by positive expansiveness or expansiveness of the natural extension.

It is known that the set of fully supported measure is either empty or residual in $\operatorname{\mathcal{M}_T}(X)$, e.g. see [@DGS Proposition 21.11]. It is easy to see that if minimal points are dense in $X$ then the set of fully supported measures is nonempty, hence fully supported measures are dense in $\operatorname{\mathcal{M}_T}(X)$. It follows that the specification property has a strong influence not only on the topological entropy but also on the space of invariant measures. Sigmund studied relations between the specification property and the structure of $\operatorname{\mathcal{M}_T}(X)$ in [@Sigmund70; @Sigmund74]. Parthasarathy [@Parthasarathy61] proved similar results for a dynamical system $(Y,T)$ where $Y=X^\infty$ is a product of countably many copies of a complete separable metric space $X$ and $T$ is the shift transformation. Sigmund’s results may be summarized as follows:

\[thm:sigmund\] If $(X,T)$ has the periodic specification property, then:

1.  \[Sig:1\] The set $\operatorname{\operatorname{\mathcal{M}}_T^{\text{co}}}(X)$ is dense, hence $\operatorname{\operatorname{\mathcal{M}}_T^e}(X)$ is arcwise connected and residual in $\operatorname{\mathcal{M}_T}(X)$, hence $\operatorname{\mathcal{M}_T}(X)$ is the Poulsen simplex.

2.  \[Sig:5\] The set $\operatorname{\operatorname{\mathcal{M}}_T^e}(X)\cap\operatorname{\operatorname{\mathcal{M}}_T^+}(X)$ is residual in $\operatorname{\mathcal{M}_T}(X)$.

3.  \[Sig:6\] The set $\operatorname{\operatorname{\mathcal{M}}_T^{\text{mix}}}(X)$ is of first category in $\operatorname{\mathcal{M}_T}(X)$.

4.  \[Sig:9\] The set of all non-atomic measures is residual 