Recently I start learning mapping class group. The Nielsen-Thurston classification says that each element in mapping class group $Mod(S\_{g,n}),g,n\geq 0$ is periodic, reducible, or pseudo-Anosov. Take any element in $Mod(S\_{g,n})$, how to determine the element is in which one of the cases?



Classification of elements in mapping class groups
An effective algorithm can be found in 



```
 Mladen Bestvina and Michael Handel. Train-tracks for surface homeomorphisms. Topology, vol. 34 (1995), no. 1, pp. 109–140

```

This has been implemented by Peter Brinkman in Xtrain (which you can get via computop.org).


