the periodic specification property then $${h_\textrm{top}}(T)=\lim_{n\to \infty}\frac{1}{n}\log |\operatorname{Per}_n(T)|.$$

Every expansive dynamical system has a measure of *maximal entropy*, since expansiveness implies that the function $\operatorname{\mathcal{M}_T}(X)\ni\mu\mapsto h_\mu(T)\in [0,\infty)$ is upper semicontinuous and every such function on a compact metric space is bounded from the above and attains its supremum. It turns out that for a system with the periodic specification property the entropy maximizing measure is unique and can be described more precisely.

For each $n\in {\mathbb{N}}$ such that $\operatorname{Per}_n(T)$ is nonempty denote by $\mu_n$ the probability measure uniformly distributed on $\operatorname{Per}_n(T)$, that is, $$\mu_n=\frac{1}{|\operatorname{Per}_n(T)|}\sum_{x\in \operatorname{Per}_n(T)}\hat\delta(x).\label{mun}$$ Clearly, each $\mu_n$ is an invariant measure. By the above observation, if $(X,T)$ is expansive and has the periodic specification property, then we can consider an infinite sequence formed by $\mu_n$’s. The proof of following result may be found in [@DGS]. It closely follows Bowen’s proofs in [@Bowen71] and [@Bowen74].

\[thm:10\] If $(X,T)$ is an invertible expansive dynamical system with the periodic specification property, then the sequence $\mu_n$ defined by converges to a fully supported ergodic measure $\mu_B\in\operatorname{\mathcal{M}_T}(X)$, which is the unique measure of maximal entropy of $T$. In particular, $(X,T)$ is intrinsically ergodic.

In Theorems \[thm:8\]–\[thm:10\] one can replace invertible by surjective and expansiveness 