Let $M\_{\phi}$ be a hyperbolic mapping torus coming from a pseudo-Anosov map $\phi$ in a surface $S$. Is there any way to estimate the length of the geodesic representing a given curve in the surface in terms of the map $\phi$? That is, knowing something like the stable and unstable foliations for the map or something equivalent, can you estimate the length of a given curve? Any references for something like this are really appreciated.


For example, if you take a mapping torus $M\_{\phi}$, drill one simple nontrivial curve $\alpha$ in the surface and re-glue by $\sigma^n$, a large Dehn twist about $\alpha$, you are going to get a hyperbolic mapping torus $M\_{\phi\sigma^{n}}$. In this manifold $\alpha$ is going to be very short. 


Another example, if you take a map $\psi = \phi\sigma^n$ where $\phi$ is pseudo-anosov in all of $S$ and $\sigma$ is a pseudo-Anosov just in a subsurface $X \subset S$, I think the curves in the complement of $X$ have to be very small for $n$ large, right?



Pseudoanosov mapping torus and length of curves.
In [this paper](http://www.ams.org/mathscinet-getitem?mr=1859018), McMullen gets a coarse description of geodesics in
a mapping torus of a punctured torus in terms of the Minsky model. I 
think one ought to get an estimate of their lengths from this. 


