# How to get a Dehn-twist presentation of a periodic map of a Riemann surface?

Let $f:S\to S$ be a given periodic map of order $n$, and $(m_i,\lambda_i, \sigma_i)$ be the valency of the multiple point $p_i$ of $f$ ( $i=1,\cdots,r$ ).

A classical result says such $f$ is isotopic to a product of Dehn twists. It is trivial when $n=1$. Now we assume that $n>1$. I want to know how to get such a Dehn twist presentation.

For a pseudo-periodic map, a similar Dehn twist presentation implies Picard-Lefschetz formula of the monodromy of a singular fiber (semistable or non-semistable). In fact, I wish to comupte the monodromy of a non-semistable fiber.

For hyperelliptic periodic maps, Ishzaka provided a method.

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How is the map given in the first place? –  Igor Rivin Jan 1 '11 at 2:48
The above question is answered in some detail by my answer to the following: mathoverflow.net/questions/142365/… –  Sam Nead Jan 13 at 21:38
So, I am voting to close. –  Sam Nead Jan 13 at 21:40
@SamNead: Do you think it is reasonable to close an old question as a duplicate of a much newer question? –  Stefan Kohl Jan 13 at 23:44
@Stefan: Well, the alternative is to cut-and-paste my answer. But I thought that would be poor form. –  Sam Nead Jan 15 at 22:28