth; see for example the following paper, which gives some results and references from the symbolic point of view.

Cyr, Van; Kra, Bryna, Counting generic measures for a subshift of linear growth, J. Eur. Math. Soc. (JEMS) 21, No. 2, 355-380 (2019). ZBL1437.37020.

Answer 2

In the most standard examples, the set of invariant ergodic measures has the cardinality of the continuum. For example, take $X=\{0,1\}^{\mathbb Z}$ equipped with the product, topology, the Borel $\sigma$-algebra and the left shift $T(\{x_n\}_{n \in {\mathbb Z}})=(\{x_{n+1}\}_{n \in {\mathbb Z}})$. Then all the product measures $(p,1-p)^{\mathbb Z}$ for $0<p<1$ are invariant and ergodic.

An example where the set of invariant ergodic measures is countable is given in [1], and the construction is easily adapted to yield $k$ invariant ergodic measures for each $k \ge 1$.

[1] Examples of topological dynamical systems with countably infinitely many ergodic invariant measures

Answer 3

I know that you were mostly asking about cardinality, but more generally, you can ask about the structure of this space $\mathcal{M}(X,T)$ of invariant measures as a topological space with the weak topology (as Vaughn mentioned). It's known that $\mathcal{M}(X,T)$ must be a nonempty convex compact metrizable Choquet simplex, meaning that every point can be written as an "average" (integral) over extreme points. (This is because extreme points are precisely the ergodic measures, and the ergodic decomposition theorem states that every invariant measure is an "average" of ergodic measures).

In fact EVERY possible nonempty compact metrizable Choquet simplex is realizable as $\mathcal{M}(X,T)$ for some topological dynamical system $(X,T)$. There are all kinds of crazy Choquet simplices (in fact one where the extreme points are dense, as Vaughn mentioned), so you can get many different structures for the ergodic measures.

There are some hypotheses that guarantee that the set of ergodic measures is small. For instance, linear growth of the word complexity function implies this for subshifts. So does a uniform bound on the topological sequence entropy for all sequences. But the examples with huge $\mathcal{M}(X)$ are quite ubiquitous; for instance, there are several results showing that all possible Choquet simplices are realizable as spaces of invariant measures for topological systems from restricted classes (e.g. minimal systems, Toeplitz subshifts, logistic maps, etc.)