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Report on D. Scheglov paper

The paper gives an important contribution to the study of the ergodic theory of flows on surfaces of higher genus. It contains key new ideas in the subject. It clearly deserves to be
published on the Annals of Mathematics.

However the paper needs revisions. These revisions are not essential from
a mathematical point of view but are needed to improve clarity of exposition
and overall readability. Here is a list of suggested modifications.

1) Improve english : there are many spelling mistakes/misprints (such as 'sence' for 'sense', 'symblos' for 'symbols', 'continios' for 'continuous', 'ergodisity' for 'ergodicity', 'Rieman-Roch' for 'Riemann-Roch', 'existance' for 'existence', etc. The grammar can also be improved.
2) p. 2 : also cite Ulcigrai-Sinai's paper where some partial results on the absence of mixing for symmetric logarithmic singularities were proved ;

3) in reference to the 'deep Diophantine-like properties' you may quote the paper by Avila, Gouezel and Yoccoz on the exponential mixing of the Teichmuller geodesic flow where the key estimate used by Ulcigrai was proved ;
4) p. 3 : please give some general reference for the basic material on IET's, Kontsevich cocycle etc. in section 1.1 ;
5) p. 4 : in the definion of Rauzy and Kontsevich cocycle, one should take the transpose of the matrices (or the inverse) ;
6) p. 6 : the partition $\epsilon$ is not defined ;
7) p. 7 : the notation 'graph X' in the statement of Lemma 1 is never introduced ;
8) p. 7 : it would be useful to add a few words pointing out the significance of Theorem 2.1 and where exactly it is used in the argument ;
9) p. 8, line -10 : Now we can prove Theorem 2.1 (instead of Lemma 1)
10) p. 9 : in the proof of Lemma 2.2 (which is an existence statement) at least indicate a cycle which satisfies the desired properties; the verification may be left to the reader (a sketch of out to proceed in the verification would be useful)
11) p. 9 : in Definition 1 part 2) : it is for any symbols a, b in {1,2, ...,5} or, as it is written, only for symbols a, b in {1,2, ...,4}
12) p. 10 : indicate more clearly that the $\omega_i$ depend on $n$ ; also words denoted by $\omega^1$, ...,$\omega^5$ appear. We guess that they are the same as the words $\omega_1$, ...,$\omega_5$
13) p. 11 : Explain more about the 'minimal adjustements' needed in the proof of Theorem 3.1 ;
14) p. 12 : Kocergin's paper referred to here is probably [10] (certainly not [14] which is Ulcigrai's) ;
15) p. 12 : the notation $... /K_n$ and $... /K_3$ for the difference between sets is not the standard one ; it would be better to use $\setminus$ in LaTeX (which looks like $ \$).
16) p. 13 : the numbering (1), (2) of the equations is not easily readable ; add horizontal spacing or put the numbering to the left ;
17) p. 14 : the sentence at the very end of the proof of Theorem 3.2, which gives some indication about the case of a general logarithmic roof function, should be expanded and put before the proof of Theorem 3.1 (since it seems relevant also for the proof of that theorem).
18) p. 14 : in the statement of Lemma 3.3, does each pair {x_i,y_i} belong to some element of $\Theta_n$ ?
19) p. 17 : the definition of the measure class on the space of conservative flows with saddle singularities on surfaces is a natural one. It would be better to explain it here rather than refer to Kontsevich's paper (which is likely not the first to have given it). If the author wants to quote Kontsevich's paper or any other paper, he should give precise indications (numbering, page) on where the definition appears in the paper.
20) p. 17 : please state the criterion for absence of mixing somewhere at the beginning of section 4.