In Proposition 3.1. in this article by John Franks, he applies the Nielsen-Thurston classification of surface homeomorphisms to a homeomorphism $ \ f:M \rightarrow M$ of an **open** surface $M$ which is a $k$-times punctured sphere with $k \geq 3$. Think about $M$ for example as an open annulus $\mathbb{R}/\mathbb{Z} \times (0,1)$ with $k-2$ points removed.

What confuses me is that I cannot seem to find any reference for the Thurston classification of homeomorphisms for **open** surfaces. Everywhere it's done only for
compact surfaces w/o boundary. I have no idea why it should still work for the said punctured spheres. Any help is much appreciated!

He uses it to prove that for an area-preserving homeo $f:M \rightarrow M$ without periodic points, there has to be a power $f^n$ which is isotopic to the identity. Following the Thurston classification, there are essentially three possibilities for $f$:

- $f$ is periodic (i.e. there is a power $f^n$ which is isotopic to id)
- $f$ is pseudo-anosov or has pseudo-anosov components (which would imply periodic points)
- $f$ is reducible only with periodic components (which also implies periodic points)

Thanks alot :)