# Limits at infinity of fellow-travelling sequences in Teichmuller space,

I have a question concerning limits of sequences of points in Teichmuller space, and how this notion is preserved under fellow-travelling.

Suppose that we have closed surface of genus $g\geq 2$, and a sequence $\{X_n\}_{n\in \mathbb{N}}$ of points in its Teichmuller space $T(S)$.

Suppose furthermore that, for $n\rightarrow \infty$ this sequence converges to a point $\lambda\in \mathbb{P}ML(S)$ in Thurston's compactification of Teichmuller space. For example, we can take a diverging sequence of points on a Teichmuller geodesic associated to a pseudo-Anosov automorphism of $S$.

Now suppose that we have another sequence $\{Y_n\}_{n\in \mathbb{N}}$ of points in $T(S)$ which fellow travels the first, meaning that $$d(X_n,Y_n)\leq r$$ for all $n$ and some $r>0$. Here, $d$ denotes the Teichmuller distance.

Here are the questions:

1. Does the sequence $\{Y_n\}$ converge to some point in $\mathbb{P}ML(S)$?

2. If (1) is true, is the limit point of $\{Y_n\}$ the same as the limit point of $\{X_n\}$?

It seems to me that the answer to both questions should be positive, but I am having trouble in proving the above statements, as convergence to a point in the boundary is expressed in terms of ratios between hyperbolic lengths and transverse measures of curves, while Teichmuller distance is related to the stretch factor of Teichmuller maps. These notions are (at least for me) not easily related.

Any help is kindly appreciated!

There are some limited situations where your question has a positive answer, which can be stated in the setting of "convex cocompact subgroups of the mapping class group" $\text{MCG}(S)$, as developed in a paper of myself and Farb with title as in the quotes.
Namely, if $\Gamma < \text{MCG}(S)$ is convex cocompact, and if $Z \in T(S)$ is any point, and if there exists $s>0$ such that the sequence $X_n$ stays within distance $s$ of the $\Gamma$-orbit of $Z$, then both of your questions 1 and 2 are answered affirmatively.