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 Hausdorff 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 $\phi$. 



teichmuller geodesics and hyperbolic mapping torus
Points in the Teichmuller geodesic $\sigma$ are given quite concretely in terms of the pseudo-Anosov data, namely the stable and unstable measured foliations, by using the standard method one uses to describe any Teichmuller geodesic from a pair of measured foliations.


I'm guessing that there's not going to be any uniform estimate in terms of genus for Hausdorff distance between a pleated surface path and the Teichmuller geodesic. There exist examples in which $M\_\phi$ has arbitrarily short geodesics, and a pleated surface passing near such a geodesic also will have arbitrarily short geodesics. I'm pretty sure in such a case that there would then be an arbitrarily large distance in Teichmuller space from the conformal structure on that pleated surface to the Teichmuller geodesic.


