TITLE: pseudo-Anosovs with given action on homology
QUESTION [6 upvotes]: It is well-known that the Mapping Class Group of a closed surface of genus $g$ surjects onto $Sp(2g, \mathbb{Z})$ (see, for example the Farb-Margalit book). However, I was wondering if there is a simple proof that the set of pseudo-Anosov elements in MCG surjects onto $Sp(2g, \mathbb{Z})$ (and also a reference for where this might have been stated first) -- I can construct a somewhat sophisticated proof, but this should be easier.

REPLY [3 votes]: Suppose that $f$ is a pseudo-Anosov and let $\lambda^\pm$ be its stable and unstable laminations.  Suppose that $g$ is any mapping class with the following property: $g(\lambda^+) \neq \lambda^-$.  Then, for all sufficiently large $n$, the composition $f^n g$ is also pseudo-Anosov.  This is proved using the north-south dynamics of $f$.
Now, as in Mosher's answer, choose $f$ in Torelli and let $g$ be a lift of the desired element of $Sp$.  If needed, compose $g$ with a bounding pair map to arrange the side-condition.  We are done.  
Added later: Here are some of the details of the dynamical argument. I've edited this several times to try and make it correct.  Full disclosure - I first heard this from Yair Minsky. 
Pick $U$, a small neighborhood of $\lambda^+$, chosen so that $\lambda^-$ is not contained in $g(U)$.  I'll also want $f|U$ to be strictly contracting. Using north-south dynamics, there is an $m^+ > 0$ so that for all $n > m^+$ we have $f^{m^+} g(U) \subset U$.  The contracting property implies that the map $f^n g$ has a unique fixed point in $U$.
On the other hand, pick a small neighborhood $V$ of $g^{-1}(\lambda^-)$, so that $\lambda^+$ is not in $V$.  Also, we require $f^{-1}|g(V)$ to be strictly contracting  There is a power $m^- > 0$ so that for all $n > m^-$ we have $V \subset f^n g(V)$.  Deduce that the map $f^n g$ has a unique fixed point in $V$. 
We place one more restriction on $n$.  We need $f^n g(V^c)$ to be contained in $U$. 
I now claim that $h = f^n g$ has no other fixed points, and so is pseudo-Anosov.