# Do there exist Markov partitions with (nearly) uniform Riemannian measures?

This question complements this one; the difference is in considering Riemannian versus SRB measures.

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

Question:

Does there ever/always exist $\mathcal{R}$ s.t. $v^{(\mathcal{R})}$ 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. $v^{(\mathcal{R}_m)}$ converges to a uniform measure in some nontrivial sense?

Of course, for hyperbolic toral automorphisms the two questions are equivalent. However I would guess that this question might be easier in general, since the classical construction of Markov partitions uses an open cover of $M$ by balls of fixed radius to form an initial (typically non-Markovian) partition that is subsequently perturbed iteratively into a Markov partition.

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Given a Markov partition you can find a R. metric so that volumes of the rectangles are the same. By changing locally volume of small balls that sit inside of rectangles. Concerning your other question. I am quite sure that the answer is that typically this distribution is not uniform. I don't have a proof of this though. – Andrey Gogolev Jun 2 '10 at 23:26
Andrey--Would the modified Riemannian metric be compatible with the hyperbolic structure? If you can sketch this out either way I'd be more than happy to accept it as an answer. – Steve Huntsman Jun 3 '10 at 0:13
Steve, it depends on what you mean by compatible. The property of being Anosov diffeomorphism is independent of the choice of R. metric on the manifold. The only thing that depends on the metric is the constant $C$ in the inequalities ($|f^nv|\le C\lambda^n|v|$). After this modification ("blowing up" R. metric in small balls) constant $C$ will become large. – Andrey Gogolev Jun 3 '10 at 3:14
Andrey--Any idea what the analogue of this would do to a Markov family for the geodesic flow on a surface of constant negative curvature? Can blowing up preserve the (ergodic properties of) orbits? I am thinking in the context of 3.1 of arxiv.org/abs/cond-mat/0507672 – Steve Huntsman Jun 3 '10 at 16:20
Sketch of an idea along these lines: blow up in directions orthogonal to the flow in some covariant way to normalize the measures of elements of a Markov family and preserve the geodesic flow. However I'm not entirely sure if this can be done. – Steve Huntsman Jun 3 '10 at 17:33