Let $M$ be a compact, finite-dimensional Riemannian manifold, let $T: M \rightarrow M$ be an Anosov diffeomorphism, and let $\mu$ be a Sinai-Ruelle-Bowen (probability) measure. Write $\mathcal{R} = \{ R_1,\dots,R_n \}$ for a Markov partition; write $p_j^{(\mathcal{R})} := \mu(R_j)$.

**Question:**

Does there ever/always exist $\mathcal{R}$ s.t. $p^{(\mathcal{R})}$ is a nontrivial uniform measure on $\{1,\dots,n\}$? If not, does there ever/always exist a sequence of partitions $\mathcal{R}_m$ s.t. $p^{(\mathcal{R}_m)}$ converges to a uniform measure in some nontrivial sense?