The content of the article well describes the subject except for some minor points. My main criticism concerns the references and complete absence of the chronological timeline - which, I think, is indispensable for this kind of publication.
Let me list my comments paragraph by paragraph.
1. Maximizing entropy.
- It is not clear why in the case of topological entropy the reference to KH95 is accompanied with a reference to the original publication, and in the case of metric entropy it is not. I think that it would be reasonable in both cases just refer to the corresponding articles of Scholarpedia.
- There is no special article on "Variational principle". I think that in this case references to a contemoprary exposition in KH95 must be accompanied with references to original publications (and, of course, the names of the people who obtained the corresponding results).
- The last sentence is misleading. Bowen's construction does not shed any light on the inequality between the topological and metric entropies and the fact that the topological entropy is the supremum of metric ones. This part should be properly referenced (maybe, with a reference to the fact that the measure of maximal entropy need not exist in general).
2. Bowen measure.
- What exactly is the claim? It should say something like: for an expansive homeomeorphism the limit exists and is a measure of maximal entropy. What exactly is the scope of the definition of the Bowen measure? Is it still called this way when the maximal entropy is not unique?
3. Margulis measure.
- It might be better to choose the chronological sequence and put Margulis' measure first.
- Where for the first time appears the fact that Bowen's and Margulis' measures coincide? Was Bowen aware of Margulis' construction when writing his own paper?
- The definition of a conditional measure is misleading. However, it is not necessary for the construction of Margulis - as the latter goes in the opposite direction. Instead of decomposing a measure he first obtains his leafwise measures from a topological consideration (so that they are defined everywhere and not a.e. as would be the case with properly defined conditional measures), and then obtaines the global measure by "multiplying" these leafwise measures in stable and unstable directions.
4. The Patterson-Sullivan construction.
- No references are given whatsoever. I suggest that the original papers by Patterson and Sullivan (definition of this measure in the constant curvature case) and Kaimanovich (relation of the PS construction with the BM measure) should be quoted.
5. Hausdorff measures.
- Ones again, no references. The original paper by Hamenstadt should be quoted.
- I am not sure whether it is worth giving here the definition of the Hausdorff measure.
6. Periodic orbits.
- Counting is just one particular application of the BM measure. I am not sure it is appropriate to discuss just one aspect in such a detailed way.
7. Generalizations of the Bowen construction.
- Again no references at all
- What about "generalizations of the Margulis construction". Why not mention Gibbs measures (measures with the prescribed Radon-Nikodym derivatives) which are a generalization of the Margulis construction in precisely the same as sense in which the author talks about generalizations of the Bowen construction.