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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 $\phi$.

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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.

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Thanks a lot for your answer, I really appreciate it, do you know where can I find a reference describing how all the surface in the geodesic between two points in teichmuller space $X$ and $Y$ are going to look geometrically, depending in how $X$ and $Y$ look like geometrically? –  shurtados Mar 19 '12 at 6:15
    
@shurtados: This is hard to answer without knowing what geometric information you are particularly looking for. But, what I was thinking of in the 1st paragraph of my answer is the standard description of Teichmuller geodesics, and the corresponding conformal structures and singular Euclidean metrics, which can be found in Farb Margalit, "A primer on mapping class groups". I am also fond of Abikoff's little book "The real analytic theory of Teichmuller space". Descriptions in terms of hyperbolic structures are implicit from the uniformization theorem, but are much harder to see directly. –  Lee Mosher Mar 22 '12 at 16:59

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