Relation between volume entropy and Hausdorff dim of limit set?

I have a very stupid question: I often see that the volume entropy of a compact Riemmannian manifold with negative curvature coincide with the Hausdorff dim of the limit set or Patterson sullivan measure on the boundary.

But the volume entropy is not invariant under scaling of the measure, and I assume the dimension of limit set is. So what is the correct normalization to make the claim above true?

• "Hausdorff dim of" what "limit set"? $\:$ – user5810 May 17 '13 at 21:42
• In this context, limit set is the ideal boundary of the universal cover. Classically, one uses this in the case of Kleinian groups. Then the right formula is that volume entropy equals Hausdorff dimension of the conical limit set of the group (which could be less than dimension of the full limit set). See Nicholls' book "Ergodic theory of discrete groups". Normalization in variable curvature was worked out in papers by Besson, Courtois and Gallot (maybe also Hammenstadt) and you only have an inequality. Equality if I remember correctly is only in curvature -1 case. – Misha May 17 '13 at 22:33
• See also math.univ-lyon1.fr/~remy/smf_sec_18_09.pdf and references therein. – Misha May 17 '13 at 22:33
• Thanks Misha, the answer is very helpful. While I'll try to look it up in the BCG papers, do you happen to remember in which direction the inequality is, between entropy and dimension, for non-constantly curved manifolds? – Jerry May 17 '13 at 23:44