(see [@Walters §6.2]). In particular, $\operatorname{\mathcal{M}_T}(X)$ is the closure of the convex hull of $\operatorname{\operatorname{\mathcal{M}}_T^e}(X)$, thus $\operatorname{\operatorname{\mathcal{M}}_T^e}(X)$ is a nonempty $G_\delta$-subset of $\operatorname{\mathcal{M}_T}(X)$. Note that $\operatorname{\mathcal{M}_T}(X)$ is a compact metric space, hence a subset of $\operatorname{\mathcal{M}_T}(X)$ is residual if, and only if, it is a dense $G_\delta$.

Recall that $\mu\in\operatorname{\mathcal{M}_T}(X)$ is *strongly mixing* if for any Borel sets $A,B\subset X$ we have $\mu(A\cap T^{-n}B)\to \mu(A)\mu(B)$ as $n\to\infty$. We denote by $\operatorname{\operatorname{\mathcal{M}}_T^{\text{mix}}}(X)$ the set of all strongly mixing measures.

Let $\operatorname{\operatorname{\mathcal{M}}_T^+}(B)$ denote the set of all $\mu\in\operatorname{\mathcal{M}_T}(X)$ such that the Borel set $B\subset X$ is a subset of $\operatorname{supp}\mu$. In particular, $\operatorname{\operatorname{\mathcal{M}}_T^+}(X)$ denote the set of all measures with full support.

We denote by $\operatorname{\operatorname{\mathcal{M}}_T^{\text{co}}}(X)$ the set of all invariant measures supported on the orbit of some periodic point.

Generic points
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Let ${\hat{\delta}({x})}$ denote the point mass measure (Dirac measure) concentrated on $x$. For any $x\in X$ and $N\in{\mathbb{N}}$ let $\operatorname{\mathfrak{m}}(x,N)=\frac 1N\sum_{n=0}^{N-1}\hat\delta(T^n(x))$. A measure $\mu\in \operatorname{\mathcal{M}_T}(X)$ is *generated* by $x\in X$ if $\mu$ is a limit of some subsequence of $\{\operatorname{\mathfrak{m}}(x,n)\}_{n=1}^\infty$. The set of all invariant measures generated by $x\in X$ is denoted by ${\hat{\omega}({x})}$. We say that $x$ is a *generic point* for $\mu\in\operatorname{\mathcal{M}_T}(X)$ if $\mu$ is the unique 