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, let's denote it by $M^\mathrm{erg}_\varphi(X)$, can become. More precisely, is $M^\mathrm{erg}_\varphi(X)$ always countable? And if not in general, are there conditions that assure that $M^\mathrm{erg}_\varphi(X)$ is countable, e.g., $\varphi$ is a homeomorphism, $X$ is metrizable or both.

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.