\operatorname{\mathbf{N}}^*$ there exist $\psi_n \in S$ and $y_n \in B\left(x, \frac{1}{n}\right)$ such that $\psi_n y_n \not \in B(\psi_n x, \varepsilon)$. Clearly $y_n \to x$ and we can assume by passing to a subsequence that $\psi_n x \to a$ and $\psi_n y_n \to b$ for some $a \neq b$. Hence, the sequences $a_n = \psi_n x \to a$ and $b_n = \psi_n y_n \to b$ are collapsed by $\{\varphi_n = \psi_n^{-1}\} \subseteq S^{-1}$. The case where $S^{-1}$ is not equicontinuous is identical but gives two sequences which are collapsed by a sequence in $(S^{-1})^{-1} = S$.

Sequences collapsed by a sequence of automorphisms
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In this section, we consider a compact polygon $\Delta$. Proposition \[proposition:closer\] states that one can deduce from non-relative compactness of a subset $S$ of $\operatorname{\mathrm{Auttop}}(\Delta)$ that there exist two sequences $a_n \to a$ and $b_n \to b$ of vertices of $\Delta$ collapsed by a sequence in $S$ or in $S^{-1}$. The goal of this subsection is to show that we can actually obtain additional constraints on these two sequences. The following result of Burns and Spatzier already goes in this direction by asserting that we can assume $\operatorname{\mathrm{D}}(a_n, b_n) = \operatorname{\mathrm{D}}(a, b) = 2$.

\[proposition:centered\] Let $\Delta$ 