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.



Count of non-trivial ergodic measures of a topological dynamical system
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](https://mathoverflow.net/questions/83981/connectedness-of-space-of-ergodic-measures) 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**](http://dx.doi.org/10.4171/JEMS/838), J. Eur. Math. Soc. (JEMS) 21, No. 2, 355-380 (2019). [ZBL1437.37020](https://zbmath.org/?q=an:1437.37020).


