not contain any proper nonempty subsystem. Given $x\in X$ we define the *orbit* of $x\in X$ as the set $\operatorname{Orb}_T(x)={\left\{x,T(x), T^2(x),\ldots\right\}}$ and the *orbit closure* of an $x$ as $\overline{\operatorname{Orb}(x,T)}$. A point $x\in X$ is *minimal* if its orbit closure is a minimal set.

Invariant measures
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Let $\operatorname{\mathcal{M}}(X)$ be the set of all Borel probability measures on $X$ equipped with the weak$^\ast$-topology. It is well known that this is a compact metrizable space (see [@Walters §6.1]). A metric inducing the weak$^\ast$-topology on $\operatorname{\mathcal{M}}(X)$ is given by $$\operatorname{\overrightarrow D}(\mu,\nu)=\inf \{{\varepsilon}>0\colon \mu(A)\le \nu (A^{\varepsilon}) +{\varepsilon},\text{ for every Borel set }A\subset X\},$$ where $\mu,\nu\in \operatorname{\mathcal{M}}(X)$ and $A^{\varepsilon}=\{x\in X\colon \rho(x,A)<{\varepsilon}\}$ denotes the ${\varepsilon}$-neighborhood of $A$ (see [@Strassen65]). The *support* of a measure $\mu\in\operatorname{\mathcal{M}}(X)$, denoted by $\operatorname{supp}\mu$, is the smallest closed set $C\subset X$ such that $\mu(C)=1$. We say that $\mu\in \operatorname{\mathcal{M}}(X)$ has *full support* if $\operatorname{supp}\mu=X$.

Let $\operatorname{\mathcal{M}_T}(X)$ denote the set of all $T$-invariant measures in $\operatorname{\mathcal{M}}(X)$. By the Krylov-Bogolyubov theorem any dynamical system admits at least one invariant Borel probability measure. We write $\operatorname{\operatorname{\mathcal{M}}_T^e}(X)$ for the subset of all ergodic measures. We say that $T$ is *uniquely ergodic* if there is exactly one $T$-invariant measure.

Recall that $\operatorname{\mathcal{M}_T}(X)$ is a Choquet simplex 