Let $M_{\phi}$ be a hyperbolic mapping torus coming from a pseudoanosov map $\phi$ in a surface $S$. Is there anyway of estimating 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 make a large dehn twist, you are going to get a hyperbolic mapping torus $M_{\phi\sigma^{n}}$, where $\sigma$ is a dehn twist in the curve, in this manifold $\alpha$ is going to be very short.

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

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?