function $\operatorname{\mathcal{M}_T}(X)\ni\nu\mapsto h_\nu(T)\in{\mathbb{R}}$ is upper semicontinuous at $\mu$, then $\mu$ is the limit in the weak$^\ast$ topology of a sequence of ergodic measures $\mu_n$ such that the entropy of $\mu$ is the limit of the entropies of the $\mu_n$. This is an important point in obtaining large deviations estimates, which was first emphasized in [@FO] (see also [@Comman09; @Yamamoto09]). Analysis of the proof of Theorem B in [@EKW] yields the following.

\[thm:8\] Let $(X,T)$ be an invertible dynamical system with the specification property. Then the ergodic measures are *entropy dense*, that is, for every measure $\mu\in \operatorname{\mathcal{M}_T}(X)$, every neighborhood $U$ of $\mu$ in $\operatorname{\mathcal{M}_T}(X)$ and every ${\varepsilon}>0$ there is an ergodic measure $\nu\in U$ with $h_\nu(T)\in (h_\mu(T)-{\varepsilon},h_\mu(T)]$.

Let $\operatorname{Per}_n(T)$ denote the set of fixed points of $T^n$, where $n\in{\mathbb{N}}$. Observe that if $(X,T)$ is expansive, then for every $n$ the set $\operatorname{Per}_n(T)$ is finite, and is nonempty for all $n$ large enough provided that $(X,T)$ has the periodic specification property.

Bowen [@Bowen71] proved that if $T$ is expansive and has periodic specification, then the topological entropy of $T$ equals the exponential growth rate of the number of fixed points of $T^n$.

If $(X,T)$ is an invertible expansive dynamical system with 