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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?

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The SRB measure is always isomorphic to a Bernoulli scheme (up to a period) and hence has positive entropy.

Regarding continuity properties of entropy, we have upper semicontinuity whenever the map is expansive, which is true for Anosov diffeos. But in general you should not expect lower semicontinuity, since (as you point out) there is a dense set of zero entropy measures, so lower semicontinuity would force every measure to be zero entropy.

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  • $\begingroup$ Thank you. Could you introduce to me a reference that shows isomorphism SRB measure and Bernoulli? $\endgroup$ – Adam Jul 10 at 14:12
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    $\begingroup$ I'm not sure where this fact was first proved, but for instance you can see Theorem 1.25 of Rufus Bowen's 1975 monograph "Equilibrium states and the ergodic theory of Anosov diffeomorphisms", which gives the Bernoulli property upon applying the 1970 result of Friedman and Ornstein ("On isomorphism of weak Bernoulli transformations", Adv. Math.). It should also be in Ruelle's 1976 paper "A measure associated with axiom-A attractors" (Amer. J. Math.). $\endgroup$ – Vaughn Climenhaga Jul 11 at 2:38
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    $\begingroup$ Maybe I should also mention an alternate proof that every SRB measure (more generally, every equilibrium state for a Holder potential over a uniformly hyperbolic system) has positive entropy, which actually gives a way to compute a lower bound; see Theorem 6.1 of arxiv.org/abs/1703.05722, which will appear in ETDS later this year. $\endgroup$ – Vaughn Climenhaga Jul 11 at 2:42

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