It is well known that many strongly chaotic dynamical systems have the property that periodic measures are (weak-star) dense in the space of all invariant probability measures.

Let $f:M \rightarrow M$ be a transitive Anosove diffeomorphism on compact metric space $M$.

An $f$-invariant probability measure $μ$ is called an SRB (or physical) measure if there exists an open set $U\subset M$ containing the support of $\mu$ such that, for every continuous function $\Phi:M \rightarrow \mathbb{R}$ and $\mu$-a.e.$x\in U$ , $$ \lim_{n \rightarrow \infty} \frac{1}{n}\sum_{i=1}^{n} \phi(f^{i}(x)) =\int_{M} \phi d\mu.$$ In other words, Lebesgue-almost every point $x\in U$ one has $\frac{1}{n}\sum_{i=0}^{n-1} \delta_{T^{i}(x)} \rightarrow \mu \hspace{0.2cm}\textrm{weak$^{\ast}$topolgy}$. The maximal open set $U$ with this property is called the basin of $f$.

It is well known that every transitive Anosov diffeomorphism carries a unique SRB measure.

Question : Can one give an example that SRB measure has zero entropy?

My solution : Let $p \in M$ be a periodic point associated to $\delta, S$ and $\{x, f(x),..., f^{n-1}(x)\}$ in the periodic exponential specification property. In particular, $f^{n+S}(p)=p$ and \begin{equation}\label{spec1} d(f^{j}(p), f^{j}(x))<\delta e^{- \epsilon min\{j,n-j\}} \hspace{0.2cm} \forall \hspace{0.1cm} j=0,1,...,n . \end{equation}There exists an accuracy function $\gamma:\mathbb{N} \to \mathbb{R}_{\geq 0}$, $\gamma(n)\to 0, n\to \infty$ such that for every natural $N$ there exists a natural $n>N$ and an $n$-periodic orbit $p$ such that $$ \rho\left( \frac{1}{n}\sum_{i=0}^{n-1} \delta_{f^i(p)} , \mu \right) < \gamma(n) , $$ where $\rho$ is some metrisation of the weak-* topology. I need continuity of entropy to finish the prove. Does one know when we have continuity of entropy?