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.