# When entropy SRB measure is zero

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?