Given a pseudoanosov map $\phi$ of a surface $S$, there is a geodesic $\sigma$ in Teichmuller space (with the teichmuller metric) that is an axis for $\phi$, In other words, $\phi$ acts as a translation along that geodesic. In the other hand, we can take the cyclic covering $\tilde M_\phi$ of the Mapping torus $M_\phi$, and get a path of pleated surfaces $S_t$ exiting both ends of $M_\phi$, that "coarsely" is going to give us a path $\pi$ in teichmuller space.

I would like to know what's the relation between these two curves $l$ and $ \sigma$in teichmuller space, if it's possible to estimate the Haussdorff distance between them in terms of the genus of $S$.

I would also like to know if there is any way of describing how the points in $\sigma$ are going to look like, that is, to give some description of the surfaces in the axis of $\phi$.