Let $M$ be a compact, finite-dimensional Riemannian manifold, let $T: M \rightarrow M$ be an [Anosov diffeomorphism](http://en.wikipedia.org/wiki/Anosov_diffeomorphism), and let $\mu$ be a [Sinai-Ruelle-Bowen (probability) measure](http://www.scholarpedia.org/article/Hyperbolic_dynamics#Measure-theoretic_properties). Write $\mathcal{R} = \{ R\_1,\dots,R\_n \}$ for a [Markov partition](http://en.wikipedia.org/wiki/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?



Do there exist Markov partitions with (nearly) uniform SRB measures?
Since (p\_1....,p\_n) is an eigenvector of the transition matrix, your uniformity assumption is satisfied iff the transition matrix associated to the partition is bistochastic. I guess this is rarely the case.


