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Whether the following statemant is correct (I guess the answer is "Yes" and I guess that maybe it is trivial for an expert about Pseudo-Anosov map)?

"For a given $n\in N$, there exists a closed orientbale surface $\Sigma^n$ such that there exists a pseudo-Anosov diffeomorphism $f_n$ on $\Sigma^n$ with an $n$-prong singularity. "

Could you pleae provide me some references or comments? Thank you.

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The answer to your question is yes and essentially the only restriction on the singularity type (=number of prongs) is the one coming from Euler-Poincare formula. For further details, I think an useful reference is, e.g., the book "A Primer on Mapping Class Groups" by B. Farb and D. Margalit (http://press.princeton.edu/titles/9495.html).

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    $\begingroup$ The result to which Matheus is referring is a theorem in the paper by Masur and Smillie entitled "Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms", MR1214233. $\endgroup$ – Lee Mosher Oct 14 '12 at 12:31
  • $\begingroup$ @Matheus and @Lee, thank you very much. Your answers are exactly what I need. $\endgroup$ – Bin Yu Oct 14 '12 at 13:11

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