(60:0.52); at (80:0.58cm) [$x_1$]{}; (80:0.48) – (80:0.52); at (103:0.58cm) [$x_2$]{}; (103:0.48) – (103:0.52); at (185:0.58cm) [$x_3$]{}; (185:0.48) – (185:0.52); at (262:0.58cm) [$x_4$]{}; (262:0.48) – (262:0.52); at (32:0.58cm) [$x_5$]{}; (32:0.48) – (32:0.52); (0,0) circle(0.5cm);

at (60:0.59cm) [$a$]{}; at (180:0.59cm) [$b$]{}; (0,0) circle(0.5cm); (60:0.48) – (60:0.52); (180:0.48) – (180:0.52); (60:0.5) arc (60:180:0.5);

\[Predecessors and successors in $\cZ$\] \[rmk:predecessors\] It follows directly from Definition \[D:admissible\] that:

-   Each $z \in {\mathscr{Z}}$ has a unique predecessor $z^- \in {\mathscr{Z}}$, i.e. a unique element $z^- \in {\mathscr{Z}}$ such that $(z^-,z) \cap {\mathscr{Z}}= \varnothing$.

-   Each $z \in {\mathscr{Z}}$ has a unique successor $z^+ \in {\mathscr{Z}}$, i.e. a unique element $z^+ \in {\mathscr{Z}}$ such that $(z,z^+) \cap {\mathscr{Z}}= \varnothing$.

Figure \[fig:admissible subset\] in the introduction shows an example of an admissible subset and of the predecessor and successor of one of its elements.

\[A dichotomy for sequences in $\cZ$\] \[R:dichotomy\] Since ${\mathscr{Z}}$ is discrete, each sequence $\{z_i\}_{i \in {\mathbb{Z}}_{\geqslant 0}}$ from ${\mathscr{Z}}$ which converges to a $z \in {\mathscr{Z}}$ has to satisfy $z_i = z$ for $i \gg 0$. Thus each convergent sequence from ${\mathscr{Z}}$ that is not constant from some step converges to an element of $L({\mathscr{Z}})$. Furthermore, since $S^1$ is compact, each sequence $\{z_i\}_{i 