Let $T$ be the 2-dimensional torus and let $S$ be $T$ minus one point. Then Birman exact sequence of mapping class groups becomes an isomorphism $$ \beta: Map(S)\to Map(T)=GL(2, {\mathbb Z}). $$ It is then essentially immediate that $\beta$ preserves Thurston's classification of elements of the mapping class group into three types: $\beta(f)$ is Anosov if and only if $f$ is pseudo-Anosov, etc.

*Question.* Did anybody bother to record this elementary observation in the literature?

*I just need a reference, since anybody who knows anything about the mapping class group knows how to prove it (in several ways).* (Please, do not write proofs, I know at least 4.)

I was nearly sure that Farb and Margalit have it, but they do not. Same for Casson and Bleiler, same for Ivanov. Of course, maybe this is one of the cases when it is easier to write a proof then to find a reference.