marked points on the boundary, see [@BZ], [@QZ], [@ZZZ] for surface type and [@HJR:tubes] for cluster tubes.

The Igusa–Todorov cluster categories $\cC( \cZ )$ of Dynkin type $A_{ \infty }$ {#subsec:A}
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To explain the categories $\cC( \cZ )$ and their combinatorial models by $\infty$-gons, we first state two definitions.

\[Admissible subsets of $S^1$\] \[D:admissible\] A subset $\cZ$ of the circle $S^1$ is called [*admissible*]{} if it satisfies the following conditions.

1.  $\cZ$ has infinitely many elements.

2.  $\cZ \subset S^1$ is a discrete subset, i.e. for each $z \in {\mathscr{Z}}$ there is an open neighbourhood of $z$ in $S^1$, equipped with its usual topology, containing no other element of ${\mathscr{Z}}$.

3.  $\cZ$ satisfies the [*two-sided limit condition*]{}, i.e. each $x \in S^1$ which is the limit of a sequence from ${\mathscr{Z}}$ is a limit of both an increasing and a decreasing sequence from ${\mathscr{Z}}$ with respect to the cyclic order.

Throughout the paper, $\cZ \subset S^1$ is a fixed admissible subset. We think of $\cZ$ as the vertices of an $\infty$-gon, see Figure \[fig:admissible subset\].

(0,0) circle(0.5cm); at (50:0.58)[$z$]{}; at (39:0.60)[$z^-$]{}; at (61:0.59)[$z^+$]{};

(128:0.48) – (128:0.52); (131:0.48) – (131:0.52); (135:0.48) – (135:0.52); (140:0.48) – (140:0.52); (146:0.48) – (146:0.52); (153:0.48) 